**5. Conclusion**

20 Magnetohydrodynamics

 − *Z* (*e<sup>ρ</sup>* + *e p h f*) ∧ *f p h f*

*q h f*) − *∂Z*

 − *Z* (*E* + *e p*

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

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

<sup>2</sup>) + <sup>H</sup>*δt*(*<sup>v</sup>*

variables are available as inputs and outputs, the balance of each decomposed energy flows

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

*jr*) <sup>∧</sup> (*<sup>f</sup> <sup>b</sup>*

where *ebi* is the boundary control input or output, *f bi* is the boundary output or input, *e*¯

*<sup>j</sup>* (*<sup>v</sup> <sup>r</sup>*) <sup>−</sup> *<sup>u</sup><sup>p</sup>*

> *<sup>j</sup>* (*<sup>v</sup> <sup>r</sup>*) <sup>−</sup> ¯ *f b <sup>j</sup>* (*<sup>v</sup> r*))

*<sup>r</sup>*)) <sup>∧</sup> (*<sup>f</sup> <sup>b</sup>*

(*H* + *e q*

(*ev* + *e q*

*he*) ∧ (*E* + *e*

, for instance, by using those on the boundary of each subdivided

<sup>5</sup>) <sup>∈</sup> <sup>X</sup>*FK*(*Z*) <sup>X</sup>*HK*(*Z*) <sup>X</sup>*CG*(*Z*) <sup>X</sup>*HG*(*Z*) <sup>X</sup>*GG*(*Z*). (77)

<sup>4</sup>) + <sup>H</sup>*δt*(*<sup>v</sup>*

5) 

*<sup>r</sup>* for 1 ≤ *r* ≤ 5. If all boundary port

*jr*), (79)

*<sup>∂</sup>Zj* , (80)

*bi* and

*<sup>∂</sup>Zi* <sup>=</sup> 0, (78)

<sup>3</sup>) + <sup>H</sup>*δt*(*<sup>v</sup>*

*h f*) ∧ (*e<sup>ρ</sup>* + *e*

*p*

*p he*)

*he*) ∧ (*J* + *f*

*p*

*h f*) = 0, (75)

*he*) = 0. (76)

*q h f*) ∧ *v<sup>t</sup>*

*q he*) ∧ *B<sup>t</sup>*

*h f*) ∧ (*g*<sup>2</sup> + *f*

(73) satisfies the power balances

−(*e<sup>ρ</sup>* + *e*

−(*E* + *e*

*p*

*p*

− *Z* (*ev* + *e q*

− *Z* (*H* + *e q he*) ∧ *f q he* + *∂Z*

transitions of systems and controlling energy flows.

*h f*) ∧ *ρ<sup>t</sup>* − (*ev* + *e*

*he*) ∧ *D<sup>t</sup>* − (*H* + *e*

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

) if the subdivision is sufficiently fine. Let

<sup>1</sup>) + <sup>H</sup>*δt*(*<sup>v</sup>*

realize an energy flow control. Indeed, the control law is

5 ∑ *r*=1 (*e b j*(*<sup>v</sup> <sup>r</sup>*) <sup>−</sup> *<sup>u</sup><sup>q</sup>*

*jr* <sup>=</sup> *<sup>g</sup>ij*(*<sup>e</sup>*

*b j*(*<sup>v</sup> <sup>r</sup>*) − *e*¯ *b j*(*<sup>v</sup>*

*f bi* are the desired energy flows, and *gij* is the feedback gain.

 *∂Zj*

*ui*

*<sup>r</sup>*) means the split energy flow generated by *<sup>v</sup>*

 *Z* 

 *Z* 

variables distributed on *∂Z<sup>i</sup>*

(*v* <sup>1</sup>, *<sup>v</sup>* <sup>2</sup>, *<sup>v</sup>* <sup>3</sup>, *<sup>v</sup>* <sup>4</sup>, *<sup>v</sup>*

Then, we can rewrite (61) as follows:

 <sup>H</sup>*δt*(*<sup>v</sup>*

H*δ<sup>t</sup>* = ∑ *i*

can be confirmed from (78).

domain *∂*(*∂Z<sup>i</sup>*

where <sup>H</sup>*δt*(*<sup>v</sup>*

¯

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 energy flows.

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