**2.1 Ideal magnetohydrodynamical equations**

Magnetohydrodynamics (MHD) is a discipline involving modeling magnetically confined plasmas (Wesson, 2004; Pironti and Walker, 2005; Ariola and Pironti, 2008). The ideal MHD system is a coupled system consisting of a single fluid and an electromagnetic field with certain constitutive relations.

The fluid is described by the two equations in three dimensions. The first is the mass conservation law,

$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0,\tag{1}$$

where *<sup>n</sup>* <sup>=</sup> 3, *<sup>ρ</sup>* <sup>∈</sup> *<sup>Ω</sup>*3(*Z*) is the mass density, *<sup>v</sup>* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*) is the fluid velocity, *<sup>J</sup>* <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) is the free current density, *<sup>B</sup>* <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) is the magnetic field induction, �*v*�, *<sup>v</sup>*�� <sup>=</sup> �*v*��<sup>2</sup> is the inner

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 171

for *<sup>ω</sup>* <sup>=</sup> *fi*1···*ik* (*x*) *dxi*<sup>1</sup> ∧···∧ *dxik* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*), where *<sup>i</sup>*<sup>1</sup> ··· *ik* is the combination of *<sup>k</sup>* different

for *<sup>ω</sup>* <sup>=</sup> <sup>∑</sup>*i*1<···<*ik fi*1···*ik* (*x*) *dxi*<sup>1</sup> ∧···∧ *dxik* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*), where *<sup>j</sup>*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *jn*−*<sup>k</sup>* is the rearrangement of the complement of *i*<sup>1</sup> < ··· < *ik* in the set {1, ··· , *n*} in ascending order, and sgn(*I*, *<sup>J</sup>*) is the sign of the permutation of *<sup>i</sup>*1, ··· , *ik*, *<sup>j</sup>*1, ··· , *jn*−*<sup>k</sup>* generated by interchanging of the basic forms *dx<sup>i</sup>* (if we interchange *dx<sup>i</sup>* and *dx<sup>j</sup>* in *ω* for arbitrary *i* and

> (−1)*m*−<sup>1</sup> *fi*1···*ik gim dxi*<sup>1</sup> ∧···∧ *dxim*−<sup>1</sup> <sup>∧</sup> *dxim*<sup>+</sup><sup>1</sup> ∧···∧ *dxik* if *<sup>j</sup>* <sup>=</sup> *im*, 0 if *j* �= *im*

• *iv*� : *<sup>Ω</sup>k*(*Z*) <sup>→</sup> *<sup>Ω</sup>k*−1(*Z*) ··· The interior product *iv*� with respect to *<sup>v</sup>*� is defined as

) and *<sup>ω</sup>* <sup>=</sup> *fi*1···*ik* (*x*)*dxi*<sup>1</sup> ∧···∧ *dxik* .

In (5), we used the formula (*v* · ∇)*v* = (1/2)∇(*v* · *v*) + Curl *v* × *v*, and the enthalpy *w*(∗*ρ*) = (*∂*/*∂*∗*ρ*)(∗*<sup>ρ</sup> <sup>U</sup>*(∗*ρ*)) is related to the pressure *<sup>p</sup>*(∗*ρ*) by (∗*ρ*)−1*dp*(∗*ρ*) = *dw*(∗*ρ*), where *U*(*ρ*) is the internal energy function of the fluid satisfying *p*(∗*ρ*) = *w*(∗*ρ*)∗*ρ* − *U*(∗*ρ*)∗*ρ*.

where *<sup>D</sup>* <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) is the electric field induction, *<sup>H</sup>* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*) is the magnetic field intensity,

Let us recall the definition of DPH systems. The advantage of these systems will be explained

*<sup>E</sup>* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*) is the electric field intensity, and *�* <sup>∈</sup> *<sup>Ω</sup>*3(*Z*) is the free charge density.

• <sup>∗</sup> : *<sup>Ω</sup>k*(*Z*) <sup>→</sup> *<sup>Ω</sup>n*−*k*(*Z*) ··· The Hodge star operator <sup>∗</sup> induced in terms of a Riemannian

*<sup>∂</sup>x<sup>j</sup> dx<sup>j</sup>* <sup>∧</sup> *dxi*<sup>1</sup> ∧···∧ *dxik* (6)

sgn(*I*, *<sup>J</sup>*)*fi*1···*ik dxj*<sup>1</sup> ∧···∧ *dxjn*−*<sup>k</sup>* <sup>∈</sup> *<sup>Ω</sup>n*−*k*(*Z*) (7)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> *<sup>d</sup>E*, *<sup>d</sup><sup>B</sup>* <sup>=</sup> 0, *<sup>d</sup><sup>D</sup>* <sup>=</sup> *�*, (9)

*B* = *μ*∗*H*, ∗(*E* + *iv*�*B*) = *ηJ*. (10)

(8)

product with respect to *v*�, and we have introduced the following operators:

*n* ∑ *j*=1

*dω* =

integers selected from 1 to *n*, and *j* �= *i*<sup>1</sup> �= ··· �= *ik*.

*i*1<···<*ik*

∗*<sup>ω</sup>* = ∑

*j*, the sign of *ω* changes, i.e., it is alternating).

Next, Maxwell's equations are defined as follows:

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*d<sup>H</sup>* <sup>+</sup> *<sup>J</sup>*, <sup>−</sup>*∂<sup>B</sup>*

from the viewpoint of passivity and boundary controls in later sections.

−*∂<sup>D</sup>*

The constitutive relations are written as follows:

**2.3 Definition of port-Hamiltonian system**

metric on *Z* is defined as

*iv*�*ω* =

for *v*� = *gj*(*x*)(*∂*/*∂x<sup>j</sup>*

• *<sup>d</sup>* : *<sup>Ω</sup>k*(*Z*) <sup>→</sup> *<sup>Ω</sup>k*+1(*Z*) ··· The exterior differential operator *<sup>d</sup>* on *<sup>Z</sup>* is defined as

*∂ fi*1···*ik*

where *ρ*(*t*, *x*) ∈ **R** is the local mass density at time *t* ∈ **R** at the spatial position *x* = (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3) <sup>∈</sup> **<sup>R</sup>**3, and *<sup>v</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is the fluid (Eulerian) velocity at *<sup>t</sup>* and *<sup>x</sup>*. The second is Newton's law applied to an infinitesimal plasma element with an electromagnetic coupling,

$$
\rho \frac{\partial \mathbf{v}}{\partial t} = -\rho \mathbf{v} \cdot \nabla \mathbf{v} - \nabla p + \mathbf{J} \times \mathbf{B},\tag{2}
$$

where *<sup>p</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>** is the kinetic pressure in plasma, *<sup>J</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is the free current density, *<sup>B</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is the magnetic field induction, and the Lorentz force term *<sup>J</sup>* <sup>×</sup> *<sup>B</sup>* means the coupling.

The electromagnetic field satisfies the Maxwell's equations consisting of Ampere's law, Faraday's law, and Gauss's law for the magnetic induction field:

$$-\frac{\partial \mathbf{D}}{\partial t} = -\nabla \times \mathbf{H} + \mathbf{J}, \quad -\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \mathbf{E}, \quad \nabla \cdot \mathbf{B} = 0,\tag{3}$$

where the time derivative of the electric field induction *<sup>D</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is neglected in MHD.

The constitutive relations are given by

$$\mathbf{B} = \mu \mathbf{H}, \quad \mathbf{E} + \mathbf{v} \times \mathbf{B} = \eta \mathbf{J}, \tag{4}$$

where *μ* is the magnetic permeability and *η* is the resistance coefficient that is assumed to be zero in an ideal MHD system.

#### **2.2 Geometric formulation of MHD**

The main framework of this chapter is the port-Hamiltonian system for PDEs called a *distributed port-Hamiltonian (DPH) system* (Van der Schaft and Maschke, 2002). DPH systems are expressed in terms of differential forms (Flanders, 1963; Morita, 2001). Moreover, a formulation using differential forms defined on Riemannian manifolds can describe the relation between the vector fields of systems and the topological properties of system domains (see *Section 4*). Thus, we shall rewrite the equations of MHD by using differential forms to derive the DPH representation of MHD.

Let *Y* be an *n*-dimensional smooth Riemannian manifold. Let *Z* be an *n*-dimensional smooth Riemannian submanifold of *Y* with a smooth boundary *∂Z*. We assume that the time coordinate *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>** is split from the spatial coordinates *<sup>x</sup>* = (*x*1, ··· , *<sup>x</sup>n*) <sup>∈</sup> *<sup>Z</sup>* in the local chart of *<sup>Z</sup>*. We denote the space of differential *<sup>k</sup>*-forms on *<sup>Z</sup>* by *<sup>Ω</sup>k*(*Z*) for 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*. We denote the infinite-dimensional vector space of all smooth vector fields in *Z* by X(*Z*). We identify the 1-from *<sup>v</sup>* with the vector field *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*). The fluid equations (1) and (2) can be rewritten as follows:

$$\begin{cases} \frac{\partial \rho}{\partial t} = -de\_{\upsilon}, & \frac{\partial \sigma}{\partial t} = -de\_{\rho} + g\_1 + g\_{2\prime} \\ e\_{\upsilon} = i\_{\mathfrak{D}^{\sharp}}\rho, & e\_{\rho} = \frac{1}{2} \langle \mathfrak{v}^{\sharp}, \mathfrak{v}^{\sharp} \rangle + w(\*\rho), \\ g\_1 = -(\*\rho)^{-1} \* (\*d\sigma \wedge \*e\_{\upsilon}), & g\_2 = (\*\rho)^{-1} \* (\*J \wedge \*\mathcal{B}), \end{cases} \tag{5}$$

4 Magnetohydrodynamics

where *ρ*(*t*, *x*) ∈ **R** is the local mass density at time *t* ∈ **R** at the spatial position *x* = (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3) <sup>∈</sup> **<sup>R</sup>**3, and *<sup>v</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is the fluid (Eulerian) velocity at *<sup>t</sup>* and *<sup>x</sup>*. The second is Newton's law applied to an infinitesimal plasma element with an electromagnetic coupling,

where *<sup>p</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>** is the kinetic pressure in plasma, *<sup>J</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is the free current density, *<sup>B</sup>*(*t*, *<sup>x</sup>*) <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is the magnetic field induction, and the Lorentz force term *<sup>J</sup>* <sup>×</sup> *<sup>B</sup>* means the

The electromagnetic field satisfies the Maxwell's equations consisting of Ampere's law,

where *μ* is the magnetic permeability and *η* is the resistance coefficient that is assumed to be

The main framework of this chapter is the port-Hamiltonian system for PDEs called a *distributed port-Hamiltonian (DPH) system* (Van der Schaft and Maschke, 2002). DPH systems are expressed in terms of differential forms (Flanders, 1963; Morita, 2001). Moreover, a formulation using differential forms defined on Riemannian manifolds can describe the relation between the vector fields of systems and the topological properties of system domains (see *Section 4*). Thus, we shall rewrite the equations of MHD by using differential forms to

Let *Y* be an *n*-dimensional smooth Riemannian manifold. Let *Z* be an *n*-dimensional smooth Riemannian submanifold of *Y* with a smooth boundary *∂Z*. We assume that the time coordinate *<sup>t</sup>* <sup>∈</sup> **<sup>R</sup>** is split from the spatial coordinates *<sup>x</sup>* = (*x*1, ··· , *<sup>x</sup>n*) <sup>∈</sup> *<sup>Z</sup>* in the local chart of *<sup>Z</sup>*. We denote the space of differential *<sup>k</sup>*-forms on *<sup>Z</sup>* by *<sup>Ω</sup>k*(*Z*) for 0 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*. We denote the infinite-dimensional vector space of all smooth vector fields in *Z* by X(*Z*). We identify the 1-from *<sup>v</sup>* with the vector field *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*). The fluid equations (1) and (2) can be rewritten as

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*de<sup>ρ</sup>* <sup>+</sup> *<sup>g</sup>*<sup>1</sup> <sup>+</sup> *<sup>g</sup>*2,

�*<sup>v</sup>*, *<sup>v</sup>*� <sup>+</sup> *<sup>w</sup>*(∗*ρ*), *<sup>g</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>(∗*ρ*)−<sup>1</sup> <sup>∗</sup>(∗*d<sup>v</sup>* ∧ ∗*ev*), *<sup>g</sup>*<sup>2</sup> = (∗*ρ*)−<sup>1</sup> <sup>∗</sup>(<sup>∗</sup> *<sup>J</sup>* ∧ ∗*B*),

2

where the time derivative of the electric field induction *<sup>D</sup>* <sup>∈</sup> **<sup>R</sup>**<sup>3</sup> is neglected in MHD.

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*ρ<sup>v</sup>* · ∇*<sup>v</sup>* − ∇*<sup>p</sup>* <sup>+</sup> *<sup>J</sup>* <sup>×</sup> *<sup>B</sup>*, (2)

*B* = *μH*, *E* + *v* × *B* = *ηJ*, (4)

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> ∇ × *<sup>E</sup>*, ∇ · *<sup>B</sup>* <sup>=</sup> 0, (3)

(5)

*ρ ∂v*

Faraday's law, and Gauss's law for the magnetic induction field:

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> −∇ × *<sup>H</sup>* <sup>+</sup> *<sup>J</sup>*, <sup>−</sup>*∂<sup>B</sup>*

−*∂<sup>D</sup>*

The constitutive relations are given by

zero in an ideal MHD system.

**2.2 Geometric formulation of MHD**

derive the DPH representation of MHD.

⎧ ⎪⎪⎪⎨

*∂ρ*

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> <sup>−</sup>*dev*, *<sup>∂</sup><sup>v</sup>*

*ev* <sup>=</sup> *iv<sup>ρ</sup>*, *<sup>e</sup><sup>ρ</sup>* <sup>=</sup> <sup>1</sup>

⎪⎪⎪⎩

coupling.

follows:

where *<sup>n</sup>* <sup>=</sup> 3, *<sup>ρ</sup>* <sup>∈</sup> *<sup>Ω</sup>*3(*Z*) is the mass density, *<sup>v</sup>* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*) is the fluid velocity, *<sup>J</sup>* <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) is the free current density, *<sup>B</sup>* <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) is the magnetic field induction, �*v*�, *<sup>v</sup>*�� <sup>=</sup> �*v*��<sup>2</sup> is the inner product with respect to *v*�, and we have introduced the following operators:

• *<sup>d</sup>* : *<sup>Ω</sup>k*(*Z*) <sup>→</sup> *<sup>Ω</sup>k*+1(*Z*) ··· The exterior differential operator *<sup>d</sup>* on *<sup>Z</sup>* is defined as

$$d\omega = \sum\_{j=1}^{n} \frac{\partial f\_{i\_1 \dots i\_k}}{\partial x^j} dx^j \wedge dx^{i\_1} \wedge \dots \wedge dx^{i\_k} \tag{6}$$

for *<sup>ω</sup>* <sup>=</sup> *fi*1···*ik* (*x*) *dxi*<sup>1</sup> ∧···∧ *dxik* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*), where *<sup>i</sup>*<sup>1</sup> ··· *ik* is the combination of *<sup>k</sup>* different integers selected from 1 to *n*, and *j* �= *i*<sup>1</sup> �= ··· �= *ik*.

• <sup>∗</sup> : *<sup>Ω</sup>k*(*Z*) <sup>→</sup> *<sup>Ω</sup>n*−*k*(*Z*) ··· The Hodge star operator <sup>∗</sup> induced in terms of a Riemannian metric on *Z* is defined as

$$\*\omega = \sum\_{i\_1 < \cdots < i\_k} \text{sgn}(I\_\prime I) f\_{\dot{i}\_1 \cdots \dot{i}\_k} dx^{\dot{\jmath}\_1} \wedge \cdots \wedge dx^{\dot{\jmath}\_{n-k}} \in \Omega^{n-k}(Z) \tag{7}$$

for *<sup>ω</sup>* <sup>=</sup> <sup>∑</sup>*i*1<···<*ik fi*1···*ik* (*x*) *dxi*<sup>1</sup> ∧···∧ *dxik* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*), where *<sup>j</sup>*<sup>1</sup> <sup>&</sup>lt; ··· <sup>&</sup>lt; *jn*−*<sup>k</sup>* is the rearrangement of the complement of *i*<sup>1</sup> < ··· < *ik* in the set {1, ··· , *n*} in ascending order, and sgn(*I*, *<sup>J</sup>*) is the sign of the permutation of *<sup>i</sup>*1, ··· , *ik*, *<sup>j</sup>*1, ··· , *jn*−*<sup>k</sup>* generated by interchanging of the basic forms *dx<sup>i</sup>* (if we interchange *dx<sup>i</sup>* and *dx<sup>j</sup>* in *ω* for arbitrary *i* and *j*, the sign of *ω* changes, i.e., it is alternating).

• *iv*� : *<sup>Ω</sup>k*(*Z*) <sup>→</sup> *<sup>Ω</sup>k*−1(*Z*) ··· The interior product *iv*� with respect to *<sup>v</sup>*� is defined as

$$\mathbf{i}\_{\mathfrak{D}^\sharp}\boldsymbol{\omega} = \begin{cases} (-1)^{m-1} f\_{\dot{\mathbf{i}}\_1 \cdots \dot{\mathbf{i}}\_k} \mathbf{g}\_{\dot{\mathbf{i}}\_m} d\mathbf{x}^{\dot{\mathbf{i}}\_1} \wedge \cdots \wedge d\mathbf{x}^{\dot{\mathbf{i}}\_{m-1}} \wedge d\mathbf{x}^{\dot{\mathbf{i}}\_{m+1}} \wedge \cdots \wedge d\mathbf{x}^{\dot{\mathbf{i}}\_k} \text{ if } \mathbf{j} = \mathbf{i}\_{\mathfrak{M}}\\ 0 & \text{if } \mathbf{j} \neq \mathbf{i}\_m \end{cases} \tag{8}$$

$$\text{for } \mathfrak{v}^{\sharp} = \mathfrak{g}\_{\not\!j}(\mathfrak{x}) (\partial \!\!/ \partial \!\!x^{\flat}) \text{ and } \omega = f\_{\not\!i\_1 \cdots \not\!i\_k}(\mathfrak{x}) d\mathfrak{x}^{\flat\_1} \wedge \cdots \wedge d\mathfrak{x}^{\flat\_k}.$$

In (5), we used the formula (*v* · ∇)*v* = (1/2)∇(*v* · *v*) + Curl *v* × *v*, and the enthalpy *w*(∗*ρ*) = (*∂*/*∂*∗*ρ*)(∗*<sup>ρ</sup> <sup>U</sup>*(∗*ρ*)) is related to the pressure *<sup>p</sup>*(∗*ρ*) by (∗*ρ*)−1*dp*(∗*ρ*) = *dw*(∗*ρ*), where *U*(*ρ*) is the internal energy function of the fluid satisfying *p*(∗*ρ*) = *w*(∗*ρ*)∗*ρ* − *U*(∗*ρ*)∗*ρ*.

Next, Maxwell's equations are defined as follows:

$$-\frac{\partial \mathbf{D}}{\partial t} = -dH + \mathbf{J}, \quad -\frac{\partial \mathbf{B}}{\partial t} = d\mathbf{E}, \quad d\mathbf{B} = 0, \quad d\mathbf{D} = \mathbf{g}.\tag{9}$$

where *<sup>D</sup>* <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) is the electric field induction, *<sup>H</sup>* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*) is the magnetic field intensity, *<sup>E</sup>* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*) is the electric field intensity, and *�* <sup>∈</sup> *<sup>Ω</sup>*3(*Z*) is the free charge density.

The constitutive relations are written as follows:

$$\mathbf{B} = \mu \ast \mathbf{H}, \quad \ast (\mathbf{E} + i\_{\mathfrak{D}^\sharp} \mathbf{B}) = \eta \mathbf{J}. \tag{10}$$

#### **2.3 Definition of port-Hamiltonian system**

Let us recall the definition of DPH systems. The advantage of these systems will be explained from the viewpoint of passivity and boundary controls in later sections.

*<sup>e</sup><sup>b</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>* <sup>a</sup> *boundary energy flow*. On the other hand, the terms *<sup>e</sup>*

The advantages of DPH systems are grounded in the following stability.

*V*(*x*(*t*1)) − *V*(*x*(*t*0)) ≤

*<sup>j</sup>* and *<sup>f</sup> <sup>i</sup>*

*<sup>j</sup>* and *<sup>e</sup><sup>b</sup>*

*p <sup>d</sup>* ∧ *f p <sup>d</sup>* and *e q <sup>d</sup>* ∧ *f q*

**2.4 Passivity and boundary integrability of energy flows**

energy flows. We call *e*

Hence, in (15), all port variables *e<sup>i</sup>*

Hamiltonian: *i*X*<sup>i</sup>*

boundary controls.

outputs for passivity-based controls. The boundary port variables *f <sup>b</sup>*

**2.5 Port-Hamiltonian representation of MHD**

H*<sup>f</sup>* = *Z* 1 2 �*v* , *v*

ideal fluid in (Van der Schaft and Maschke, 2002).

Hamiltonian densities of the fluid and the electromagnetic field

non-boundary-integrable; therefore, we cannot detect changes in them from the boundary

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 173

**Definition 2.3.** *Consider a system with an input vector u*(*t*) *and an output vector y*(*t*)*. The system is called* passive *if there exists a C*<sup>0</sup> *class non-negative function V*(*x*) *such that V*(0) = 0 *and*

*for all inputs u*(*t*) *and an initial value x*(*t*0)*, where t*<sup>0</sup> ≤ *t*<sup>1</sup> *and* � *means the transpose of vectors.*

*V*(*x*) can be regarded as the internal energy of the systems, which is an extended Lyapunov function. The inequality in (16) means that the energy always decreases; therefore, the system is stable in the sense of Lyapunov. Controls using the relation (16) are called *passivity-based controls*. Standard control systems with pairs of inputs/outputs satisfying (16) are called *port-Hamiltonian systems*. In this case, *V*(*x*) corresponds with the Hamiltonian of the system.

controls (Van der Schaft, 2000; Duindam et al., 2009) (details are given in *Section 2.6*). In (15), the first integral means the time variation of Hamiltonian; i.e., it is calculated by taking the interior product between a possible variational vector field and the variational derivative of

field and *α<sup>j</sup>* is the variational variable. The power of the first integral can be transformed into that of the third integral by appealing to boundary integrability of Stokes theorem. The second integral means non-boundary-integrable energy flows. Hence, if the second integral is zero, we can detect the variation of energies distributed on system domains from the variation on the boundary. In this sense, the power balance (15) is the principle of passivity-based

In this section, we derive the DPH representation of MHD from the geometric formulation presented in *Section 2.2*, which has been partially treated as Maxwell's equations and as an

Let *n* = 3. The DPH representation can be systematically constructed in terms of the

�*ρ* + *U*(∗*ρ*)*ρ*, H*<sup>e</sup>* =

*d*H*<sup>i</sup>* for *i* ∈ { *f* ,*e*}, where X*<sup>i</sup>* = ∑*j*(*∂αj*/*∂t*)(*∂*/*∂αj*) is the variational vector

 *Z* 1

<sup>2</sup> (*<sup>E</sup>* <sup>∧</sup> *<sup>D</sup>* <sup>+</sup> *<sup>H</sup>* <sup>∧</sup> *<sup>B</sup>*) (17)

*<sup>d</sup> distributed energy flows*.

 *t*<sup>1</sup> *t*0

*p <sup>d</sup>* ∧ *f p <sup>d</sup>* and *e*

*u*�(*s*)*y*(*s*) *ds* (16)

*<sup>j</sup>* for *i* ∈ {*p*, *q*, *b*} and *j* ∈ { *f* ,*e*} might be inputs and

*<sup>j</sup>* in (15) can be used as passivity-based boundary

*q <sup>d</sup>* ∧ *f q <sup>d</sup>* are

The inner product of *k*-forms can be defined on *Z* as

$$
\langle \omega, \eta \rangle = \omega \wedge \ast \eta, \quad \langle \omega, \eta \rangle\_{\mathbb{Z}} = \int\_{\mathbb{Z}} \langle \omega, \eta \rangle \tag{11}
$$

for *<sup>ω</sup>*, *<sup>η</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*). Moreover, we can identify the 1-from *<sup>v</sup>* with the vector field *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*); therefore, (11) can be defined as the inner product of vector fields, as in (5). DPH systems are defined by Stokes-Dirac structures (Van der Schaft and Maschke, 2002; Courant, 1990) with respect to the inner product (11).

#### **Definition 2.1.** *Let*

$$\begin{cases} (f^p, f^q, f^b) \in \Omega^p(\mathbf{Z}) \times \Omega^q(\mathbf{Z}) \times \Omega^{n-p}(\partial \mathbf{Z}),\\ (e^p, e^q, e^b) \in \Omega^{n-p}(\mathbf{Z}) \times \Omega^{n-q}(\mathbf{Z}) \times \Omega^{n-q}(\partial \mathbf{Z}),\\ (f^p\_d, f^d\_d) \in \Omega^p(\mathbf{Z}) \times \Omega^q(\mathbf{Z}),\\ (e^p\_d, e^d\_d) \in \Omega^{n-p}(\mathbf{Z}) \times \Omega^{n-q}(\mathbf{Z}),\end{cases} \tag{12}$$

*where all f <sup>i</sup> and e<sup>i</sup> for i* ∈ {*p*, *<sup>q</sup>*, *<sup>b</sup>*} *and all f <sup>i</sup> <sup>d</sup> and e<sup>i</sup> <sup>d</sup> for i* ∈ {*p*, *q*} *constitute the pairs with respect to the inner product* � · , · �*Z. The Stokes-Dirac structure is defined as follows:*

$$
\begin{bmatrix} f^p \\ f^q \end{bmatrix} = \begin{bmatrix} 0 \ (-1)^r d \\ d & 0 \end{bmatrix} \begin{bmatrix} e^p \\ e^q \end{bmatrix} - \begin{bmatrix} f\_d^p \\ f\_d^q \end{bmatrix},
\begin{bmatrix} e\_d^p \\ e\_d^q \end{bmatrix} = \begin{bmatrix} e^p \\ e^q \end{bmatrix},
\begin{bmatrix} f^b \\ e^b \end{bmatrix} = \begin{bmatrix} e^p|\_{\partial \mathcal{Z}} \\ (-1)^p e^q|\_{\partial \mathcal{Z}} \end{bmatrix},\tag{13}
$$

*where r* <sup>=</sup> *pq* <sup>+</sup> <sup>1</sup>*, p* <sup>+</sup> *<sup>q</sup>* <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>*,* <sup>|</sup>*∂<sup>Z</sup> is the restriction of differential forms to <sup>∂</sup>Z, df <sup>p</sup> <sup>d</sup>* �= 0*, and d f <sup>q</sup> <sup>d</sup>* �= 0*.*

A DPH system is formed by substituting the following variables obtained from a Hamiltonian density in the above Stokes-Dirac structure.

**Definition 2.2.** *Let* <sup>H</sup>(*αp*, *<sup>α</sup>q*) <sup>∈</sup> *<sup>Ω</sup>n*(*Z*) *be a Hamiltonian density, where <sup>α</sup><sup>i</sup>* <sup>∈</sup> *<sup>Ω</sup><sup>i</sup>* (*Z*) *for i* ∈ {*p*, *q*}*. A DPH system is defined by substituting*

$$f^p = -\frac{\partial a^p}{\partial t}, \quad f^q = -\frac{\partial a^q}{\partial t}, \quad e^p = \frac{\partial \mathcal{H}}{\partial a^p}, \quad e^q = \frac{\partial \mathcal{H}}{\partial a^q} \tag{14}$$

*into (13), where ∂*/*∂α<sup>i</sup> means the variational derivative with respect to α<sup>i</sup> . The variables f <sup>p</sup> <sup>d</sup> and f <sup>q</sup> d cannot be derived from any Hamiltonian.*

DPH systems satisfy the following boundary integrable relation that comes from Stokes theorem (Flanders, 1963; Morita, 2001).

**Proposition 2.1** (Van der Schaft and Maschke (2002))**.** *A DPH system satisfies the following power balance:*

$$\int\_{Z} \left( \varepsilon^{p} \wedge f^{p} + \varepsilon^{q} \wedge f^{q} \right) + \int\_{Z} \left( \varepsilon^{p}\_{d} \wedge f^{p}\_{d} + \varepsilon^{q}\_{d} \wedge f^{q}\_{d} \right) + \int\_{\partial Z} \varepsilon^{b} \wedge f^{b} = 0. \tag{15}$$

*where each term e<sup>i</sup>* <sup>∧</sup> *<sup>f</sup> <sup>i</sup> for i* ∈ {*p*, *<sup>q</sup>*, *<sup>b</sup>*} *has the dimension of power.*

In DPH systems, each *<sup>f</sup> <sup>i</sup>* and *<sup>e</sup><sup>i</sup>* for *<sup>i</sup>* ∈ {*p*, *<sup>q</sup>*} are called *port variables*, and *<sup>f</sup> <sup>b</sup>* and *<sup>e</sup><sup>b</sup>* are called *boundary port variables* that are a pair of boundary inputs and outputs. We call *<sup>e</sup><sup>b</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>* <sup>a</sup> *boundary energy flow*. On the other hand, the terms *<sup>e</sup> p <sup>d</sup>* ∧ *f p <sup>d</sup>* and *e q <sup>d</sup>* ∧ *f q <sup>d</sup>* are non-boundary-integrable; therefore, we cannot detect changes in them from the boundary energy flows. We call *e p <sup>d</sup>* ∧ *f p <sup>d</sup>* and *e q <sup>d</sup>* ∧ *f q <sup>d</sup> distributed energy flows*.

### **2.4 Passivity and boundary integrability of energy flows**

6 Magnetohydrodynamics

for *<sup>ω</sup>*, *<sup>η</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*). Moreover, we can identify the 1-from *<sup>v</sup>* with the vector field *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*); therefore, (11) can be defined as the inner product of vector fields, as in (5). DPH systems are defined by Stokes-Dirac structures (Van der Schaft and Maschke, 2002; Courant, 1990) with

> (*<sup>f</sup> <sup>p</sup>*, *<sup>f</sup> <sup>q</sup>*, *<sup>f</sup> <sup>b</sup>*) <sup>∈</sup> *<sup>Ω</sup>p*(*Z*) <sup>×</sup> *<sup>Ω</sup>q*(*Z*) <sup>×</sup> *<sup>Ω</sup>n*−*p*(*∂Z*), (*ep*,*eq*,*eb*) <sup>∈</sup> *<sup>Ω</sup>n*−*p*(*Z*) <sup>×</sup> *<sup>Ω</sup>n*−*q*(*Z*) <sup>×</sup> *<sup>Ω</sup>n*−*q*(*∂Z*),

*<sup>d</sup>* ) <sup>∈</sup> *<sup>Ω</sup>p*(*Z*) <sup>×</sup> *<sup>Ω</sup>q*(*Z*),

*the inner product* � · , · �*Z. The Stokes-Dirac structure is defined as follows:*

� �*e<sup>p</sup> eq* � − � *f p d f q d*

*<sup>d</sup>*) <sup>∈</sup> *<sup>Ω</sup>n*−*p*(*Z*) <sup>×</sup> *<sup>Ω</sup>n*−*q*(*Z*),

*<sup>d</sup> and e<sup>i</sup>*

� , � *e p d e q d*

*where r* <sup>=</sup> *pq* <sup>+</sup> <sup>1</sup>*, p* <sup>+</sup> *<sup>q</sup>* <sup>=</sup> *<sup>n</sup>* <sup>+</sup> <sup>1</sup>*,* <sup>|</sup>*∂<sup>Z</sup> is the restriction of differential forms to <sup>∂</sup>Z, df <sup>p</sup>*

**Definition 2.2.** *Let* <sup>H</sup>(*αp*, *<sup>α</sup>q*) <sup>∈</sup> *<sup>Ω</sup>n*(*Z*) *be a Hamiltonian density, where <sup>α</sup><sup>i</sup>* <sup>∈</sup> *<sup>Ω</sup><sup>i</sup>*

*<sup>∂</sup><sup>t</sup>* , *<sup>f</sup> <sup>q</sup>* <sup>=</sup> <sup>−</sup>*∂α<sup>q</sup>*

*into (13), where ∂*/*∂α<sup>i</sup> means the variational derivative with respect to α<sup>i</sup>*

*<sup>q</sup>* <sup>∧</sup> *<sup>f</sup> <sup>q</sup>*) <sup>+</sup>

*where each term e<sup>i</sup>* <sup>∧</sup> *<sup>f</sup> <sup>i</sup> for i* ∈ {*p*, *<sup>q</sup>*, *<sup>b</sup>*} *has the dimension of power.*

� *Z* � *e p <sup>d</sup>* ∧ *f p <sup>d</sup>* + *e q <sup>d</sup>* ∧ *f q d* � + � *∂Z e*

� = � *ep eq* � , � *f b eb* � =

A DPH system is formed by substituting the following variables obtained from a Hamiltonian

DPH systems satisfy the following boundary integrable relation that comes from Stokes

**Proposition 2.1** (Van der Schaft and Maschke (2002))**.** *A DPH system satisfies the following power*

In DPH systems, each *<sup>f</sup> <sup>i</sup>* and *<sup>e</sup><sup>i</sup>* for *<sup>i</sup>* ∈ {*p*, *<sup>q</sup>*} are called *port variables*, and *<sup>f</sup> <sup>b</sup>* and *<sup>e</sup><sup>b</sup>* are called *boundary port variables* that are a pair of boundary inputs and outputs. We call

*<sup>∂</sup><sup>t</sup>* , *<sup>e</sup><sup>p</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>H</sup>

*∂α<sup>p</sup>* , *<sup>e</sup>*

*<sup>q</sup>* <sup>=</sup> *<sup>∂</sup>*<sup>H</sup>

� *Z*

�*ω*, *η*� (11)

*<sup>d</sup> for i* ∈ {*p*, *q*} *constitute the pairs with respect to*

� *<sup>e</sup>p*|*∂<sup>Z</sup>* (−1)*peq*|*∂<sup>Z</sup>*

�

, (13)

*<sup>d</sup>* �= 0*, and*

(*Z*) *for i* ∈ {*p*, *q*}*.*

*<sup>d</sup> and f <sup>q</sup> d*

*∂α<sup>q</sup>* (14)

*<sup>b</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>* <sup>=</sup> 0. (15)

*. The variables f <sup>p</sup>*

(12)

�*ω*, *η*� = *ω* ∧ ∗*η*, �*ω*, *η*�*<sup>Z</sup>* =

The inner product of *k*-forms can be defined on *Z* as

respect to the inner product (11).

⎧ ⎪⎪⎨

⎪⎪⎩

*where all f <sup>i</sup> and e<sup>i</sup> for i* ∈ {*p*, *<sup>q</sup>*, *<sup>b</sup>*} *and all f <sup>i</sup>*

density in the above Stokes-Dirac structure.

*<sup>f</sup> <sup>p</sup>* <sup>=</sup> <sup>−</sup>*∂α<sup>p</sup>*

*A DPH system is defined by substituting*

*cannot be derived from any Hamiltonian.*

theorem (Flanders, 1963; Morita, 2001).

(*e<sup>p</sup>* <sup>∧</sup> *<sup>f</sup> <sup>p</sup>* <sup>+</sup> *<sup>e</sup>*

� *Z* (*f p <sup>d</sup>* , *f q*

(*e p d* ,*e q*

<sup>0</sup> (−1)*rd d* 0

**Definition 2.1.** *Let*

� *f p f q* � = �

*d f <sup>q</sup> <sup>d</sup>* �= 0*.*

*balance:*

The advantages of DPH systems are grounded in the following stability.

**Definition 2.3.** *Consider a system with an input vector u*(*t*) *and an output vector y*(*t*)*. The system is called* passive *if there exists a C*<sup>0</sup> *class non-negative function V*(*x*) *such that V*(0) = 0 *and*

$$V(\mathbf{x}(t\_1)) - V(\mathbf{x}(t\_0)) \le \int\_{t\_0}^{t\_1} u^\top(\mathbf{s}) y(\mathbf{s}) \, d\mathbf{s} \tag{16}$$

*for all inputs u*(*t*) *and an initial value x*(*t*0)*, where t*<sup>0</sup> ≤ *t*<sup>1</sup> *and* � *means the transpose of vectors.*

*V*(*x*) can be regarded as the internal energy of the systems, which is an extended Lyapunov function. The inequality in (16) means that the energy always decreases; therefore, the system is stable in the sense of Lyapunov. Controls using the relation (16) are called *passivity-based controls*. Standard control systems with pairs of inputs/outputs satisfying (16) are called *port-Hamiltonian systems*. In this case, *V*(*x*) corresponds with the Hamiltonian of the system. Hence, in (15), all port variables *e<sup>i</sup> <sup>j</sup>* and *<sup>f</sup> <sup>i</sup> <sup>j</sup>* for *i* ∈ {*p*, *q*, *b*} and *j* ∈ { *f* ,*e*} might be inputs and outputs for passivity-based controls.

The boundary port variables *f <sup>b</sup> <sup>j</sup>* and *<sup>e</sup><sup>b</sup> <sup>j</sup>* in (15) can be used as passivity-based boundary controls (Van der Schaft, 2000; Duindam et al., 2009) (details are given in *Section 2.6*). In (15), the first integral means the time variation of Hamiltonian; i.e., it is calculated by taking the interior product between a possible variational vector field and the variational derivative of Hamiltonian: *i*X*<sup>i</sup> d*H*<sup>i</sup>* for *i* ∈ { *f* ,*e*}, where X*<sup>i</sup>* = ∑*j*(*∂αj*/*∂t*)(*∂*/*∂αj*) is the variational vector field and *α<sup>j</sup>* is the variational variable. The power of the first integral can be transformed into that of the third integral by appealing to boundary integrability of Stokes theorem. The second integral means non-boundary-integrable energy flows. Hence, if the second integral is zero, we can detect the variation of energies distributed on system domains from the variation on the boundary. In this sense, the power balance (15) is the principle of passivity-based boundary controls.

#### **2.5 Port-Hamiltonian representation of MHD**

In this section, we derive the DPH representation of MHD from the geometric formulation presented in *Section 2.2*, which has been partially treated as Maxwell's equations and as an ideal fluid in (Van der Schaft and Maschke, 2002).

Let *n* = 3. The DPH representation can be systematically constructed in terms of the Hamiltonian densities of the fluid and the electromagnetic field

$$\mathcal{H}\_f = \int\_Z \frac{1}{2} \langle \mathbf{v}^\dagger, \mathbf{v}^\dagger \rangle \rho + \mathcal{U}(\*\rho)\rho, \quad \mathcal{H}\_\varepsilon = \int\_Z \frac{1}{2} \left( \mathbf{E} \wedge \mathbf{D} + \mathbf{H} \wedge \mathbf{B} \right) \tag{17}$$

bond graph theory (Karnopp et al., 2006), which is a generalized circuit theory for describing physical systems from the viewpoint of energy flows. For instance, the following diagram is

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 175

*∂Z eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

> ±*d* :

*DTF*:

*eb <sup>f</sup> f <sup>b</sup> f* ❴❴✤ 

where *∂Z* is the boundary of the systems, and we have defined the following bond graph

• The causal stroke | at the edge of the arrows indicates the direction in which the effort

• The *n* pairs of variables *ei* and *fi* around the 0-junction satisfy *e*<sup>1</sup> = *e*<sup>2</sup> = ··· = *en* and

• The *n* pairs of variables *ei* and *fi* around the 1-junction satisfy *f*<sup>1</sup> = *f*<sup>2</sup> = ··· = *fn* and

• The *GY* element with a parameter *M* means the gyrator satisfies *e*<sup>2</sup> = *M f*<sup>1</sup> and *e*<sup>1</sup> = *M f*2. • The *DTF* element means the differential transformer that has a Stokes-Dirac structure. In the case with the symbol *<sup>d</sup>*, *<sup>e</sup>*<sup>2</sup> <sup>=</sup> *de*1, *<sup>f</sup>*<sup>1</sup> <sup>=</sup> *d f*2, *<sup>f</sup> <sup>b</sup>* <sup>=</sup> *<sup>e</sup>*1|*∂<sup>Z</sup>* and *<sup>e</sup><sup>b</sup>* <sup>=</sup> <sup>−</sup>*f*2|*∂Z*. In the case

The Hamiltonian is shaped by connecting new systems to it through the pairs of boundary

*<sup>e</sup>*). For example, we can connect an electromagnetic system as

*<sup>i</sup>*=<sup>1</sup> *si fi* = 0, where *si* = 1 if the arrow is directed towards the junction and *si* = −1

*d*

*DTF* 1 ✤ ✤ *H* ❴ *dE Bt*

*H* ❴✤ ✤ *I* : *μ*

0 *dev <sup>e</sup><sup>ρ</sup>* ✤ ❴ ✤ ✤

✤ *eρ ρt* ❴

*C* : *ρ*−1*e<sup>ρ</sup>*

<sup>−</sup><sup>∞</sup> *f dt*.

<sup>−</sup><sup>∞</sup> *edt*.

(22)

the bound graph representation of the DPH system of MHD:

*�*−<sup>1</sup> : *C* 0

*ρ*−<sup>1</sup> : *I* 1

• The direction arrow indicates the sign of the energy flow.

elements:

∑*n*

∑*n*

otherwise.

port variables (*f <sup>b</sup>*

*<sup>f</sup>* ,*e<sup>b</sup>*

*<sup>f</sup>*) and (*<sup>f</sup> <sup>b</sup>*

*<sup>e</sup>* ,*e<sup>b</sup>*

a controller on the boundary *∂Z* of the upper part of (22) as follows:

*<sup>i</sup>*=<sup>1</sup> *siei* = 0.

signal is directed.

*ev* ❴ *vt* ✤ ✤ ✤ ✤ *de<sup>ρ</sup> ev* ❴ ❴❴

✤ ✤ *Dt* ❴ *<sup>E</sup>*

> *J E* ✤ ❴❴

*J vB* ❴❴✤ 

❴❴ *ev* −*g*<sup>2</sup> ✤ 

*ev* ✤ −*g*<sup>1</sup> 

<sup>−</sup>*ρ*(*dv*)−<sup>1</sup> : *<sup>R</sup> <sup>∂</sup><sup>Z</sup>*

• The arrow with the pair of variables *e* and *f* means the energy flow *e* ∧ *f* .

• The *C* element with a parameter *K* means the capacitor satisfies *e* = *K <sup>t</sup>*

• The *I* element with a parameter *K* means the inductor satisfies *f* = *K*−<sup>1</sup> *<sup>t</sup>*

• The *R* element with a parameter *K* means the resister satisfies *e* = *K f* .

with the symbol <sup>±</sup>*d*, *<sup>e</sup>*<sup>2</sup> <sup>=</sup> *de*1, *<sup>f</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>*d f*2, *<sup>f</sup> <sup>b</sup>* <sup>=</sup> *<sup>e</sup>*1|*∂<sup>Z</sup>* and *<sup>e</sup><sup>b</sup>* <sup>=</sup> *<sup>f</sup>*2|*∂Z*.

*E* −*dH* ❴✤ ✤

<sup>1</sup> *<sup>E</sup>*+*vB J* ❴✤ ✤ *R* : *η*−<sup>1</sup>

*GY* : *ρ*−1*B*

under constraints defined by the system equations (5), (9) and (10). Indeed, the DPH system of MHD can be constructed as

$$\begin{cases} \begin{bmatrix} -\boldsymbol{\rho}\_{t} \\ -\boldsymbol{\nu}\_{t} \end{bmatrix} = \begin{bmatrix} 0 \ d \\ d \ 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{e}\_{\rho} \\ \boldsymbol{e}\_{\boldsymbol{v}} \end{bmatrix} - \begin{bmatrix} 0 \\ \boldsymbol{g}\_{1} + \boldsymbol{g}\_{2} \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{f}^{b} \\ \boldsymbol{e}\_{f}^{b} \end{bmatrix} = \begin{bmatrix} \boldsymbol{e}\_{\rho} \vert\_{\partial \mathcal{Z}} \\ -\boldsymbol{e}\_{\boldsymbol{v}} \vert\_{\partial \mathcal{Z}} \end{bmatrix}, \\\ \begin{bmatrix} -\boldsymbol{\mathcal{D}}\_{t} \\ -\boldsymbol{\mathcal{B}}\_{t} \end{bmatrix} = \begin{bmatrix} 0 - \boldsymbol{d} \\ \boldsymbol{d} \ 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{E} \\ \boldsymbol{H} \end{bmatrix} + \begin{bmatrix} \boldsymbol{J} \\ 0 \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{\varepsilon}^{b} \\ \boldsymbol{e}\_{\varepsilon}^{b} \end{bmatrix} = \begin{bmatrix} \boldsymbol{E} \vert\_{\partial \mathcal{Z}} \\ \boldsymbol{H} \vert\_{\partial \mathcal{Z}} \end{bmatrix} \end{cases} \tag{18}$$

where the subscript *t* means the partial derivative with respect to *t*, and we have defined

$$\begin{cases} \boldsymbol{e}\_{\upsilon} = \dot{\boldsymbol{\iota}}\_{\mathfrak{D}^{\sharp}} \boldsymbol{\rho}, & \boldsymbol{e}\_{\rho} = \frac{1}{2} \langle \boldsymbol{\sigma}^{\sharp}, \boldsymbol{\sigma}^{\sharp} \rangle + \boldsymbol{w} (\ast \boldsymbol{\rho}), \\ \boldsymbol{g}\_{1} = -(\ast \boldsymbol{\rho})^{-1} \ast (\ast d \boldsymbol{\sigma} \wedge \ast \boldsymbol{e}\_{\upsilon}), & \boldsymbol{g}\_{2} = (\ast \boldsymbol{\rho})^{-1} \ast (\ast \boldsymbol{I} \wedge \ast \boldsymbol{B}), \\ \boldsymbol{B} = \mu \ast \boldsymbol{H}, & \ast (\boldsymbol{E} + \dot{\boldsymbol{\iota}}\_{\mathfrak{D}^{\sharp}} \mathbf{B}) = \eta \boldsymbol{I}\_{\mathsf{A}} \end{cases} \tag{19}$$

having set *p* = 3, *q* = 1, and *r* = 3 · 1 + 1 for the fluid, and *p* = 2, *q* = 2, and *r* = 2 · 2 + 1 for the electromagnetic field. The DPH system satisfies the following power balance equations:

$$\int\_{Z} \left( -\mathbf{e}\_{\rho} \wedge \mathbf{p}\_{t} - \mathbf{e}\_{\upsilon} \wedge \mathbf{v}\_{t} \right) - \int\_{Z} \mathbf{e}\_{\upsilon} \wedge \mathbf{g}\_{2} - \int\_{\partial Z} \mathbf{e}\_{\upsilon} \wedge \mathbf{e}\_{\rho} = \mathbf{0},\tag{20}$$

$$
\int\_{Z} \left( -\mathbf{E} \wedge \mathbf{D}\_{t} - \mathbf{H} \wedge \mathbf{B}\_{t} \right) - \int\_{Z} \mathbf{E} \wedge \mathbf{J} + \int\_{\partial Z} \mathbf{H} \wedge \mathbf{E} = \mathbf{0},\tag{21}
$$

where *ev* <sup>∧</sup> *<sup>g</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>(∗*ρ*)−1*ev* ∧ ∗(∗*d<sup>v</sup>* ∧ ∗*ev*) = <sup>−</sup>(∗*ρ*)−<sup>1</sup> <sup>∗</sup>*ev* ∧ ∗*d<sup>v</sup>* ∧ ∗*ev* <sup>=</sup> 0 and (21) which corresponds to Poynting's theorem. Note that the definition of the boundary energy flow in (21) is invariant even if *D<sup>t</sup>* is assumed to be zero, as is done in the standard theory of MHD. The first integrals of (20) and (21) correspond to the total change in energy of the system defined on *Z*, and the third integral is equal to the energy flowing across *∂Z*.

#### **2.6 Passivity-based boundary controls**

The basic strategy of passivity-based controls is to connect controllers through pairs of port variables, e.g., new port-Hamiltonian systems for changing the total Hamiltonians, or dissipative elements for stabilizing the system to the global minimum of the shaped Hamiltonian. The passivity-based boundary controls for DPH systems are applied to the boundary port variables *f <sup>b</sup> <sup>j</sup>* and *<sup>e</sup><sup>b</sup> <sup>j</sup>* for *<sup>j</sup>* ∈ { *<sup>f</sup>* ,*e*}. The product *<sup>e</sup><sup>b</sup> <sup>j</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup> <sup>j</sup>* has the dimension of power; therefore, *f <sup>b</sup> <sup>j</sup>* and *<sup>e</sup><sup>b</sup> <sup>j</sup>* can be considered to be a generalized velocity and a generalized force in analogy to mechanical systems (the correspondence might be the inverse in some cases).

Applying the output *f <sup>b</sup> <sup>j</sup>* magnified by a negative gain to the input *<sup>e</sup><sup>b</sup> <sup>j</sup>* means velocity feedback. This is one of most important passivity-based controls, i.e, damping assignment. Moreover, the boundary energy flow *e<sup>b</sup> <sup>j</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup> <sup>j</sup>* balances the internal energy of DPH systems; therefore, the total energy of the controlled system decreases, and the system becomes stable in the sense of passivity (16).

On the other hand, the Hamiltonian of the original DPH system can be changed by connecting other DPH systems to the original. The connection by means of port variables is expressed by 8 Magnetohydrodynamics

under constraints defined by the system equations (5), (9) and (10). Indeed, the DPH system

� 0 *g*<sup>1</sup> + *g*<sup>2</sup>

where the subscript *t* means the partial derivative with respect to *t*, and we have defined

�*<sup>v</sup>*, *<sup>v</sup>*� <sup>+</sup> *<sup>w</sup>*(∗*ρ*), *<sup>g</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>(∗*ρ*)−<sup>1</sup> <sup>∗</sup>(∗*d<sup>v</sup>* ∧ ∗*ev*), *<sup>g</sup>*<sup>2</sup> = (∗*ρ*)−<sup>1</sup> <sup>∗</sup>(<sup>∗</sup> *<sup>J</sup>* ∧ ∗*B*),

having set *p* = 3, *q* = 1, and *r* = 3 · 1 + 1 for the fluid, and *p* = 2, *q* = 2, and *r* = 2 · 2 + 1 for the electromagnetic field. The DPH system satisfies the following power balance equations:

> � *Z*

where *ev* <sup>∧</sup> *<sup>g</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>(∗*ρ*)−1*ev* ∧ ∗(∗*d<sup>v</sup>* ∧ ∗*ev*) = <sup>−</sup>(∗*ρ*)−<sup>1</sup> <sup>∗</sup>*ev* ∧ ∗*d<sup>v</sup>* ∧ ∗*ev* <sup>=</sup> 0 and (21) which corresponds to Poynting's theorem. Note that the definition of the boundary energy flow in (21) is invariant even if *D<sup>t</sup>* is assumed to be zero, as is done in the standard theory of MHD. The first integrals of (20) and (21) correspond to the total change in energy of the system

The basic strategy of passivity-based controls is to connect controllers through pairs of port variables, e.g., new port-Hamiltonian systems for changing the total Hamiltonians, or dissipative elements for stabilizing the system to the global minimum of the shaped Hamiltonian. The passivity-based boundary controls for DPH systems are applied to the

force in analogy to mechanical systems (the correspondence might be the inverse in some

This is one of most important passivity-based controls, i.e, damping assignment. Moreover,

total energy of the controlled system decreases, and the system becomes stable in the sense of

On the other hand, the Hamiltonian of the original DPH system can be changed by connecting other DPH systems to the original. The connection by means of port variables is expressed by

*<sup>j</sup>* magnified by a negative gain to the input *<sup>e</sup><sup>b</sup>*

*<sup>j</sup>* for *<sup>j</sup>* ∈ { *<sup>f</sup>* ,*e*}. The product *<sup>e</sup><sup>b</sup>*

*ev* ∧ *g*<sup>2</sup> −

*E* ∧ *J* +

� *∂Z*

� *∂Z*

� − � *Z* � , � *f b f eb f*

� =

� *<sup>e</sup>ρ*|*∂<sup>Z</sup>* −*ev*|*∂<sup>Z</sup>*

> � ,

*<sup>j</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>*

*<sup>j</sup>* can be considered to be a generalized velocity and a generalized

*<sup>j</sup>* balances the internal energy of DPH systems; therefore, the

� ,

*ev* ∧ *e<sup>ρ</sup>* = 0, (20)

*H* ∧ *E* = 0, (21)

*<sup>j</sup>* has the dimension of

*<sup>j</sup>* means velocity feedback.

(18)

(19)

� �*e<sup>ρ</sup> ev* � −

> � � *E H* � + � *J* 0 � , � *f b e eb e* � = � *E*|*∂<sup>Z</sup> H*|*∂<sup>Z</sup>*

2

*B* = *μ*∗*H*, ∗(*E* + *ivB*) = *ηJ*,

−*e<sup>ρ</sup>* ∧ *ρ<sup>t</sup>* − *ev* ∧ *v<sup>t</sup>*

*<sup>j</sup>* and *<sup>e</sup><sup>b</sup>*

*<sup>j</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>*

(−*E* ∧ *D<sup>t</sup>* − *H* ∧ *Bt*) −

defined on *Z*, and the third integral is equal to the energy flowing across *∂Z*.

of MHD can be constructed as

⎧ ⎪⎪⎪⎨

� −*ρ<sup>t</sup>* −*v<sup>t</sup>* � = � 0 *d d* 0

� −*D<sup>t</sup>* −*B<sup>t</sup>*

� = � 0 −*d d* 0

*ev* <sup>=</sup> *iv<sup>ρ</sup>*, *<sup>e</sup><sup>ρ</sup>* <sup>=</sup> <sup>1</sup>

⎪⎪⎪⎩

⎧ ⎪⎨

⎪⎩

� *Z* �

� *Z*

**2.6 Passivity-based boundary controls**

*<sup>j</sup>* and *<sup>e</sup><sup>b</sup>*

boundary port variables *f <sup>b</sup>*

power; therefore, *f <sup>b</sup>*

Applying the output *f <sup>b</sup>*

the boundary energy flow *e<sup>b</sup>*

cases).

passivity (16).

bond graph theory (Karnopp et al., 2006), which is a generalized circuit theory for describing physical systems from the viewpoint of energy flows. For instance, the following diagram is the bound graph representation of the DPH system of MHD:

*∂Z eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴ *�*−<sup>1</sup> : *C* 0 ✤ ✤ *Dt* ❴ *<sup>E</sup> J E* ✤ ❴❴ *E* −*dH* ❴✤ ✤ ±*d* : *DTF* 1 ✤ ✤ *H* ❴ *dE Bt H* ❴✤ ✤ *I* : *μ* <sup>1</sup> *<sup>E</sup>*+*vB J* ❴✤ ✤ *R* : *η*−<sup>1</sup> *GY* : *ρ*−1*B J vB* ❴❴✤ *ρ*−<sup>1</sup> : *I* 1 ❴❴ *ev* −*g*<sup>2</sup> ✤ *ev* ❴ *vt* ✤ ✤ ✤ ✤ *de<sup>ρ</sup> ev* ❴ ❴❴ *ev* ✤ −*g*<sup>1</sup> *DTF*: *d* 0 *dev <sup>e</sup><sup>ρ</sup>* ✤ ❴ ✤ ✤ ✤ *eρ ρt* ❴ *C* : *ρ*−1*e<sup>ρ</sup>* <sup>−</sup>*ρ*(*dv*)−<sup>1</sup> : *<sup>R</sup> <sup>∂</sup><sup>Z</sup> eb <sup>f</sup> f <sup>b</sup> f* ❴❴✤ (22)

where *∂Z* is the boundary of the systems, and we have defined the following bond graph elements:


The Hamiltonian is shaped by connecting new systems to it through the pairs of boundary port variables (*f <sup>b</sup> <sup>f</sup>* ,*e<sup>b</sup> <sup>f</sup>*) and (*<sup>f</sup> <sup>b</sup> <sup>e</sup>* ,*e<sup>b</sup> <sup>e</sup>*). For example, we can connect an electromagnetic system as a controller on the boundary *∂Z* of the upper part of (22) as follows:

where the inputs *u<sup>p</sup>*

system.

(*eb <sup>f</sup>* , *<sup>f</sup> <sup>b</sup>* *<sup>d</sup>* and *<sup>u</sup><sup>q</sup>*

(*e b <sup>f</sup>* , *<sup>f</sup> <sup>b</sup>*

*<sup>f</sup>* ) can be transformed as follows:

(*e b <sup>f</sup>* <sup>1</sup>, *<sup>f</sup> <sup>b</sup>*

**2.7 Port representation of balanced MHD**

*∂ρ*

*<sup>∂</sup><sup>t</sup>* <sup>=</sup> 0, *<sup>∂</sup><sup>v</sup>*

W*δ<sup>t</sup>* =

an infinitesimal variation. From (9), we obtain

 *Z δx δ δt*

the DPH system.

displacement:

can be reduced as follows:

*ev* ∧ *e<sup>ρ</sup>* =

 *∂Z*

term means external work. Hence, the altered port variables are

= *∂Z iv* 1 2 �*v* , *v*

 *∂Z*

A practical problem is whether the boundary port variables *e<sup>b</sup>*

passivity-based boundary control of MHD are the boundary port variables

*ivρ* ∧

*<sup>f</sup>* <sup>1</sup>)=(H*<sup>f</sup>* |*∂Z*, *v*|*∂Z*), (*e*

1 2 �*v* , *v*

where the first term corresponds to the boundary energy flow of convections and the second

This section discusses the stability of the DPH systems of MHD (18) with (19) in a balanced state. If the change in the potential energy of MHD caused by physically admissible perturbation is positive, then the equilibrium of MHD is stable. This fact is called *the energy principle of MHD* (Wesson, 2004). We derive the basic equation of the energy principle from

If the 2-form *dv* is zero at a certain time *t* = *t*0, it continues to be zero after *t*0. Accordingly, (5)

Now, let us consider the variation in energy with respect to an infinitesimal variation in

where the subscript *δt* means the variational derivative with respect to the time, and *δ* means

*δH*

*δJ <sup>δ</sup><sup>t</sup>* <sup>=</sup> *<sup>d</sup>*

*<sup>f</sup>* )=(−*ev*|*∂Z*,*eρ*|*∂Z*), (*e*

*<sup>d</sup>* distributed on *<sup>Z</sup>*. Moreover, in (23), *<sup>R</sup>*: *<sup>η</sup>*�−<sup>1</sup> distributed on *<sup>Z</sup>*� is

*<sup>i</sup>* and *<sup>f</sup> <sup>b</sup>*

*<sup>e</sup>* )=(*H*|*∂Z*, *E*|*∂Z*). (28)

*<sup>f</sup>* <sup>2</sup>)=(*p*|*∂Z*, *v*|*∂Z*). (30)

*<sup>i</sup>* can actually be used

*iv* (∗ *p*), (29)

considered as an element to create energy flowing across the boundary of the original MHD

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 177

as inputs and outputs. In this section, we show all possible boundary port variables of MHD regardless of whether they are actually usable or not. The input/output pairs for the

> *b <sup>e</sup>*, *<sup>f</sup> <sup>b</sup>*

� + *w*(∗*ρ*)

�*ρ* + *U*(∗*ρ*)*ρ*

*b <sup>f</sup>* <sup>2</sup>, *<sup>f</sup> <sup>b</sup>*  + *∂Z*

*<sup>∂</sup><sup>t</sup>* = (∗*ρ*)−<sup>1</sup> {−*dp*(∗*ρ*) + <sup>∗</sup>(<sup>∗</sup> *<sup>J</sup>* ∧ ∗*B*)} <sup>=</sup> 0. (31)

{−*dp*(∗*ρ*) + ∗(∗ *J* ∧ ∗*B*)} , (32)

*<sup>δ</sup><sup>t</sup>* , (33)

where *Z*� is the domain of the new electromagnetic system and each system is connected through the common boundary *∂Z* = *∂Z*� . In this case, the original Hamiltonian H*<sup>f</sup>* + H*<sup>e</sup>* is changed into the controlled Hamiltonian H*<sup>f</sup>* + H*<sup>e</sup>* + H� *<sup>e</sup>*, where H� *<sup>e</sup>* is the Hamiltonian of the new electromagnetic system. Note that the Hamiltonians can only be shaped to control the energy flows of boundary port variables or energy levels of the original system, not to control the distributed states in the sense of boundary value problems.

The energy flow through the boundary *∂Z* = *∂Z*� can be described as

$$\mathcal{H}\_{\delta t} = \int\_{\partial Z} e^b \wedge f^b - e^{b\prime} \wedge f^{b\prime} \,. \tag{24}$$

where *eb*� and *f <sup>b</sup>*� are the pair of the boundary port variables defined on *∂Z*� . In general, when the port variable *e<sup>b</sup>* is regarded as an input, the power balance (15) is changed into

$$\int\_{Z} \left( \varepsilon^{p} \wedge f^{p} + \varepsilon^{q} \wedge f^{q} \right) + \int\_{Z} \left\{ \varepsilon^{p}\_{d} \wedge \left( f^{p}\_{d} + \mathfrak{u}^{p}\_{d} \right) + \varepsilon^{q}\_{d} \wedge \left( f^{q}\_{d} + \mathfrak{u}^{q}\_{d} \right) \right\} + \int\_{\partial Z} \mathfrak{u}^{b} \wedge f^{b} = 0,\tag{25}$$

where *e<sup>b</sup>* = *u<sup>b</sup>* is the boundary control, and *u<sup>p</sup> <sup>d</sup>* and *<sup>u</sup><sup>q</sup> <sup>d</sup>* are the distributed controls. If *<sup>f</sup> <sup>b</sup>* is regarded as an input, then the boundary control is replaced by *f <sup>b</sup>* = *ub*.

Damping terms are assigned by connecting of resisters to the pair on the system domain; they are illustrated as *R* elements in the bond graph. If systems with dissipative elements are connected to the boundary of a controlled system, it corresponds to a boundary damping assignment that absorbs the energy of the original system through the boundary. For example, in (25), the controls

$$
\mu^b = -\mathbf{K}^b f^b, \quad \mathfrak{u}\_d^p = -\mathbf{K}\_d^p \mathfrak{a}^p, \quad \mathfrak{u}\_d^q = -\mathbf{K}\_d^q \mathfrak{a}^q \tag{26}
$$

are equivalent to connecting an *R* element to the port variables, where *K<sup>b</sup>* is the gain function defined on *<sup>∂</sup>Z*, *<sup>K</sup><sup>p</sup> <sup>d</sup>* and *<sup>K</sup><sup>p</sup> <sup>d</sup>* are the gain functions defined on *<sup>Z</sup>*, and *<sup>f</sup> <sup>i</sup>* <sup>=</sup> <sup>−</sup>(*∂α<sup>i</sup>* /*∂t*). For eliminating distributed energy flows *f p <sup>d</sup>* and *f q <sup>d</sup>* that are exactly known, we can use the controls

$$
\mu\_d^p = -f\_{d'}^p \quad \mu\_d^q = -f\_{d'}^q \tag{27}
$$

where the inputs *u<sup>p</sup> <sup>d</sup>* and *<sup>u</sup><sup>q</sup> <sup>d</sup>* distributed on *<sup>Z</sup>*. Moreover, in (23), *<sup>R</sup>*: *<sup>η</sup>*�−<sup>1</sup> distributed on *<sup>Z</sup>*� is considered as an element to create energy flowing across the boundary of the original MHD system.

A practical problem is whether the boundary port variables *e<sup>b</sup> <sup>i</sup>* and *<sup>f</sup> <sup>b</sup> <sup>i</sup>* can actually be used as inputs and outputs. In this section, we show all possible boundary port variables of MHD regardless of whether they are actually usable or not. The input/output pairs for the passivity-based boundary control of MHD are the boundary port variables

$$(\epsilon^{b}\_{f'}f^{b}\_{f}) = (-e\_{\upsilon}|\_{\partial \mathbb{Z}'}e\_{\rhd}|\_{\partial \mathbb{Z}}), \quad (\epsilon^{b}\_{\varepsilon'}f^{b}\_{\varepsilon}) = (\mathbf{H}|\_{\partial \mathbb{Z}'}\mathbf{E}|\_{\partial \mathbb{Z}}).\tag{28}$$

(*eb <sup>f</sup>* , *<sup>f</sup> <sup>b</sup> <sup>f</sup>* ) can be transformed as follows:

10 Magnetohydrodynamics

*DTF*:

1 *eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

> ±*d* :

*DTF* 1 ✤ ✤ *H* ❴ *dE Bt*

where *Z*� is the domain of the new electromagnetic system and each system is connected

new electromagnetic system. Note that the Hamiltonians can only be shaped to control the energy flows of boundary port variables or energy levels of the original system, not to control

*<sup>b</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>* <sup>−</sup> *<sup>e</sup>*

*<sup>d</sup>* ) + *e q <sup>d</sup>* ∧ (*f q <sup>d</sup>* <sup>+</sup> *<sup>u</sup><sup>q</sup> d*) + *∂Z*

Damping terms are assigned by connecting of resisters to the pair on the system domain; they are illustrated as *R* elements in the bond graph. If systems with dissipative elements are connected to the boundary of a controlled system, it corresponds to a boundary damping assignment that absorbs the energy of the original system through the boundary. For example,

*<sup>d</sup>* <sup>=</sup> <sup>−</sup>*K<sup>p</sup>*

are equivalent to connecting an *R* element to the port variables, where *K<sup>b</sup>* is the gain function

*<sup>d</sup>* and *<sup>u</sup><sup>q</sup>*

*<sup>d</sup> <sup>α</sup>p*, *<sup>u</sup><sup>q</sup>*

*<sup>d</sup>* are the gain functions defined on *<sup>Z</sup>*, and *<sup>f</sup> <sup>i</sup>* <sup>=</sup> <sup>−</sup>(*∂α<sup>i</sup>*

*<sup>d</sup>* = −*f q*

*<sup>d</sup>* <sup>=</sup> <sup>−</sup>*K<sup>q</sup>*

*<sup>d</sup>* that are exactly known, we can use the controls

*<sup>b</sup>*� <sup>∧</sup> *<sup>f</sup> <sup>b</sup>*�

*eb*� *<sup>e</sup> f <sup>b</sup>*� *e* ✤ ❴❴

1 ✤ ✤ *H* ❴ *dE Bt*

*H* ❴✤ ✤

*H* ❴✤ ✤

*I* : *μ*� *Z*

*I* : *μ Z*

. In this case, the original Hamiltonian H*<sup>f</sup>* + H*<sup>e</sup>* is

*<sup>e</sup>* is the Hamiltonian of the

, (24)

*<sup>d</sup>* are the distributed controls. If *<sup>f</sup> <sup>b</sup>* is

*<sup>d</sup>α<sup>q</sup>* (26)

*<sup>d</sup>* , (27)

/*∂t*). For

. In general, when

*<sup>u</sup><sup>b</sup>* <sup>∧</sup> *<sup>f</sup> <sup>b</sup>* <sup>=</sup> 0, (25)

*<sup>e</sup>*, where H�

*∂Z*

(23)

±*d*

*R* : *η*�−<sup>1</sup>

−*dH* ❴✤ ✤

*E* −*dH* ❴✤ ✤

*�*�−<sup>1</sup> : *C* 0

*�*−<sup>1</sup> : *C* 0

changed into the controlled Hamiltonian H*<sup>f</sup>* + H*<sup>e</sup>* + H�

the distributed states in the sense of boundary value problems.

The energy flow through the boundary *∂Z* = *∂Z*� can be described as

H*δ<sup>t</sup>* =

regarded as an input, then the boundary control is replaced by *f <sup>b</sup>* = *ub*.

*<sup>u</sup><sup>b</sup>* <sup>=</sup> <sup>−</sup>*K<sup>b</sup> <sup>f</sup> <sup>b</sup>*, *<sup>u</sup><sup>p</sup>*

*p <sup>d</sup>* and *f q*

*up <sup>d</sup>* = −*f p <sup>d</sup>* , *<sup>u</sup><sup>q</sup>*

where *eb*� and *f <sup>b</sup>*� are the pair of the boundary port variables defined on *∂Z*�

 *∂Z e*

the port variable *e<sup>b</sup>* is regarded as an input, the power balance (15) is changed into

through the common boundary *∂Z* = *∂Z*�

 *Z*

(*e<sup>p</sup>* <sup>∧</sup> *<sup>f</sup> <sup>p</sup>* <sup>+</sup> *<sup>e</sup>*

in (25), the controls

defined on *<sup>∂</sup>Z*, *<sup>K</sup><sup>p</sup>*

*<sup>q</sup>* <sup>∧</sup> *<sup>f</sup> <sup>q</sup>*) <sup>+</sup>

where *e<sup>b</sup>* = *u<sup>b</sup>* is the boundary control, and *u<sup>p</sup>*

*<sup>d</sup>* and *<sup>K</sup><sup>p</sup>*

eliminating distributed energy flows *f*

 *Z e p <sup>d</sup>* ∧ (*f p <sup>d</sup>* <sup>+</sup> *<sup>u</sup><sup>p</sup>*

✤ ✤ *Dt* ❴ *<sup>E</sup> <sup>E</sup>*

✤ ✤ *Dt* ❴ *<sup>E</sup>*

> *J E* ✤ ❴❴

. . .

*J E* ❴❴✤ 

$$\begin{split} \int\_{\partial \mathbb{Z}} \mathbf{e}\_{\mathcal{D}} \wedge \mathbf{e}\_{\mathcal{P}} &= \int\_{\partial \mathbb{Z}} \dot{\mathbf{i}}\_{\mathcal{D}^{\sharp}} \boldsymbol{\rho} \wedge \left( \frac{1}{2} \langle \mathbf{v}^{\sharp}, \mathbf{v}^{\sharp} \rangle + w(\ast \boldsymbol{\rho}) \right) \\ &= \int\_{\partial \mathbb{Z}} \dot{\mathbf{i}}\_{\mathcal{D}^{\sharp}} \left( \frac{1}{2} \langle \mathbf{v}^{\sharp}, \mathbf{v}^{\sharp} \rangle \boldsymbol{\rho} + \mathcal{U}(\ast \boldsymbol{\rho}) \boldsymbol{\rho} \right) + \int\_{\partial \mathbb{Z}} \dot{\mathbf{i}}\_{\mathcal{D}^{\sharp}}(\ast \boldsymbol{\rho}), \end{split} \tag{29}$$

where the first term corresponds to the boundary energy flow of convections and the second term means external work. Hence, the altered port variables are

$$(\boldsymbol{e}\_{f\boldsymbol{1}'}^{b}\boldsymbol{f}\_{f\boldsymbol{1}}^{b}) = (\mathcal{H}\_{f}|\_{\partial\mathcal{Z}}, \boldsymbol{\mathfrak{v}}|\_{\partial\mathcal{Z}}), \quad (\boldsymbol{e}\_{f\mathcal{Z}'}^{b}\boldsymbol{f}\_{f\boldsymbol{2}}^{b}) = (\boldsymbol{p}|\_{\partial\mathcal{Z}'}\boldsymbol{\mathfrak{v}}|\_{\partial\mathcal{Z}}).\tag{30}$$

#### **2.7 Port representation of balanced MHD**

This section discusses the stability of the DPH systems of MHD (18) with (19) in a balanced state. If the change in the potential energy of MHD caused by physically admissible perturbation is positive, then the equilibrium of MHD is stable. This fact is called *the energy principle of MHD* (Wesson, 2004). We derive the basic equation of the energy principle from the DPH system.

If the 2-form *dv* is zero at a certain time *t* = *t*0, it continues to be zero after *t*0. Accordingly, (5) can be reduced as follows:

$$\frac{\partial \rho}{\partial t} = 0, \quad \frac{\partial \sigma}{\partial t} = (\ast \rho)^{-1} \left\{-d\rho (\ast \rho) + \ast (\ast \mathbf{J} \wedge \ast \mathbf{B})\right\} = 0. \tag{31}$$

Now, let us consider the variation in energy with respect to an infinitesimal variation in displacement:

$$\mathcal{W}\_{\delta t} = \int\_{Z} \delta \mathbf{x} \frac{\delta}{\delta t} \left\{-d p(\*\rho) + \*(\*\mathbf{J} \wedge \*\mathbf{B})\right\},\tag{32}$$

where the subscript *δt* means the variational derivative with respect to the time, and *δ* means an infinitesimal variation. From (9), we obtain

$$\frac{\delta \mathbf{J}}{\delta t} = d \frac{\delta \mathbf{H}}{\delta t} \,' \,. \tag{33}$$

extended so as to have perturbations as follows:

� = � 0 −*d d* 0

� �*e<sup>ρ</sup> ev* � −

> *j* (*u<sup>a</sup>*

*<sup>t</sup>* , *<sup>u</sup><sup>a</sup> <sup>y</sup>*, *u<sup>a</sup> <sup>z</sup>*, *u<sup>a</sup> tt*, *<sup>u</sup><sup>a</sup> ty*, *<sup>u</sup><sup>a</sup> tz*, *<sup>u</sup><sup>a</sup> yy*, *u<sup>a</sup> yz*, *u<sup>a</sup>*

*<sup>j</sup>* is a *pseudo potential* derived from

*γi*

*ϕ*˜*i*

0

*hv*(*ω*) = � <sup>1</sup>

*<sup>I</sup>* <sup>−</sup> *<sup>u</sup><sup>a</sup>*

**3.2 Decomposition of model perturbations of DPH systems**

includes up to second-order derivatives: *r* = 2. Accordingly, Δ*<sup>i</sup>*

*<sup>j</sup> du* <sup>=</sup> <sup>Δ</sup> *du* <sup>−</sup> *<sup>d</sup>ϕ*˜*<sup>i</sup>*

*cI*), and usually *<sup>u</sup><sup>a</sup>*

*ϕ*˜*i <sup>j</sup>* = *w* +

*<sup>j</sup>* a *dual exact system*, which corresponds to a distributed energy variable.

*<sup>j</sup>* is calculated as

*<sup>j</sup>* <sup>=</sup> *hv*(Δ*<sup>i</sup>*

� � *E H* � + � *J* 0 � + � Δ*p e* Δ*q e* � , � *f b e eb e* � = � *E*|*∂<sup>Z</sup> H*|*∂<sup>Z</sup>*

� 0 *g*<sup>1</sup> + *g*<sup>2</sup>

� + � Δ*p f* Δ*q f*

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 179

function defined by the local coordinates *<sup>x</sup><sup>k</sup>* of *<sup>Y</sup>* for 1 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*, and we denote all possible

Consider the DPH system (39) of MHD with perturbations. We assume that the DPH system

*<sup>j</sup>* <sup>+</sup> *<sup>γ</sup><sup>i</sup> j*

*<sup>j</sup> du* <sup>=</sup> <sup>Δ</sup> *du* <sup>−</sup> *<sup>γ</sup><sup>i</sup>*

*a*,*I* (*u<sup>a</sup> <sup>I</sup>* <sup>−</sup> *<sup>u</sup><sup>a</sup> cI*) *<sup>∂</sup> ∂u<sup>a</sup> I*

*cI* <sup>=</sup> 0. In (40), we call *<sup>d</sup>ϕ<sup>i</sup>*

1 2

*<sup>j</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *wt* <sup>+</sup> *wtt* for some *<sup>i</sup>* and *<sup>j</sup>*, where *<sup>u</sup>*<sup>1</sup> <sup>=</sup> *<sup>w</sup>* and *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *<sup>t</sup>*. The

Δ*i <sup>j</sup>* <sup>=</sup> *<sup>d</sup>ϕ<sup>i</sup>*

> *j* , *ϕ<sup>i</sup>*

*<sup>j</sup> dua*) = � <sup>1</sup>

*hv* is *the homotopy operator* for *<sup>ω</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) with respect to an equilibrium point *<sup>u</sup><sup>a</sup>*

*<sup>i</sup>ν*¯*ω*(*x*, *<sup>λ</sup>u*¯*I*) *<sup>λ</sup>*−1*dλ*, *<sup>ν</sup>*¯ <sup>=</sup> ∑

1 2 *wwt* +

0

*<sup>u</sup><sup>a</sup>* · <sup>Δ</sup>*<sup>i</sup> j* (*xk*, *λu<sup>a</sup>*

� , � *f b f eb f*

*<sup>j</sup>* for *i* ∈ {*p*, *q*} and *j* ∈ { *f* ,*e*} means a perturbation. Now, let us consider the

� =

*<sup>I</sup>*), where *<sup>i</sup>* ∈ {*p*, *<sup>q</sup>*}, *<sup>j</sup>* ∈ { *<sup>f</sup>* ,*e*}, *<sup>u</sup><sup>a</sup>* for 1 <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *<sup>l</sup>* is the

*<sup>I</sup>* and denote the order by 0 ≤ |*I*| ≤ *r*. For example,

� *<sup>e</sup>ρ*|*∂<sup>Z</sup>* −*ev*|*∂<sup>Z</sup>*

> � ,

*zz*} for (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3)=(*t*, *<sup>y</sup>*, *<sup>z</sup>*), and the

, (40)

*<sup>j</sup>* can be uniquely decomposed

*<sup>j</sup> du*, (41)

*<sup>I</sup>*) *dλ*, (42)

*cI*, called a

, (43)

*<sup>j</sup>* an *exact system* and call

*wwtt* (44)

� ,

(39)

⎧ ⎪⎪⎪⎨

� −*ρ<sup>t</sup>* −*v<sup>t</sup>* � = � 0 *d d* 0

� −*D<sup>t</sup>* −*B<sup>t</sup>*

subsystem of DPH systems, Δ*<sup>i</sup>*

*<sup>I</sup>* for *<sup>r</sup>* <sup>=</sup> 2 means {*ua*, *<sup>u</sup><sup>a</sup>*

derivatives up to the order *r* of *u<sup>a</sup>* by *u<sup>a</sup>*

subscript means the partial derivative.

⎪⎪⎪⎩

where each Δ*<sup>i</sup>*

*ua*

into

where *ϕ<sup>i</sup>*

where *u*¯*<sup>a</sup>*

*γi*

the temporal variable *ϕ*˜*<sup>i</sup>*

*homotopy center*, defined by

*<sup>I</sup>* <sup>=</sup> *<sup>u</sup><sup>a</sup>*

temporal variable

*cI* <sup>+</sup> *<sup>λ</sup>*(*u<sup>a</sup>*

For example, let us consider Δ*<sup>i</sup>*

where we have assumed that *D<sup>t</sup>* = 0 and *η* = 0; therefore,

$$d\mathbf{D}\_t = \varrho\_t = 0, \quad \varrho\_t = d\mathbf{J} = 0, \quad \mathbf{E} = -i\_{\mathbf{D}^\dagger} \mathbf{B}. \tag{34}$$

The DPH system of balanced MHD can be constructed as follows:

$$\begin{cases} \begin{bmatrix} -\boldsymbol{\rho}\_{t} \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \ d \\ d \, 0 \end{bmatrix} \begin{bmatrix} w\_{\delta t}(\ast \boldsymbol{\rho}) \\ \dot{i}\_{\mathsf{v}\boldsymbol{\theta}} \boldsymbol{\rho} \end{bmatrix} - \begin{bmatrix} 0 \\ (\ast \boldsymbol{\rho})^{-1} d \boldsymbol{p}\_{\delta t} \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{fs}^{b} \\ \boldsymbol{e}\_{fs}^{b} \end{bmatrix} = \begin{bmatrix} w\_{\delta t}(\ast \boldsymbol{\rho})|\_{\partial \mathcal{Z}} \\ -\dot{i}\_{\mathsf{v}\boldsymbol{\theta}} \boldsymbol{\rho}|\_{\partial \mathcal{Z}} \end{bmatrix}, \\\ \begin{bmatrix} 0 \\ -\mathsf{B}\_{t} \end{bmatrix} = \begin{bmatrix} 0 - d \\ d \; 0 \end{bmatrix} \begin{bmatrix} -\boldsymbol{i}\_{\mathsf{v}\boldsymbol{\theta}} \mathsf{B} \\ \mathbf{H}\_{\delta t} \end{bmatrix} + \begin{bmatrix} \boldsymbol{f}\_{\delta t} \\ 0 \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{\mathsf{c}\boldsymbol{\theta}}^{b} \\ \boldsymbol{e}\_{\mathsf{c}\boldsymbol{\theta}}^{b} \end{bmatrix} = \begin{bmatrix} -\mathsf{i}\_{\mathsf{v}\boldsymbol{\theta}} \mathsf{B}|\_{\partial \mathcal{Z}} \\ \mathbf{H}\_{\delta t}|\_{\partial \mathcal{Z}} \end{bmatrix}, \end{cases} \tag{35}$$

where *δx* = *v*. The DPH system (35) satisfies the power balance equations,

$$-\int\_{Z} w\_{\delta t}(\*\rho) \wedge \rho\_{t} + \int\_{Z} i\_{\mathfrak{D}^{\sharp}} \rho \wedge (\*\rho)^{-1} d\rho\_{\delta t} - \int\_{\partial Z} i\_{\mathfrak{D}^{\sharp}} \rho \wedge w\_{\delta t}(\*\rho) = 0,\tag{36}$$

$$-\int\_{Z} \mathbf{H}\_{\delta t} \wedge \mathbf{B}\_{t} + \int\_{Z} \mathbf{i}\_{\mathfrak{D}^{\sharp}} \mathbf{B} \wedge \mathbf{J}\_{\delta t} - \int\_{\partial Z} \mathbf{H}\_{\delta t} \wedge \mathbf{i}\_{\mathfrak{D}^{\sharp}} \mathbf{B} = \mathbf{0}.\tag{37}$$

As a result, we obtain the boundary port variables

$$(f\_{fs'}^{b}e\_{fs}^{b}) = (w\_{\delta t}(\*\rho)|\_{\partial \mathbb{Z}^{\prime}} - i\_{\mathfrak{D}^{\sharp}}\mathfrak{p}|\_{\partial \mathbb{Z}})\_{\prime} \quad (f\_{\mathfrak{es}}^{b}, e\_{\mathfrak{es}}^{b}) = (-i\_{\mathfrak{D}^{\sharp}}\mathfrak{B}|\_{\partial \mathbb{Z}^{\prime}}H\_{\delta t}|\_{\partial \mathbb{Z}}) \tag{38}$$

from (35).

The energy principle is frequently used to analyze the stability of MHD. The DPH system of MHD generates the power balance equation (37) for an analysis. The boundary port variables of (35) correspond to those of the DPH system of dynamical MHD (18) except for the term depending on *v*. Hence, (18) can be considered to be a generalized system following the energy principle of MHD. If active controls are used in MHD systems, e.g., in Tokamaks, the control side of the DPH system able to be used, e.g., as a boundary control for subdivided MHD systems.

#### **3. Construction pseudo potentials for non-Hamiltonian subsystems**

#### **3.1 DPH systems of MHD with perturbations**

*Section 2* discussed the energy structure of the DPH system of MHD on the basis of its physical meaning. However, model perturbations caused by, for instance, disturbances, additional terms derived by using system identification methods for model refinements, or controllers designed by a control theory do not always have physical interpretations. In this section, we show a method of determining the energy structure of such perturbations. Precisely speaking, we decompose a given perturbation into a Hamiltonian subsystem and a non-Hamiltonian subsystem that can be regarded as an external force in terms of Euler-Lagrange equations (Nishida et al., 2007a).

In this section, we consider an *n*-dimensional smooth Riemannian manifold *Y* that is homeomorphic to an *n*-dimensional Euclidian space (i.e., topologically same, and one can be deformed into the other). Let *Z* be an *n*-dimensional smooth Riemannian submanifold of *Y* with a smooth boundary *∂Z*. The DPH system (18) of MHD defined on a domain *Z* is extended so as to have perturbations as follows:

12 Magnetohydrodynamics

� 0 (∗*ρ*)−1*dpδ<sup>t</sup>*

*iv<sup>ρ</sup>* <sup>∧</sup> (∗*ρ*)−1*dpδ<sup>t</sup>* <sup>−</sup>

The energy principle is frequently used to analyze the stability of MHD. The DPH system of MHD generates the power balance equation (37) for an analysis. The boundary port variables of (35) correspond to those of the DPH system of dynamical MHD (18) except for the term depending on *v*. Hence, (18) can be considered to be a generalized system following the energy principle of MHD. If active controls are used in MHD systems, e.g., in Tokamaks, the control side of the DPH system able to be used, e.g., as a boundary control for subdivided

*Section 2* discussed the energy structure of the DPH system of MHD on the basis of its physical meaning. However, model perturbations caused by, for instance, disturbances, additional terms derived by using system identification methods for model refinements, or controllers designed by a control theory do not always have physical interpretations. In this section, we show a method of determining the energy structure of such perturbations. Precisely speaking, we decompose a given perturbation into a Hamiltonian subsystem and a non-Hamiltonian subsystem that can be regarded as an external force in terms of

In this section, we consider an *n*-dimensional smooth Riemannian manifold *Y* that is homeomorphic to an *n*-dimensional Euclidian space (i.e., topologically same, and one can be deformed into the other). Let *Z* be an *n*-dimensional smooth Riemannian submanifold of *Y* with a smooth boundary *∂Z*. The DPH system (18) of MHD defined on a domain *Z* is

� *∂Z*

*dD<sup>t</sup>* = *�<sup>t</sup>* = 0, *�<sup>t</sup>* = *dJ* = 0, *E* = −*ivB*. (34)

� = �

−*ivB*|*∂<sup>Z</sup> Hδt*|*∂<sup>Z</sup>*

*wδt*(∗*ρ*)|*∂<sup>Z</sup>* −*ivρ*|*∂<sup>Z</sup>*

*ivρ* ∧ *wδt*(∗*ρ*) = 0, (36)

� ,

*Hδ<sup>t</sup>* ∧ *ivB* = 0. (37)

*es*)=(−*ivB*|*∂Z*, *Hδt*|*∂Z*) (38)

� ,

(35)

� , � *f b f s eb f s*

> � *∂Z*

*es*,*e b*

where we have assumed that *D<sup>t</sup>* = 0 and *η* = 0; therefore,

⎧ ⎪⎪⎪⎨

� −*ρ<sup>t</sup>* 0 � = � 0 *d d* 0

� 0 −*B<sup>t</sup>*

− � *Z*

− � *Z*

(*f <sup>b</sup> f s*,*e b*

from (35).

MHD systems.

� = � 0 −*d d* 0

*wδt*(∗*ρ*) ∧ *ρ<sup>t</sup>* +

As a result, we obtain the boundary port variables

� *Z*

*Hδ<sup>t</sup>* ∧ *B<sup>t</sup>* +

**3.1 DPH systems of MHD with perturbations**

Euler-Lagrange equations (Nishida et al., 2007a).

⎪⎪⎪⎩

The DPH system of balanced MHD can be constructed as follows:

� �*wδt*(∗*ρ*) *ivρ*

> � �−*iv<sup>B</sup> Hδ<sup>t</sup>*

> > � *Z*

� −

> � + � *Jδt* 0 � , � *f b es eb es* � = �

where *δx* = *v*. The DPH system (35) satisfies the power balance equations,

*ivB* ∧ *Jδ<sup>t</sup>* −

*f s*)=(*wδt*(∗*ρ*)|*∂Z*, <sup>−</sup>*iv<sup>ρ</sup>*|*∂Z*), (*<sup>f</sup> <sup>b</sup>*

**3. Construction pseudo potentials for non-Hamiltonian subsystems**

$$\begin{cases} \begin{bmatrix} -\boldsymbol{\rho}\_{t} \\ -\boldsymbol{\nu}\_{t} \end{bmatrix} = \begin{bmatrix} 0 \ d \\ d \ 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{e}\_{\boldsymbol{\rho}} \\ \boldsymbol{e}\_{\boldsymbol{\nu}} \end{bmatrix} - \begin{bmatrix} 0 \\ \boldsymbol{g}\_{1} + \boldsymbol{g}\_{2} \end{bmatrix} + \begin{bmatrix} \boldsymbol{\Delta}\_{f}^{p} \\ \boldsymbol{\Delta}\_{f}^{q} \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{f}^{b} \\ \boldsymbol{e}\_{f}^{b} \end{bmatrix} = \begin{bmatrix} \boldsymbol{e}\_{\boldsymbol{\rho}} \vert\_{\partial \mathcal{Z}} \\ -\boldsymbol{e}\_{\boldsymbol{\nu}} \vert\_{\partial \mathcal{Z}} \end{bmatrix}, \\\ \begin{bmatrix} -\boldsymbol{\mathcal{D}}\_{t} \\ -\boldsymbol{\mathcal{B}}\_{t} \end{bmatrix} = \begin{bmatrix} 0 - \boldsymbol{d} \\ \boldsymbol{d} & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{E} \\ \boldsymbol{H} \end{bmatrix} + \begin{bmatrix} \boldsymbol{J} \\ 0 \end{bmatrix} + \begin{bmatrix} \boldsymbol{\Delta}\_{e}^{p} \\ \boldsymbol{\Delta}\_{e}^{q} \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{\boldsymbol{\varepsilon}}^{b} \\ \boldsymbol{e}\_{\boldsymbol{\varepsilon}}^{b} \end{bmatrix} = \begin{bmatrix} \boldsymbol{E}|\_{\partial \mathcal{Z}} \\ \boldsymbol{H}|\_{\partial \mathcal{Z}} \end{bmatrix} \end{cases} \tag{39}$$

where each Δ*<sup>i</sup> <sup>j</sup>* for *i* ∈ {*p*, *q*} and *j* ∈ { *f* ,*e*} means a perturbation. Now, let us consider the subsystem of DPH systems, Δ*<sup>i</sup> j* (*u<sup>a</sup> <sup>I</sup>*), where *<sup>i</sup>* ∈ {*p*, *<sup>q</sup>*}, *<sup>j</sup>* ∈ { *<sup>f</sup>* ,*e*}, *<sup>u</sup><sup>a</sup>* for 1 <sup>≤</sup> *<sup>a</sup>* <sup>≤</sup> *<sup>l</sup>* is the function defined by the local coordinates *<sup>x</sup><sup>k</sup>* of *<sup>Y</sup>* for 1 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *<sup>n</sup>*, and we denote all possible derivatives up to the order *r* of *u<sup>a</sup>* by *u<sup>a</sup> <sup>I</sup>* and denote the order by 0 ≤ |*I*| ≤ *r*. For example, *ua <sup>I</sup>* for *<sup>r</sup>* <sup>=</sup> 2 means {*ua*, *<sup>u</sup><sup>a</sup> <sup>t</sup>* , *<sup>u</sup><sup>a</sup> <sup>y</sup>*, *u<sup>a</sup> <sup>z</sup>*, *u<sup>a</sup> tt*, *<sup>u</sup><sup>a</sup> ty*, *<sup>u</sup><sup>a</sup> tz*, *<sup>u</sup><sup>a</sup> yy*, *u<sup>a</sup> yz*, *u<sup>a</sup> zz*} for (*x*1, *<sup>x</sup>*2, *<sup>x</sup>*3)=(*t*, *<sup>y</sup>*, *<sup>z</sup>*), and the subscript means the partial derivative.

#### **3.2 Decomposition of model perturbations of DPH systems**

Consider the DPH system (39) of MHD with perturbations. We assume that the DPH system includes up to second-order derivatives: *r* = 2. Accordingly, Δ*<sup>i</sup> <sup>j</sup>* can be uniquely decomposed into

$$
\Delta^i\_j = d\varphi^i\_j + \gamma^i\_{j\prime} \tag{40}
$$

where *ϕ<sup>i</sup> <sup>j</sup>* is a *pseudo potential* derived from

$$
\gamma^{\dot{i}}\_{\dot{j}} \, d\mu = \Delta \, d\mu - d\tilde{\varphi}^{\dot{i}}\_{\dot{j}'} \quad \varphi^{\dot{i}}\_{\dot{j}} \, d\mu = \Delta \, d\mu - \gamma^{\dot{i}}\_{\dot{j}} \, d\mu,\tag{41}
$$

the temporal variable *ϕ*˜*<sup>i</sup> <sup>j</sup>* is calculated as

$$\mathfrak{d}\_{\dot{j}}^{i} = h\_{\upsilon}(\Delta\_{\dot{j}}^{i} du^{a}) = \int\_{0}^{1} u^{a} \cdot \Delta\_{\dot{j}}^{i} (\mathfrak{x}^{k}, \lambda u\_{I}^{a}) \, d\lambda,\tag{42}$$

*hv* is *the homotopy operator* for *<sup>ω</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) with respect to an equilibrium point *<sup>u</sup><sup>a</sup> cI*, called a *homotopy center*, defined by

$$\hbar\hbar\_{\mathbb{D}}(\omega) = \int\_0^1 \dot{\mathbf{u}}\_{\overline{\mathbb{V}}} \omega(\mathbf{x}, \lambda \ddot{\mathbf{u}}\_I) \, \lambda^{-1} d\lambda, \quad \ddot{\nu} = \sum\_{a,I} (u\_I^a - u\_{cI}^a) \frac{\partial}{\partial u\_I^a}, \tag{43}$$

where *u*¯*<sup>a</sup> <sup>I</sup>* <sup>=</sup> *<sup>u</sup><sup>a</sup> cI* <sup>+</sup> *<sup>λ</sup>*(*u<sup>a</sup> <sup>I</sup>* <sup>−</sup> *<sup>u</sup><sup>a</sup> cI*), and usually *<sup>u</sup><sup>a</sup> cI* <sup>=</sup> 0. In (40), we call *<sup>d</sup>ϕ<sup>i</sup> <sup>j</sup>* an *exact system* and call *γi <sup>j</sup>* a *dual exact system*, which corresponds to a distributed energy variable.

For example, let us consider Δ*<sup>i</sup> <sup>j</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *wt* <sup>+</sup> *wtt* for some *<sup>i</sup>* and *<sup>j</sup>*, where *<sup>u</sup>*<sup>1</sup> <sup>=</sup> *<sup>w</sup>* and *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *<sup>t</sup>*. The temporal variable

$$
\vec{\varphi}\_{\dot{j}}^{\dot{i}} = w + \frac{1}{2} w w\_t + \frac{1}{2} w w\_{tt} \tag{44}
$$

potential can be defined for a perturbation, the perturbation can be included in the variables *e<sup>p</sup>* or *e<sup>q</sup>* of the Stokes-Dirac structure. Hence, such a perturbation can be detected in terms of

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 181

*<sup>f</sup>*) ∧ *v<sup>t</sup>* − *Z*

*<sup>e</sup>* ) ∧ *B<sup>t</sup>* − *Z*

Moreover, from these relations, we can see that the exact subsystem of perturbations can be controlled by boundary port variables. Indeed, we can construct the boundary controls in the

− *∂Z*

+ *∂Z*

*<sup>f</sup>*) <sup>∧</sup> (*e<sup>ρ</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>q</sup>*

*<sup>e</sup>*) <sup>∧</sup> (*<sup>E</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>q</sup>*

*<sup>e</sup>* <sup>=</sup> *<sup>ϕ</sup><sup>p</sup>*

*ev* <sup>∧</sup> (*g*<sup>2</sup> <sup>+</sup> *<sup>γ</sup><sup>q</sup>*

*<sup>H</sup>* <sup>∧</sup> (*γ<sup>q</sup>*

*de* <sup>=</sup> <sup>−</sup>*γ<sup>p</sup>*

*<sup>f</sup>* , *<sup>u</sup><sup>q</sup>*

On the other hand, the decomposed perturbations corresponding dual exact subsystems cannot be eliminated by boundary controls. Hence, we should introduce the distributed controls in the second and third integrals of the power balance equations (51) and (52) as

*<sup>e</sup><sup>ρ</sup>* <sup>∧</sup> *<sup>γ</sup><sup>p</sup> f* − *Z*

(*ev* <sup>+</sup> *<sup>ϕ</sup><sup>p</sup>*

*<sup>E</sup>* <sup>∧</sup> (*<sup>J</sup>* <sup>+</sup> *<sup>γ</sup><sup>p</sup>*

(*<sup>H</sup>* <sup>−</sup> *<sup>ϕ</sup><sup>p</sup>*

*<sup>f</sup>* <sup>+</sup> *<sup>u</sup><sup>p</sup>*

*<sup>e</sup>* <sup>+</sup> *<sup>u</sup><sup>p</sup>*

*<sup>e</sup>* , *<sup>u</sup><sup>p</sup>*

*<sup>e</sup>* <sup>=</sup> <sup>−</sup>*ϕ<sup>q</sup>*

*<sup>f</sup>* <sup>+</sup> *<sup>u</sup><sup>q</sup>*

*<sup>e</sup>* <sup>+</sup> *<sup>u</sup><sup>q</sup>*

*<sup>e</sup>* , *<sup>u</sup><sup>q</sup>*

*de* <sup>=</sup> <sup>−</sup>*γ<sup>q</sup>*

*ev* <sup>∧</sup> (*g*<sup>2</sup> <sup>+</sup> *<sup>γ</sup><sup>q</sup>*

*<sup>H</sup>* <sup>∧</sup> *<sup>γ</sup><sup>q</sup> e*

*<sup>f</sup>*), (53)

*<sup>e</sup>* ), (54)

*<sup>e</sup>*. (55)

*d f*), (56)

*de*), (57)

*<sup>e</sup>*. (58)

*<sup>f</sup>*) <sup>∧</sup> (*e<sup>ρ</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>q</sup>*

*<sup>e</sup>* ) − *Z*

*<sup>e</sup>* ) <sup>∧</sup> (*<sup>E</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>q</sup>*

*f*)

*<sup>f</sup>*) = 0, (51)

*<sup>e</sup>*) = 0. (52)

the following boundary power balances:

<sup>−</sup>(*e<sup>ρ</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>q</sup>*

<sup>−</sup>(*<sup>E</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>q</sup>*

*<sup>f</sup>*) <sup>∧</sup> *<sup>ρ</sup><sup>t</sup>* <sup>−</sup> (*ev* <sup>+</sup> *<sup>ϕ</sup><sup>p</sup>*

*<sup>e</sup>*) <sup>∧</sup> *<sup>D</sup><sup>t</sup>* <sup>−</sup> (*<sup>H</sup>* <sup>−</sup> *<sup>ϕ</sup><sup>p</sup>*

fourth integrals of the power balance equations (51) and (52) as follows:

(*ev* <sup>+</sup> *<sup>ϕ</sup><sup>p</sup>*

(*<sup>H</sup>* <sup>−</sup> *<sup>ϕ</sup><sup>p</sup>*

*<sup>f</sup>* , *<sup>u</sup><sup>p</sup>*

*<sup>e</sup><sup>ρ</sup>* <sup>∧</sup> (*γ<sup>p</sup>*

*<sup>E</sup>* <sup>∧</sup> (*<sup>J</sup>* <sup>+</sup> *<sup>γ</sup><sup>p</sup>*

*<sup>f</sup>* , *<sup>u</sup><sup>q</sup>*

*<sup>f</sup>* <sup>+</sup> *<sup>u</sup><sup>p</sup> d f*) − *Z*

> *<sup>e</sup>* <sup>+</sup> *<sup>u</sup><sup>p</sup> de*) − *Z*

*d f* <sup>=</sup> <sup>−</sup>*γ<sup>q</sup>*

**4. Boundary observer for detecting topological symmetry breaking**

*dj* is the distributed input for eliminating dual exact subsystems such that

*<sup>f</sup>* , *<sup>u</sup><sup>p</sup>*

In this section, we first discuss the influence of topological variations in the system domains on the power balance equation of DPH systems. We can detect such changes by checking the boundary power balance of the original system; if there is an imbalance. According to Noether's theorem (Olver, 1993), conservation laws are associated with symmetries present in systems. That is, our purpose is to construct a boundary observer for detecting symmetry

*<sup>f</sup>* <sup>+</sup> *<sup>u</sup><sup>q</sup>*

*<sup>e</sup>* <sup>+</sup> *<sup>u</sup><sup>q</sup>*

*<sup>j</sup>* is the boundary input for compensating pseudo potentials such that

*<sup>f</sup>* <sup>=</sup> <sup>−</sup>*ϕ<sup>q</sup>*

 *∂Z*

> *∂Z*

*uq <sup>f</sup>* <sup>=</sup> <sup>−</sup>*ϕ<sup>p</sup>*

> − *Z*

> − *Z*

*d f* <sup>=</sup> <sup>−</sup>*γ<sup>p</sup>*

*up*

**4.1 Symmetry and power balance equations**

 *Z* 

 *Z* 

where *u<sup>i</sup>*

follows:

where *u<sup>i</sup>*

is derived from *hv*(Δ*<sup>i</sup> <sup>j</sup> dw*). Hence,

$$d\vec{\varphi}\_{\dot{\jmath}}^{\dot{i}} = (1 + w\_{l\ell}) \, dw, \quad \gamma\_{\dot{\jmath}}^{\dot{i}} = \Delta\_{\dot{\jmath}}^{\dot{i}} \, dw - d\vec{\varphi}\_{\dot{\jmath}}^{\dot{i}} = w\_{l} \, dw. \tag{45}$$

On the other hand, from the relation

$$\begin{split} \Delta\_{\rangle}^{l} dw &= \left(1 + w\_{l} + w\_{lt}\right) dw \\ &= \left(1 + \frac{1}{2} w\_{l} + w\_{lt}\right) dw + \left(-\frac{1}{2} w - w\_{t}\right) dw\_{l} \end{split} \tag{46}$$

that is transformed in terms of an integration by parts, we obtain

$$
\tilde{\varphi}\_{\dot{j}}^i = w - \frac{1}{2} w\_1^2. \tag{47}
$$

This result yields the same relation *dϕ*˜*<sup>i</sup> <sup>j</sup>* = (<sup>1</sup> <sup>+</sup> *wtt*) *dw*. Thus, the expression *<sup>ϕ</sup>*˜*<sup>i</sup> <sup>j</sup>* has variations generated by an integration by parts; therefore, we should recalculate *ϕ<sup>i</sup> <sup>j</sup>* as in (41).

#### **3.3 Necessary and sufficient condition of decomposition**

We can check whether a given Δ*<sup>i</sup> <sup>j</sup>* is an exact system or a dual exact system from the self-adjointness of the differential operator DΔ*<sup>i</sup> j* defining Δ*<sup>i</sup> j* : D<sup>∗</sup> Δ*i j* = DΔ*<sup>i</sup> j* (Olver, 1993, pp. 109, 307, 329 and 364). Here, *the Fréchet derivative* DF of a second-order subsystem F(*uI*) is an (*l* × *k*)-matrix with elements

$$(\mathcal{D}\_{\mathcal{F}})\_{ab}(h) = \left(\frac{\partial \mathcal{F}\_a}{\partial u^b} + \sum\_{i=0}^n \frac{\partial \mathcal{F}\_a}{\partial u^b\_{\mathbf{x}^i}} \frac{\partial}{\partial \mathbf{x}^i} + \sum\_{i=0}^n \sum\_{j=0}^n \frac{\partial \mathcal{F}\_a}{\partial u^b\_{\mathbf{x}^i \mathbf{x}^j}} \frac{\partial}{\partial \mathbf{x}^i} \frac{\partial}{\partial \mathbf{x}^j}\right) h \tag{48}$$

and the adjoint operator D<sup>∗</sup> <sup>F</sup> of DF is a (*<sup>k</sup>* × *<sup>l</sup>*)-matrix with elements

$$(\mathcal{D}\_{\mathcal{F}}^{\*})\_{ba}(h) = \frac{\partial \mathcal{F}\_{a}}{\partial u^{b}}h - \sum\_{i=0}^{n} \frac{\partial}{\partial \mathbf{x}^{i}} \left(\frac{\partial \mathcal{F}\_{a}}{\partial u^{b}\_{\mathbf{x}^{i}}}h\right) + \sum\_{i=0}^{n} \sum\_{j=0}^{n} \frac{\partial}{\partial \mathbf{x}^{i}} \frac{\partial}{\partial \mathbf{x}^{j}} \left(\frac{\partial \mathcal{F}\_{a}}{\partial u^{b}\_{\mathbf{x}^{i} \mathbf{x}^{j}}}h\right) \tag{49}$$

for *a* = 1, ··· , *k* and *b* = 1, ··· , *l*, where *h* = *h*(*uI*) is any function and we assume *k* = *l*. For example, consider Δ*<sup>q</sup> <sup>f</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>ν</sup><sup>v</sup>* <sup>+</sup> *<sup>v</sup><sup>t</sup>* in (39), where *<sup>u</sup>*<sup>1</sup> <sup>=</sup> *<sup>w</sup>*, *wt* <sup>=</sup> *<sup>v</sup>* and *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *<sup>t</sup>*. Then, *ϕq <sup>f</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>v</sup><sup>t</sup>* and *<sup>γ</sup><sup>q</sup> <sup>f</sup>* = *νv*, because *g* = *νv* is non-self-adjoint: D<sup>∗</sup> *<sup>g</sup>* �= D*g*, and we have used (48) and (49) with *a* = *b* = 1, i.e.,

$$\mathcal{D}\_{\mathcal{S}}(h) = \frac{\partial \mathcal{g}}{\partial u\_{\mathcal{X}^0}} \frac{\partial}{\partial \mathbf{x}^0}(h) = \nu \frac{\partial h}{\partial t}, \quad \mathcal{D}\_{\mathcal{S}}^\*(h) = -\frac{\partial}{\partial \mathbf{x}^0} \left(\frac{\partial \mathcal{g}}{\partial u\_{\mathcal{X}^0}} h\right) = -\nu \frac{\partial h}{\partial t}.\tag{50}$$

#### **3.4 Elimination of decomposed perturbations**

The uniqueness of the decomposition is determined by the topology of *Y*. That is, differential *k*-forms for *k* ≥ 1 defined on such a domain can be always described as in (40). If a pseudo 14 Magnetohydrodynamics

*<sup>j</sup>* <sup>=</sup> <sup>Δ</sup>*<sup>i</sup>*

 *dw* + −1 2 *w* − *wt*

*<sup>j</sup>* <sup>=</sup> *<sup>w</sup>* <sup>−</sup> <sup>1</sup> 2 *w*2

*j*

*∂ <sup>∂</sup>x<sup>i</sup>* <sup>+</sup>

<sup>F</sup> of DF is a (*<sup>k</sup>* × *<sup>l</sup>*)-matrix with elements

109, 307, 329 and 364). Here, *the Fréchet derivative* DF of a second-order subsystem F(*uI*) is

*∂*F*<sup>a</sup> ∂u<sup>b</sup> xi*

*n* ∑ *i*=0

*∂ ∂x<sup>i</sup>*  *∂*F*<sup>a</sup> ∂u<sup>b</sup> xi h* + *n* ∑ *i*=0

for *a* = 1, ··· , *k* and *b* = 1, ··· , *l*, where *h* = *h*(*uI*) is any function and we assume *k* = *l*.

*<sup>f</sup>* = *νv*, because *g* = *νv* is non-self-adjoint: D<sup>∗</sup>

The uniqueness of the decomposition is determined by the topology of *Y*. That is, differential *k*-forms for *k* ≥ 1 defined on such a domain can be always described as in (40). If a pseudo

*∂h ∂t* , D<sup>∗</sup> *<sup>j</sup> dw* <sup>−</sup> *<sup>d</sup>ϕ*˜*<sup>i</sup>*

*<sup>j</sup>* = (<sup>1</sup> <sup>+</sup> *wtt*) *dw*. Thus, the expression *<sup>ϕ</sup>*˜*<sup>i</sup>*

defining Δ*<sup>i</sup>*

*n* ∑ *i*=0

*n* ∑ *j*=0

> *n* ∑ *j*=0

*<sup>f</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>ν</sup><sup>v</sup>* <sup>+</sup> *<sup>v</sup><sup>t</sup>* in (39), where *<sup>u</sup>*<sup>1</sup> <sup>=</sup> *<sup>w</sup>*, *wt* <sup>=</sup> *<sup>v</sup>* and *<sup>x</sup>*<sup>0</sup> <sup>=</sup> *<sup>t</sup>*. Then,

*∂x*<sup>0</sup>

*<sup>g</sup>*(*h*) = <sup>−</sup> *<sup>∂</sup>*

*<sup>j</sup>* is an exact system or a dual exact system from the

*∂*F*<sup>a</sup> ∂u<sup>b</sup> xixj*

> *∂ ∂x<sup>i</sup>*

 *∂g ∂ux*<sup>0</sup> *h* 

*∂ ∂x<sup>j</sup>*  *∂*F*<sup>a</sup> ∂u<sup>b</sup> xixj h* 

*∂ ∂x<sup>i</sup>*

*j* : D<sup>∗</sup> Δ*i j*

*<sup>j</sup>* = *wt dw*. (45)

*dwt* (46)

*<sup>j</sup>* has variations

(Olver, 1993, pp.

*h* (48)

*<sup>g</sup>* �= D*g*, and we have used

. (50)

= −*ν ∂h ∂t*

(49)

*<sup>t</sup>* . (47)

= DΔ*<sup>i</sup> j*

> *∂ ∂x<sup>j</sup>*

*<sup>j</sup>* as in (41).

is derived from *hv*(Δ*<sup>i</sup>*

*<sup>j</sup> dw*). Hence,

= 1 + 1 2

**3.3 Necessary and sufficient condition of decomposition**

 *∂*F*<sup>a</sup> <sup>∂</sup>u<sup>b</sup>* <sup>+</sup>

*<sup>∂</sup>u<sup>b</sup> <sup>h</sup>* <sup>−</sup>

*n* ∑ *i*=0

self-adjointness of the differential operator DΔ*<sup>i</sup>*

(DF )*ab*(*h*) =

<sup>F</sup> )*ba*(*h*) = *<sup>∂</sup>*F*<sup>a</sup>*

*<sup>j</sup>* = (<sup>1</sup> <sup>+</sup> *wtt*) *dw*, *<sup>γ</sup><sup>i</sup>*

*<sup>j</sup> dw* = (1 + *wt* + *wtt*) *dw*

that is transformed in terms of an integration by parts, we obtain

*wt* + *wtt*

*ϕ*˜*i*

generated by an integration by parts; therefore, we should recalculate *ϕ<sup>i</sup>*

*dϕ*˜*<sup>i</sup>*

Δ*i*

On the other hand, from the relation

This result yields the same relation *dϕ*˜*<sup>i</sup>*

We can check whether a given Δ*<sup>i</sup>*

an (*l* × *k*)-matrix with elements

and the adjoint operator D<sup>∗</sup>

(D<sup>∗</sup>

For example, consider Δ*<sup>q</sup>*

(48) and (49) with *a* = *b* = 1, i.e.,

<sup>D</sup>*g*(*h*) = *<sup>∂</sup><sup>g</sup>*

*∂ux*<sup>0</sup>

**3.4 Elimination of decomposed perturbations**

*∂ <sup>∂</sup>x*<sup>0</sup> (*h*) = *<sup>ν</sup>*

*<sup>f</sup>* <sup>=</sup> <sup>1</sup> <sup>+</sup> *<sup>v</sup><sup>t</sup>* and *<sup>γ</sup><sup>q</sup>*

*ϕq*

potential can be defined for a perturbation, the perturbation can be included in the variables *e<sup>p</sup>* or *e<sup>q</sup>* of the Stokes-Dirac structure. Hence, such a perturbation can be detected in terms of the following boundary power balances:

$$\begin{split} \int\_{Z} \left( - (\mathbf{e}\_{\rho} + \boldsymbol{\varphi}\_{f}^{q}) \wedge \boldsymbol{\rho}\_{t} - (\mathbf{e}\_{v} + \boldsymbol{\varphi}\_{f}^{p}) \wedge \boldsymbol{\nu}\_{t} \right) - \int\_{Z} \boldsymbol{\varepsilon}\_{\rho} \wedge \boldsymbol{\gamma}\_{f}^{p} - \int\_{Z} \boldsymbol{\varepsilon}\_{v} \wedge (\boldsymbol{\varrho}\_{2} + \boldsymbol{\gamma}\_{f}^{q}) \\ - \int\_{\partial Z} (\boldsymbol{\varepsilon}\_{v} + \boldsymbol{\varphi}\_{f}^{p}) \wedge (\boldsymbol{\varepsilon}\_{\rho} + \boldsymbol{\varphi}\_{f}^{q}) = 0, \end{split} \tag{51}$$

$$\begin{split} \int\_{Z} \left\{ - \left( \mathbf{E} + \boldsymbol{\varphi}\_{\varepsilon}^{q} \right) \wedge \mathbf{D}\_{t} - \left( \mathbf{H} - \boldsymbol{\varphi}\_{\varepsilon}^{p} \right) \wedge \mathbf{B}\_{t} \right\} - \int\_{Z} \mathbf{E} \wedge \left( \mathbf{J} + \boldsymbol{\gamma}\_{\varepsilon}^{p} \right) - \int\_{Z} \mathbf{H} \wedge \boldsymbol{\gamma}\_{\varepsilon}^{q} \\ + \int\_{\partial Z} \left( \mathbf{H} - \boldsymbol{\varphi}\_{\varepsilon}^{p} \right) \wedge \left( \mathbf{E} + \boldsymbol{\varphi}\_{\varepsilon}^{q} \right) = 0. \end{split} \tag{52}$$

Moreover, from these relations, we can see that the exact subsystem of perturbations can be controlled by boundary port variables. Indeed, we can construct the boundary controls in the fourth integrals of the power balance equations (51) and (52) as follows:

$$\int\_{\partial Z} (e\_{\upsilon} + \varrho\_f^p + \mathfrak{u}\_f^q) \wedge (e\_{\theta} + \mathfrak{q}\_f^q + \mathfrak{u}\_f^p),\tag{53}$$

$$\int\_{\partial Z} (\mathbf{H} - \boldsymbol{\varrho}\_{\varepsilon}^{p} + \boldsymbol{u}\_{\varepsilon}^{q}) \wedge (\mathbf{E} + \boldsymbol{\varrho}\_{\varepsilon}^{q} + \boldsymbol{u}\_{\varepsilon}^{p}),\tag{54}$$

where *u<sup>i</sup> <sup>j</sup>* is the boundary input for compensating pseudo potentials such that

$$\mathbf{u}^{q}\_{f} = -\boldsymbol{\varrho}^{p}\_{f'} \quad \mathbf{u}^{p}\_{f} = -\boldsymbol{\varrho}^{q}\_{f'} \quad \mathbf{u}^{q}\_{\varepsilon} = \boldsymbol{\varrho}^{p}\_{\varepsilon} \quad \mathbf{u}^{p}\_{\varepsilon} = -\boldsymbol{\varrho}^{q}\_{\varepsilon}. \tag{55}$$

On the other hand, the decomposed perturbations corresponding dual exact subsystems cannot be eliminated by boundary controls. Hence, we should introduce the distributed controls in the second and third integrals of the power balance equations (51) and (52) as follows:

$$-\int\_{Z} \mathfrak{e}\_{\rho} \wedge (\gamma\_f^p + \mathfrak{u}\_{df}^p) - \int\_{Z} \mathfrak{e}\_{\upsilon} \wedge (\mathfrak{g}\_2 + \gamma\_f^q + \mathfrak{u}\_{df}^q),\tag{56}$$

$$-\int\_{Z} \mathbf{E} \wedge (\mathbf{J} + \gamma\_{\varepsilon}^{p} + \boldsymbol{u}\_{d\varepsilon}^{p}) - \int\_{Z} \mathbf{H} \wedge (\gamma\_{\varepsilon}^{q} + \boldsymbol{u}\_{d\varepsilon}^{q})\_{\prime} \tag{57}$$

where *u<sup>i</sup> dj* is the distributed input for eliminating dual exact subsystems such that

$$\mathbf{u}\_{df}^p = -\gamma\_{f'}^p \quad \mathbf{u}\_{df}^q = -\gamma\_{f'}^q \quad \mathbf{u}\_{de}^p = -\gamma\_e^p \quad \mathbf{u}\_{de}^q = -\gamma\_e^q. \tag{58}$$

#### **4. Boundary observer for detecting topological symmetry breaking**

#### **4.1 Symmetry and power balance equations**

In this section, we first discuss the influence of topological variations in the system domains on the power balance equation of DPH systems. We can detect such changes by checking the boundary power balance of the original system; if there is an imbalance. According to Noether's theorem (Olver, 1993), conservation laws are associated with symmetries present in systems. That is, our purpose is to construct a boundary observer for detecting symmetry

where � means topological equivalence (i.e., homeomorphic), \ means subtraction of sets,

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 183

This section discusses the relation between the topology of the domain *Z* of DPH systems and the decomposable components of vector fields on *Z*. After this discussion, the symmetry breaking explained in the previous section will be extended to a change in energy flows of

In *Section 2*, we assumed that the system domain *Z* is a subdomain of a manifold that is topologically the same as a Euclidian space. Actually, this assumption restricted the form of diffrential forms. In this case, differential *k*-forms for *k* ≥ 1 can be decomposed into two types, i.e., an exact form and a dual exact form as in (40). That is, differential forms *<sup>ω</sup><sup>e</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) are called *exact forms* if there exists some *<sup>η</sup>* <sup>∈</sup> *<sup>Ω</sup>k*−1(*Z*) such that *<sup>ω</sup><sup>e</sup>* <sup>=</sup> *<sup>d</sup>η*, i.e., *<sup>d</sup>ω<sup>e</sup>* <sup>=</sup> *<sup>d</sup>*(*dη*) = <sup>0</sup> because of the nature of exterior differentiation. The forms *<sup>ω</sup><sup>d</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) such that *<sup>d</sup>ω<sup>d</sup>* �<sup>=</sup> <sup>0</sup> are called *dual exact forms*. In general, there might also exist *harmonic forms <sup>ω</sup><sup>h</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) satisfying �*ω<sup>h</sup>* <sup>=</sup> 0, where � <sup>=</sup> *dd*† <sup>+</sup> *<sup>d</sup>*†*<sup>d</sup>* is the Laplacian and *<sup>d</sup>*† = (−1)*n*(*k*+1)+<sup>1</sup> <sup>∗</sup>*d*<sup>∗</sup> is the adjoint operator of exterior differentiation. The components of differential forms depend on the topology of domains. All classifications of differential forms defined on a compact domain with a smooth boundary are given by *the Hodge decomposition theorem* (Morita, 2001); i.e., an arbitrary differential form on an oriented compact Riemannian manifold can be uniquely

Moreover, a unique harmonic form on an oriented compact Riemannian manifold corresponds to a topological quantity of the manifolds called a *homology*. Precisely speaking, from *Hodge theorem*, *Poincaré duality thorem* and the duality between homology and (de Rham)

Kotiuga, 2004, pp. 102), where *Hk*(*Z*) is the vector space with real coefficients of the *k*-th

• *H*0(*Z*) ··· The vector space is generated by such equivalence classes of points in *Z* as two points are equivalent if they can be connected by a path in *Z*. dim *H*0(*Z*) is the number of components of *Z*. Note that *H*0(*Z*) ∼= **R** for a connected *Z* and the element of *H*0(*Z*) is a

• *H*1(*Z*) ··· The vector space is generated by such equivalence classes of oriented loops in *Z* as two loops are equivalent if their difference is the boundary of an oriented surface in *Z*. The number of holes of closed surfaces is called a *genus*. dim *H*1(*Z*) is the number of total

• *H*2(*Z*) ··· The vector space is generated by such equivalence surfaces of points in *Z* as two surfaces are equivalent if their difference is the boundary of some oriented subregion of *Z*. dim *H*2(*Z*) is the number of the difference between components of *∂Z* and those of *Z*.

*<sup>h</sup>*(*Z*) is the space of harmonic forms.

*<sup>ω</sup>* <sup>=</sup> *<sup>ω</sup><sup>e</sup>* <sup>+</sup> *<sup>ω</sup><sup>d</sup>* <sup>+</sup> *<sup>ω</sup><sup>h</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*). (63)

*<sup>h</sup>* (*Z*) (Morita, 2001; Gross and

**4.2 Topological decomposition of differential forms and vector fields**

decomposed into an exact form, a dual exact form, and a harmonic form:

cohomology, we obtain the isomorphism *Hk*(*Z*, *∂Z*) ∼= *Ωn*−*<sup>k</sup>*

If *n* = 3, the homology of *Z* consists of the following vector spaces:

DPH systems defined on compact manifolds.

and {0} is a point.

homology of *Z*, and *Ω<sup>k</sup>*

constant function.

• *H*3(*Z*) ··· dim *H*3(*Z*) is always 0.

genus of *Z*.

breaking (Nishida et al., 2009). Finally, we derive a boundary control for creating desired energy flows from topological properties of manifolds.

We shall clarify the first problem by means of the following example. Consider a DPH system defined on a 2-dimensional domain *Z*. We assume that the energy flow of the system can be split along the *x*- and *y*-axis. Next, we divide the domain *Z* into subdomains, i.e., *Z* = *i Zi* , where *Z<sup>i</sup>* is the *i*-th subdomain of *Z*. We denote the common boundary between *Z<sup>i</sup>* and *Z<sup>j</sup>* by *∂Zij*. The following power balance holds:

$$\mathcal{H}\_{\delta t} = \int\_{\partial Z^{ij}} \sum\_{i,j} \left( e^{b\mathbf{i}} \wedge f^{b\mathbf{i}} - e^{bj} \wedge f^{bj} \right) = 0,\tag{59}$$

where *ebi* and *f bi* are the boundary port variables defined on *∂Z<sup>i</sup>* . The DPH system can be regarded as a connected structure of DPH systems defined on *Z<sup>i</sup>* in terms of boundary port variables of *∂Zij*. We shall simplify the shapes of *Z* and each *Z<sup>i</sup>* to be squares as in the left diagram below:

Accordingly, we can split the original boundary *∂Z* and denote the boundaries with respect to the *x*- and *y*-axis by *∂Zx* and *∂Zy*, respectively. Hence, the following power balance holds:

$$\mathcal{H}\_{\delta t} = \mathcal{H}\_{\delta t}|\_{\partial \mathbb{Z}\_x} + \mathcal{H}\_{\delta t}|\_{\partial \mathbb{Z}\_y} = 0. \tag{61}$$

Now, let us assume that a structural change occurs in the inner part of *Z* on a segment along *x*-axis that we denote as *∂Z*� *<sup>y</sup>* in the right diagram of (60). Such changes are caused by, for instance, energy dissipations, or energy transformations to other physical systems, and they can be illustrated as a new element connected to *∂Z*� *<sup>y</sup>* in the bond graph. This means the energy preserving symmetry is broken along the *x*-axis. In this case, (61) should be revised to

$$
\mathcal{H}'\_{\delta t} = \mathcal{H}\_{\delta t}|\_{\partial Z\_x} + \mathcal{H}\_{\delta t}|\_{\partial Z'\_y} + \mathcal{H}\_{\delta t}|\_{\partial Z\_y} = 0. \tag{62}
$$

Hence, we can detect that the power on *∂Zy*: H*δt*|*∂Zy* = 0 becomes imbalanced if the port variables in (61) are observable. In other words, this change can be regarded as a change in the topology of the system domain, i.e., a deformation from *<sup>Z</sup>* � **<sup>R</sup>***<sup>n</sup>* to *<sup>Z</sup>* \ *<sup>∂</sup>Z*� *<sup>y</sup>* � **<sup>R</sup>***<sup>n</sup>* \ {0}, where � means topological equivalence (i.e., homeomorphic), \ means subtraction of sets, and {0} is a point.

### **4.2 Topological decomposition of differential forms and vector fields**

16 Magnetohydrodynamics

breaking (Nishida et al., 2009). Finally, we derive a boundary control for creating desired

We shall clarify the first problem by means of the following example. Consider a DPH system defined on a 2-dimensional domain *Z*. We assume that the energy flow of the system can be split along the *x*- and *y*-axis. Next, we divide the domain *Z* into subdomains, i.e., *Z* =

where *Z<sup>i</sup>* is the *i*-th subdomain of *Z*. We denote the common boundary between *Z<sup>i</sup>* and *Z<sup>j</sup>* by

*bi* <sup>∧</sup> *<sup>f</sup> bi* <sup>−</sup> *<sup>e</sup>*

regarded as a connected structure of DPH systems defined on *Z<sup>i</sup>* in terms of boundary port variables of *∂Zij*. We shall simplify the shapes of *Z* and each *Z<sup>i</sup>* to be squares as in the left

*bj* <sup>∧</sup> *<sup>f</sup> bj*

✤ ❴❴ ✤

*DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

❴❴ ❴✤

*eb e f b e* ❴✤ ✤ *DTFeb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*∂Z y* ✤

H*δ<sup>t</sup>* = H*δt*|*∂Zx* + H*δt*|*∂Zy* = 0. (61)

*<sup>y</sup>* in the right diagram of (60). Such changes are caused by, for

❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

❴✤ ✤

> ✤ *DTFeb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*∂Zx*

Accordingly, we can split the original boundary *∂Z* and denote the boundaries with respect to the *x*- and *y*-axis by *∂Zx* and *∂Zy*, respectively. Hence, the following power balance holds:

Now, let us assume that a structural change occurs in the inner part of *Z* on a segment along

instance, energy dissipations, or energy transformations to other physical systems, and they

Hence, we can detect that the power on *∂Zy*: H*δt*|*∂Zy* = 0 becomes imbalanced if the port variables in (61) are observable. In other words, this change can be regarded as a change in

preserving symmetry is broken along the *x*-axis. In this case, (61) should be revised to

*<sup>δ</sup><sup>t</sup>* = H*δt*|*∂Zx* + H*δt*|*∂Z*�

the topology of the system domain, i.e., a deformation from *<sup>Z</sup>* � **<sup>R</sup>***<sup>n</sup>* to *<sup>Z</sup>* \ *<sup>∂</sup>Z*�

*i Zi* ,

(60)

= 0, (59)

❴❴ ✤

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTFeb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*∂Zy*

*<sup>y</sup>* in the bond graph. This means the energy

*<sup>y</sup>* + H*δt*|*∂Zy* = 0. (62)

*<sup>y</sup>* � **<sup>R</sup>***<sup>n</sup>* \ {0},

*∂Zy*

❴❴

*eb e f b e* ❴✤ ✤

*eb e f b e* ❴✤ ✤ *∂Zx*

*eb e f b e* ❴✤ ✤

. The DPH system can be

energy flows from topological properties of manifolds.

H*δ<sup>t</sup>* =

❴❴ ✤

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTFeb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*∂Zy*

can be illustrated as a new element connected to *∂Z*�

H�

*∂Zy*

❴❴

*eb e f b e* ❴✤ ✤

*eb e f b e* ❴✤ ✤ *∂Zx* →

*eb e f b e* ❴✤ ✤

 *<sup>∂</sup>Zij* ∑ *i*,*j e*

where *ebi* and *f bi* are the boundary port variables defined on *∂Z<sup>i</sup>*

*∂Zij*. The following power balance holds:

diagram below:

❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

❴✤ ✤

❴✤ ✤ *DTFeb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*∂Zx*

✤ ❴❴ ✤

*DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTF eb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*eb e f b e* ❴✤ ✤ *DTFeb <sup>e</sup> f <sup>b</sup> e* ✤ ❴❴

*x*-axis that we denote as *∂Z*�

This section discusses the relation between the topology of the domain *Z* of DPH systems and the decomposable components of vector fields on *Z*. After this discussion, the symmetry breaking explained in the previous section will be extended to a change in energy flows of DPH systems defined on compact manifolds.

In *Section 2*, we assumed that the system domain *Z* is a subdomain of a manifold that is topologically the same as a Euclidian space. Actually, this assumption restricted the form of diffrential forms. In this case, differential *k*-forms for *k* ≥ 1 can be decomposed into two types, i.e., an exact form and a dual exact form as in (40). That is, differential forms *<sup>ω</sup><sup>e</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) are called *exact forms* if there exists some *<sup>η</sup>* <sup>∈</sup> *<sup>Ω</sup>k*−1(*Z*) such that *<sup>ω</sup><sup>e</sup>* <sup>=</sup> *<sup>d</sup>η*, i.e., *<sup>d</sup>ω<sup>e</sup>* <sup>=</sup> *<sup>d</sup>*(*dη*) = <sup>0</sup> because of the nature of exterior differentiation. The forms *<sup>ω</sup><sup>d</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) such that *<sup>d</sup>ω<sup>d</sup>* �<sup>=</sup> <sup>0</sup> are called *dual exact forms*. In general, there might also exist *harmonic forms <sup>ω</sup><sup>h</sup>* <sup>∈</sup> *<sup>Ω</sup>k*(*Z*) satisfying �*ω<sup>h</sup>* <sup>=</sup> 0, where � <sup>=</sup> *dd*† <sup>+</sup> *<sup>d</sup>*†*<sup>d</sup>* is the Laplacian and *<sup>d</sup>*† = (−1)*n*(*k*+1)+<sup>1</sup> <sup>∗</sup>*d*<sup>∗</sup> is the adjoint operator of exterior differentiation. The components of differential forms depend on the topology of domains. All classifications of differential forms defined on a compact domain with a smooth boundary are given by *the Hodge decomposition theorem* (Morita, 2001); i.e., an arbitrary differential form on an oriented compact Riemannian manifold can be uniquely decomposed into an exact form, a dual exact form, and a harmonic form:

$$
\omega = \omega\_{\ell} + \omega\_{d} + \omega\_{\mathbb{h}} \in \Omega^{k}(\mathbb{Z}).\tag{63}
$$

Moreover, a unique harmonic form on an oriented compact Riemannian manifold corresponds to a topological quantity of the manifolds called a *homology*. Precisely speaking, from *Hodge theorem*, *Poincaré duality thorem* and the duality between homology and (de Rham) cohomology, we obtain the isomorphism *Hk*(*Z*, *∂Z*) ∼= *Ωn*−*<sup>k</sup> <sup>h</sup>* (*Z*) (Morita, 2001; Gross and Kotiuga, 2004, pp. 102), where *Hk*(*Z*) is the vector space with real coefficients of the *k*-th homology of *Z*, and *Ω<sup>k</sup> <sup>h</sup>*(*Z*) is the space of harmonic forms.

If *n* = 3, the homology of *Z* consists of the following vector spaces:


<sup>X</sup>*HG*(*Z*) = �

<sup>X</sup>*GG*(*Z*) = �

*that is locally constant.*

the small sphere.

as follows: ⎧ ⎪⎪⎪⎪⎪⎨

� −*ρ<sup>t</sup>* −*v<sup>t</sup>*

� −*D<sup>t</sup>* −*B<sup>t</sup>*

� = � 0 *d d* 0

� = � 0 −*d d* 0

called a *Poincaré dual*: *Ω<sup>k</sup>*

� (*f p h f* ,*e p*

> (*f p he*,*e p*

Note that *Hk*(*Z*) <sup>∼</sup><sup>=</sup> *Hn*−*k*(*Z*, *<sup>∂</sup>Z*) <sup>∼</sup><sup>=</sup> *<sup>Ω</sup><sup>k</sup>*

⎪⎪⎪⎪⎪⎩

**4.3 DPH systems with harmonic energy flows**

� ⎡ ⎣ *e<sup>ρ</sup>* + *e p h f*

*ev* + *e q h f*

� � *E* + *e*

*H* + *e q he*

⎤ ⎦ −

*p he*

*<sup>h</sup>*(*Z*) <sup>∼</sup><sup>=</sup> *<sup>Ω</sup>n*−*<sup>k</sup>*

� + � *J* 0

� 0

*g*<sup>1</sup> + *g*<sup>2</sup>

� + � *f p he f q he*

where we defined the following harmonic forms yielding harmonic energy flows:

*<sup>h</sup>* (*Z*), and *f*

*h f*) <sup>∈</sup> *<sup>Ω</sup>*3(*Z*) <sup>×</sup> *<sup>Ω</sup>*0(*Z*), (*<sup>f</sup>*

*he*) <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) <sup>×</sup> *<sup>Ω</sup>*1(*Z*), (*<sup>f</sup>*

originating from topological shapes of manifolds.

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): *<sup>v</sup>* <sup>=</sup> *<sup>d</sup>ϕ*, <sup>∗</sup>*d*∗*<sup>v</sup>* <sup>=</sup> 0, *<sup>ϕ</sup>* <sup>=</sup> *<sup>C</sup>*

dim *H*1(*Z*) = dim X*HK*(*Z*), dim *H*2(*Z*) = dim X*HG*(*Z*). (72)

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): *<sup>v</sup>* <sup>=</sup> *<sup>d</sup>ϕ*, *<sup>ϕ</sup>*|*∂<sup>Z</sup>* <sup>=</sup> <sup>0</sup>

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 185

*which are respectively called* fluxless knots*,* harmonic knots*,* curly gradients*,* harmonic gradients *and* grounded gradients*, and m means all unit vector fields normal to W, and C is a function on ∂Z*

For example, consider a vector field defined on a three-dimensional disc. There is no *<sup>v</sup>* <sup>∈</sup> X*HK*(*Z*) on the disc, because the genus is 0 and dim *H*1(*Z*) = dim X*HK*(*Z*) = 0. Thus, all rotation vector fields on the disc are *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*FK*(*Z*) that is the rotating vector field whose axis is an inner point of the disc. *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*CG*(*Z*) is a constant vector field flowing across the disc; therefore, it is divergence-free and zero flux through the one and only component of *<sup>∂</sup>Z*. *<sup>v</sup>* <sup>∈</sup> X*GG*(*Z*) is a radiational vector field flowing from an inner point of the disc, where the potential *<sup>ϕ</sup>* is constant on *<sup>∂</sup>Z*. There is no *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*HG*(*Z*) on the disc, because the numbers of components of *∂Z* and *Z* are each 1, i.e., dim X*HG*(*Z*) = 0. However, a three-dimensional solid torus has a hole; therefore, dim <sup>X</sup>*HK*(*Z*) = 1, but dim <sup>X</sup>*HG*(*Z*) = 0. *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*HK*(*Z*) is a circulative vector field flowing around the hole. Moreover, for a region between two concentric round spheres, dim <sup>X</sup>*HG*(*Z*) = 1. *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*HK*(*Z*) is a radiational vector field flowing from a common center in

In this section, we extend the DPH system of MHD to include the global energy flows

Let *Z* be a three-dimensional smooth Riemannian submanifold of *Y* with a smooth boundary *∂Z*. The DPH system (18) of MHD defined on a domain *Z* is extended to have energy flows regarding harmonic knots and harmonic gradients that we call *harmonic energy flows*

> � + ⎡ ⎣ *f p h f f q h f*

> > � , � *f b e eb e*

> > > *q h f* ,*e q*

*q he*,*e q*

*p h f* and *e*

⎤ ⎦ , ⎡ ⎣ *f b f eb f*

� = � (*E* + *e p he*)|*∂<sup>Z</sup>*

*p*

⎤ ⎦ = ⎡ ⎣

(*H* + *e q he*)|*∂<sup>Z</sup>*

*h f*) <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) <sup>×</sup> *<sup>Ω</sup>*1(*Z*),

*<sup>h</sup>*(*Z*), there is the dual from of *ω<sup>h</sup>* with respect to � , �*Z*,

(*e<sup>ρ</sup>* + *e p h f*)|*∂<sup>Z</sup>*

−(*ev* + *e*

*he*) <sup>∈</sup> *<sup>Ω</sup>*2(*Z*) <sup>×</sup> *<sup>Ω</sup>*1(*Z*). (74)

*h f* are constant functions. The system

*q h f*)|*∂<sup>Z</sup>*

� , ⎤ ⎦ ,

(73)

�

�

, (70)

(71)

On the other hand, the dual space of *Hk*(*Z*) is *Hn*−*k*(*Z*, *<sup>∂</sup>Z*), where *Hk*(*Z*, *<sup>∂</sup>Z*) is called *the k-th relative homology of Z modulo ∂Z*. In *n* = 3, the relative homology of *Z* modulo *∂Z* consists of the following vector spaces with real coefficients:


Hence, *Hk*(*Z*, *<sup>∂</sup>Z*) ∼= *<sup>H</sup>*3−*k*(*Z*) for 0 ≤ *<sup>k</sup>* ≤ 3.

As we mentioned before, the space of vector fields X can be identified with that of 1-forms *Ω*<sup>1</sup> in the sense of a Riemannian metric. Thus, vector fields are affected by the decomposition of differential forms. Indeed, the space of vector fields on a compact domain *Z* in three-dimensional space with a smooth boundary can be decomposed as follows.

**Theorem 4.1** (Cantarella et al. (2002))**.** *Consider vector fields v* <sup>∈</sup> <sup>X</sup>(*Z*) *on a compact domain Z with a smooth boundary ∂Z in three-dimensional space. Let W denote any smooth orientable surface in Z whose boundary ∂W lies on the boundary ∂Z: W* ⊂ *Z and ∂W* ⊂ *∂Z, and called it a* cross-sectional surface*. The space* X(*Z*) *is the direct sum of five mutually orthogonal subspaces:*

$$\mathfrak{X}(Z) = \mathfrak{X}\_{\mathcal{K}}(Z) \oplus \mathfrak{X}\_{\mathcal{G}}(Z), \tag{64}$$

*where v* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*)*, v* <sup>∈</sup> <sup>X</sup>(*Z*)*, <sup>ϕ</sup>* <sup>∈</sup> *<sup>Ω</sup>*0(*Z*)*,*

$$\mathfrak{X}\_{\mathbb{K}}(Z) = \left\{ \upsilon^{\sharp} \in \mathfrak{X}(Z) \colon \*d \* \upsilon = 0, \ \upsilon^{\sharp} \cdot \mathfrak{n}^{\sharp} = 0 \right\}, \quad \mathfrak{X}\_{\mathbb{G}}(Z) = \left\{ \upsilon^{\sharp} \in \mathfrak{X}(Z) \colon \upsilon = d\rho \right\}, \tag{65}$$

*which are called* knots *and* gradients*, respectively, and n means all unit vector fields normal to ∂Z. Furthermore,*

$$\mathfrak{X}\_{\mathcal{K}}(Z) = \mathfrak{X}\_{\mathcal{FK}}(Z) \oplus \mathfrak{X}\_{\text{HK}}(Z), \quad \mathfrak{X}\_{\mathcal{G}}(Z) = \mathfrak{X}\_{\mathcal{CG}}(Z) \oplus \mathfrak{X}\_{\text{HG}}(Z) \oplus \mathfrak{X}\_{\text{GG}}(Z), \tag{66}$$

*where*

$$\mathcal{X}\_{FK}(Z) = \left\{ v^\sharp \in \mathfrak{X}(Z) \colon \*d \ast v = 0, \ \langle v, \mathfrak{n} \rangle\_{\partial Z} = 0, \ \langle v, \mathfrak{m} \rangle\_{W} = 0 \right\},\tag{67}$$

$$\mathcal{X}\_{HK}(Z) = \left\{ v^\dagger \in \mathfrak{X}(Z) \colon \*d \ast v = 0, \ \langle v, \mathfrak{n} \rangle\_{\partial Z} = 0, \ dv = 0 \right\},\tag{68}$$

$$\mathfrak{X}\_{\mathbb{C}G}(Z) = \left\{ v^{\sharp} \in \mathfrak{X}(Z) \colon v = d\varphi, \; \*d \ast v = 0, \; \langle v, \mathfrak{n} \rangle\_{\partial \mathbb{Z}} = 0 \right\},\tag{69}$$

$$\mathfrak{X}\_{HG}(Z) = \left\{ \upsilon^{\sharp} \in \mathfrak{X}(Z) \colon \upsilon = d\varphi, \ \*d\*\upsilon = 0, \ \varrho = \mathcal{C} \right\},\tag{70}$$

$$\mathfrak{X}\_{GG}(Z) = \left\{ v^\sharp \in \mathfrak{X}(Z) \colon v = d\varphi, \,\,\phi|\_{\mathfrak{Y}Z} = 0 \right\} \tag{71}$$

$$\dim H\_1(Z) = \dim \mathfrak{X}\_{HK}(Z), \quad \dim H\_2(Z) = \dim \mathfrak{X}\_{HG}(Z). \tag{72}$$

*which are respectively called* fluxless knots*,* harmonic knots*,* curly gradients*,* harmonic gradients *and* grounded gradients*, and m means all unit vector fields normal to W, and C is a function on ∂Z that is locally constant.*

For example, consider a vector field defined on a three-dimensional disc. There is no *<sup>v</sup>* <sup>∈</sup> X*HK*(*Z*) on the disc, because the genus is 0 and dim *H*1(*Z*) = dim X*HK*(*Z*) = 0. Thus, all rotation vector fields on the disc are *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*FK*(*Z*) that is the rotating vector field whose axis is an inner point of the disc. *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*CG*(*Z*) is a constant vector field flowing across the disc; therefore, it is divergence-free and zero flux through the one and only component of *<sup>∂</sup>Z*. *<sup>v</sup>* <sup>∈</sup> X*GG*(*Z*) is a radiational vector field flowing from an inner point of the disc, where the potential *<sup>ϕ</sup>* is constant on *<sup>∂</sup>Z*. There is no *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*HG*(*Z*) on the disc, because the numbers of components of *∂Z* and *Z* are each 1, i.e., dim X*HG*(*Z*) = 0. However, a three-dimensional solid torus has a hole; therefore, dim <sup>X</sup>*HK*(*Z*) = 1, but dim <sup>X</sup>*HG*(*Z*) = 0. *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*HK*(*Z*) is a circulative vector field flowing around the hole. Moreover, for a region between two concentric round spheres, dim <sup>X</sup>*HG*(*Z*) = 1. *<sup>v</sup>* <sup>∈</sup> <sup>X</sup>*HK*(*Z*) is a radiational vector field flowing from a common center in the small sphere.

#### **4.3 DPH systems with harmonic energy flows**

18 Magnetohydrodynamics

On the other hand, the dual space of *Hk*(*Z*) is *Hn*−*k*(*Z*, *<sup>∂</sup>Z*), where *Hk*(*Z*, *<sup>∂</sup>Z*) is called *the k-th relative homology of Z modulo ∂Z*. In *n* = 3, the relative homology of *Z* modulo *∂Z* consists of

• *H*1(*Z*, *∂Z*) ··· The vector space is generated by such equivalence classes of oriented paths whose endpoints lie on *∂Z* as two such paths are equivalent if their difference (possibly

• *H*2(*Z*, *∂Z*) ··· The vector space is generated by such equivalence classes of oriented surface whose boundaries lie on *∂Z* as two such surfaces are equivalent if their difference (possibly

• *H*3(*Z*, *∂Z*) ··· The vector space has the oriented components of *Z* as a basis. Thus, dim *H*3(*Z*, *∂Z*) is the number of components of the subregions of *Z* whose boundaries lie on *∂Z*. Note that *H*3(*Z*, *∂Z*) ∼= **R** for a connected *Z* and the element of *H*3(*Z*, *∂Z*) is a

As we mentioned before, the space of vector fields X can be identified with that of 1-forms *Ω*<sup>1</sup> in the sense of a Riemannian metric. Thus, vector fields are affected by the decomposition of differential forms. Indeed, the space of vector fields on a compact domain *Z* in three-dimensional space with a smooth boundary can be decomposed as follows.

**Theorem 4.1** (Cantarella et al. (2002))**.** *Consider vector fields v* <sup>∈</sup> <sup>X</sup>(*Z*) *on a compact domain Z with a smooth boundary ∂Z in three-dimensional space. Let W denote any smooth orientable surface in Z whose boundary ∂W lies on the boundary ∂Z: W* ⊂ *Z and ∂W* ⊂ *∂Z, and called it a* cross-sectional

<sup>X</sup>*K*(*Z*) = <sup>X</sup>*FK*(*Z*) <sup>X</sup>*HK*(*Z*), <sup>X</sup>*G*(*Z*) = <sup>X</sup>*CG*(*Z*) <sup>X</sup>*HG*(*Z*) <sup>X</sup>*GG*(*Z*), (66)

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): <sup>∗</sup>*d*∗*<sup>v</sup>* <sup>=</sup> 0, �*v*, *<sup>n</sup>*�*∂<sup>Z</sup>* <sup>=</sup> 0, �*v*, *<sup>m</sup>*�*<sup>W</sup>* <sup>=</sup> <sup>0</sup>

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): <sup>∗</sup>*d*∗*<sup>v</sup>* <sup>=</sup> 0, �*v*, *<sup>n</sup>*�*∂<sup>Z</sup>* <sup>=</sup> 0, *dv* <sup>=</sup> <sup>0</sup>

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): *<sup>v</sup>* <sup>=</sup> *<sup>d</sup>ϕ*, <sup>∗</sup>*d*∗*<sup>v</sup>* <sup>=</sup> 0, �*v*, *<sup>n</sup>*�*∂<sup>Z</sup>* <sup>=</sup> <sup>0</sup>

*which are called* knots *and* gradients*, respectively, and n means all unit vector fields normal to ∂Z.*

, X*G*(*Z*) =

<sup>X</sup>(*Z*) = <sup>X</sup>*K*(*Z*) <sup>X</sup>*G*(*Z*), (64)

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): *<sup>v</sup>* <sup>=</sup> *<sup>d</sup><sup>ϕ</sup>*

, (67)

, (68)

, (69)

, (65)

the following vector spaces with real coefficients:

paths on *∂Z*) is the boundary of an oriented surface in *Z*.

portions of *∂Z*) is the boundary of some oriented subregion of *Z*.

surface*. The space* X(*Z*) *is the direct sum of five mutually orthogonal subspaces:*

*<sup>v</sup>* <sup>∈</sup> <sup>X</sup>(*Z*): <sup>∗</sup>*d*∗*<sup>v</sup>* <sup>=</sup> 0, *<sup>v</sup>* · *<sup>n</sup>* <sup>=</sup> <sup>0</sup>

• *H*0(*Z*, *∂Z*) ··· dim *H*0(*Z*) is always 0.

Hence, *Hk*(*Z*, *<sup>∂</sup>Z*) ∼= *<sup>H</sup>*3−*k*(*Z*) for 0 ≤ *<sup>k</sup>* ≤ 3.

*where v* <sup>∈</sup> *<sup>Ω</sup>*1(*Z*)*, v* <sup>∈</sup> <sup>X</sup>(*Z*)*, <sup>ϕ</sup>* <sup>∈</sup> *<sup>Ω</sup>*0(*Z*)*,*

X*FK*(*Z*) =

X*HK*(*Z*) =

X*CG*(*Z*) =

X*K*(*Z*) =

*Furthermore,*

*where*

constant function.

In this section, we extend the DPH system of MHD to include the global energy flows originating from topological shapes of manifolds.

Let *Z* be a three-dimensional smooth Riemannian submanifold of *Y* with a smooth boundary *∂Z*. The DPH system (18) of MHD defined on a domain *Z* is extended to have energy flows regarding harmonic knots and harmonic gradients that we call *harmonic energy flows* as follows:

$$\begin{cases} \begin{bmatrix} -\boldsymbol{\rho}\_{t} \\ -\boldsymbol{\nu}\_{l} \end{bmatrix} = \begin{bmatrix} 0 \ d \\ d \ 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{e}\_{\rho} + \boldsymbol{e}\_{hf}^{p} \\ \boldsymbol{e}\_{\boldsymbol{\nu}} + \boldsymbol{e}\_{hf}^{q} \end{bmatrix} - \begin{bmatrix} 0 \\ g\_{1} + g\_{2} \end{bmatrix} + \begin{bmatrix} \boldsymbol{f}\_{hf}^{p} \\ \boldsymbol{f}\_{hf}^{q} \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{f}^{b} \\ \boldsymbol{e}\_{f}^{b} \end{bmatrix} = \begin{bmatrix} (\boldsymbol{e}\_{\rho} + \boldsymbol{e}\_{hf}^{p}) \vert\_{\partial \mathcal{Z}} \\ -(\boldsymbol{e}\_{\boldsymbol{\nu}} + \boldsymbol{e}\_{hf}^{q}) \vert\_{\partial \mathcal{Z}} \end{bmatrix}, \\\ \begin{bmatrix} -\boldsymbol{\mathcal{D}}\_{t} \\ -\boldsymbol{\mathcal{B}}\_{t} \end{bmatrix} = \begin{bmatrix} 0 - \boldsymbol{d} \\ d & 0 \end{bmatrix} \begin{bmatrix} \boldsymbol{E} + \boldsymbol{e}\_{\boldsymbol{\mu}}^{q} \\ \boldsymbol{H} + \boldsymbol{e}\_{\boldsymbol{\mu}}^{q} \end{bmatrix} + \begin{bmatrix} \boldsymbol{J} \\ 0 \end{bmatrix} + \begin{bmatrix} \boldsymbol{f}\_{\mu}^{p} \\ \boldsymbol{f}\_{\mu}^{q} \end{bmatrix}, \quad \begin{bmatrix} \boldsymbol{f}\_{e}^{b} \\ \boldsymbol{e}\_{e}^{b} \end{bmatrix} = \begin{bmatrix} (\boldsymbol{E} + \boldsymbol{e}\_{\mu}^{p}) \vert\_{\partial \mathcal{Z}} \\ (\boldsymbol{H} + \boldsymbol{e}\_{\boldsymbol{\mu}}^{q}) \vert\_{\partial \mathcal{Z}} \end{bmatrix}, \end{cases} \tag{73}$$

where we defined the following harmonic forms yielding harmonic energy flows:

$$\begin{cases} (f\_{\hbar f}^p e\_{\hbar f}^p) \in \Omega^3(\mathbb{Z}) \times \Omega^0(\mathbb{Z}), \quad (f\_{\hbar f}^q, e\_{\hbar f}^q) \in \Omega^2(\mathbb{Z}) \times \Omega^1(\mathbb{Z}),\\ (f\_{\hbar e}^p, e\_{\hbar e}^p) \in \Omega^2(\mathbb{Z}) \times \Omega^1(\mathbb{Z}), \quad (f\_{\hbar e}^q, e\_{\hbar e}^q) \in \Omega^2(\mathbb{Z}) \times \Omega^1(\mathbb{Z}). \end{cases} \tag{74}$$

Note that *Hk*(*Z*) <sup>∼</sup><sup>=</sup> *Hn*−*k*(*Z*, *<sup>∂</sup>Z*) <sup>∼</sup><sup>=</sup> *<sup>Ω</sup><sup>k</sup> <sup>h</sup>*(*Z*), there is the dual from of *ω<sup>h</sup>* with respect to � , �*Z*, called a *Poincaré dual*: *Ω<sup>k</sup> <sup>h</sup>*(*Z*) <sup>∼</sup><sup>=</sup> *<sup>Ω</sup>n*−*<sup>k</sup> <sup>h</sup>* (*Z*), and *f p h f* and *e p h f* are constant functions. The system

**5. Conclusion**

energy flows.

**6. Acknowledgement**

Springer.

*Physics*, Vol. 42, pp. 166–194.

Springer-Verlag, London.

J. Wesson, (2004). *Tokamaks*, 3rd ed., Oxford Univ. Press.

A. Pironti and M. Walker, (2005). *Control System Magazine*, Vol. 25, No. 5. M. Ariola and A. Pironti, (2008). *Magnetic Control of Tokamak Plasmas*, Springer.

**7. References**

This chapter derived the boundary controls based on passivity for ideal magnetohydrodynamics (MHD) systems in terms of distributed port-Hamiltonian (DPH) representations. In *Section 2*, We first rewrote the geometric formulation of MHD as a DPH system. Next, we explained the passivity-based controls for the DPH system of MHD by using collocated input/output pairs, i.e., port variables for stabilizing and assigning a global stable point. The boundary power balance equation of the DPH system could be considered as an extended energy principle of MHD in the sense of dynamical systems and boundary controls. In *Section 3*, we considered the DPH model of MHD with model perturbations. The perturbation can be uniquely decomposed into a Hamiltonian subsystem, called an exact subsystem, and a non-Hamiltonian subsystem, called a dual-exact subsystem. We presented the method of creating a pseudo potential for an exact subsystem of the DPH model. In *Section 4*, we explained a symmetry breaking of conservation laws associated with the DPH system. The breaking can be detected by checking quantities with the boundary port variables of the DPH system. Finally, we showed that the boundary port variables can detect the topological change of the domain of DPH systems and can create desired topological

Hamiltonian Representation of Magnetohydrodynamics for Boundary Energy Controls 187

These results open the way to active disturbance rejections or plasma shape controls. If an actual MHD system is not ideal or includes modeling errors, the power balance equations should be revised. In this case, the pseudo potential construction might be used for improving the model. The boundary control using the boundary port variables might be approximated

The authors would like to thank Professor Bernhard Maschke for fruitful discussions with us.

A.J. van der Schaft and B.M. Maschke, (2002). "Hamiltonian formulation of

S. Arimoto, (1996). *Control Theory of Non-linear Mechanical Systems: A Passivity-based and*

R. Ortega, J.A.L. Perez, P.J. Nichlasson and H.J. Sira-Ramirez, (1998). *Passivity-based Control*

A.J. van der Schaft, (2000). *L*2*-Gain and Passivity Techniques in Nonlinear Control*, 2nd revised

V. Duindam, A. Macchelli, S. Stramigioli and H. Bruyninckx (Eds.), (2009). *Modeling and Control of Complex Physical Systems - The Port-Hamiltonian Approach*, Springer.

distributed-parameter systems with boundary energy flow", *Journal of Geometry and*

*of Euler-Lagrange Systems: Mechanical, Electrical and Electromechanical Applications*,

and enlarged edition, Springer Communications and Control Engineering series,

by the discretization of port-Hamiltonian systems (Golo et al., 2004).

*Circuit-Theoretic Approach*, Oxford Univ. Press.

(73) satisfies the power balances

$$\begin{split} \int\_{Z} \left( - (\boldsymbol{e}\_{\boldsymbol{\rho}} + \boldsymbol{e}\_{\boldsymbol{h}f}^{p}) \wedge \boldsymbol{\rho}\_{t} - (\boldsymbol{e}\_{\boldsymbol{\upsilon}} + \boldsymbol{e}\_{\boldsymbol{h}f}^{q}) \wedge \boldsymbol{\upsilon}\_{t} \right) - \int\_{Z} (\boldsymbol{e}\_{\boldsymbol{\rho}} + \boldsymbol{e}\_{\boldsymbol{h}f}^{p}) \wedge f\_{\boldsymbol{h}f}^{p} \\ - \int\_{Z} (\boldsymbol{e}\_{\boldsymbol{\upsilon}} + \boldsymbol{e}\_{\boldsymbol{h}f}^{q}) \wedge (\boldsymbol{g}\_{2} + f\_{\boldsymbol{h}f}^{q}) - \int\_{\partial Z} (\boldsymbol{e}\_{\boldsymbol{\upsilon}} + \boldsymbol{e}\_{\boldsymbol{h}f}^{q}) \wedge (\boldsymbol{e}\_{\boldsymbol{\rho}} + \boldsymbol{e}\_{\boldsymbol{h}f}^{p}) = 0, \\ \int \left\{ - (\boldsymbol{E} + \boldsymbol{e}\_{\boldsymbol{h}s}^{p}) \wedge \mathbf{D}\_{t} - (\mathbf{H} + \boldsymbol{e}\_{\boldsymbol{h}s}^{q}) \wedge \mathbf{B}\_{t} \right\} - \int (\boldsymbol{E} + \boldsymbol{e}\_{\boldsymbol{h}s}^{p}) \wedge (\boldsymbol{I} + f\_{\boldsymbol{h}s}^{p}) \end{split} \tag{75}$$

$$
\begin{split} \int\_{Z} \left\{-\left(\mathbf{E} + \mathbf{e}\_{\mathrm{he}}^{r}\right) \wedge \mathbf{D}\_{l} - \left(\mathbf{H} + \mathbf{e}\_{\mathrm{he}}^{q}\right) \wedge \mathbf{B}\_{l} \right\} - \int\_{Z} \left(\mathbf{E} + \mathbf{e}\_{\mathrm{he}}^{p}\right) \wedge \left(\mathbf{J} + f\_{\mathrm{he}}^{r}\right) \\ - \int\_{Z} \left(\mathbf{H} + \mathbf{e}\_{\mathrm{he}}^{q}\right) \wedge f\_{\mathrm{he}}^{q} + \int\_{\partial Z} \left(\mathbf{H} + \mathbf{e}\_{\mathrm{he}}^{q}\right) \wedge \left(\mathbf{E} + \mathbf{e}\_{\mathrm{he}}^{p}\right) = \mathbf{0}. \end{split} \tag{76}
$$

#### **4.4 Boundary detection and control of topological transitions**

In fact, it is difficult to determine specific harmonic forms in (74). Hence, let us apply the classification of vector fields to the power balance equation for detecting topological transitions of systems and controlling energy flows.

Consider the cross-sectional surface *<sup>W</sup>* of *<sup>Z</sup>* such that *<sup>W</sup>* <sup>⊂</sup> *<sup>Z</sup>* and *<sup>∂</sup><sup>W</sup>* <sup>⊂</sup> *<sup>∂</sup>Z*. Let *<sup>∂</sup><sup>Z</sup>* <sup>=</sup> <sup>∪</sup>*i∂Z<sup>i</sup>* be a set of subdivided domains of *∂Z* or *W* in which each *∂Z<sup>i</sup>* is homeomorphic to Euclidian spaces (e.g., each component of *∂Zx* and *∂Zy* in (60)). In this setting, we can approximate port variables distributed on *∂Z<sup>i</sup>* , for instance, by using those on the boundary of each subdivided domain *∂*(*∂Z<sup>i</sup>* ) if the subdivision is sufficiently fine. Let

$$(v\_{1'}^\sharp, v\_{2'}^\sharp, v\_{3'}^\sharp, v\_{4'}^\sharp, v\_5^\sharp) \in \mathfrak{X}\_{\text{FK}}(\mathbf{Z}) \oplus \mathfrak{X}\_{\text{HK}}(\mathbf{Z}) \oplus \mathfrak{X}\_{\text{GG}}(\mathbf{Z}) \oplus \mathfrak{X}\_{\text{HG}}(\mathbf{Z}) \oplus \mathfrak{X}\_{\text{GG}}(\mathbf{Z}).\tag{77}$$

Then, we can rewrite (61) as follows:

$$\mathcal{H}\_{\delta t} = \sum\_{i} \left\{ \mathcal{H}\_{\delta t}(v\_1^\sharp) + \mathcal{H}\_{\delta t}(v\_2^\sharp) + \mathcal{H}\_{\delta t}(v\_3^\sharp) + \mathcal{H}\_{\delta t}(v\_4^\sharp) + \mathcal{H}\_{\delta t}(v\_5^\sharp) \right\} \Big|\_{\partial \mathcal{Z}^l} = 0,\tag{78}$$

where <sup>H</sup>*δt*(*<sup>v</sup> <sup>r</sup>*) means the split energy flow generated by *<sup>v</sup> <sup>r</sup>* for 1 ≤ *r* ≤ 5. If all boundary port variables are available as inputs and outputs, the balance of each decomposed energy flows can be confirmed from (78).

On the other hand, desired energy flows depending on the topology of system domain can be reinforced by servo feedback in terms of boundary port variables. If the cause of a change is a known structural perturbation and the boundary surrounds all energy flows generated by the perturbation, we can use the power balance defined on such appropriate boundaries to realize an energy flow control. Indeed, the control law is

$$\int\_{\partial \mathcal{Q}^j} \sum\_{r=1}^5 (e\_j^b(v\_r^\sharp) - u\_{jr}^q) \wedge (f\_j^b(v\_r^\sharp) - u\_{jr}^p), \tag{79}$$

$$\mu^i\_{jr} = g^{ij} (e^b\_j(v^\sharp\_r) - \overline{e}^b\_j(v^\sharp\_r)) \wedge (f^b\_j(v^\sharp\_r) - f^b\_j(v^\sharp\_r))|\_{\partial Z^{\flat \prime}} \tag{80}$$

where *ebi* is the boundary control input or output, *f bi* is the boundary output or input, *e*¯ *bi* and ¯ *f bi* are the desired energy flows, and *gij* is the feedback gain.
