**2. Methodological model**

of substances appear to pose such threats. Among them is crude oil spillage, which first came

The risk of crude oil spillage to the sea presents a major threat to the marine ecology compared with other sources of pollution in the oceans. Before now, it was earlier reported that oil spillage impacts negatively on wildlife and their environments in various ways, which include the alteration of the ecological conditions, and can result into alterations of the environmental physical and chemical composition, destruction of nutritional capita of the marine biomass, changes in the biological equilibrium of the habitat, and as a threat to human health [2]. The same can also be said about Nigeria, where oil spillage is a major environmental problem and its coastal zone is rated as one of the most polluted spots on the planet in the year 2006 [3]. For instance, from 1976 to 2007, over 1,896,960 barrels of oil were sunk into the Nigerian coastal waters resulting in a serious pollution of drinkable water and destruction of resort centers, properties, and lives along the coastal zone. This was seen to be a major contributor to the

As a case in point, after a spill in the ocean, oil in water body, regardless of whether it originated as surface or subsurface spill, forms a thin film called oil slick as it spreads in water. The oil slick movement is governed by the advection and blustery diffusion as a result of water current and wind action. The slick always spreads over the water surface due to gravitational, inertia, gluey, and interfacial strain force equilibrium. The oil composition also changes from the early time of the spill. Thus, the water-soluble components of the oil dissolve in the water column, whereas the immiscible components emulsified and disperse in the water column as small

droplets and light (low molecular weight) fractions evaporate (for example, see [4]).

worldwide while the threat of oil pollution is also likely to increase accordingly.

In essence, the frequency of accidental oil spills in aquatic environments has presented a growing global concern and awareness of the risks of oil spills and the damage they do to the environment. However, it is widely known that oil exploration is a necessity in our industrial society and a major sustainer of our lifestyle, as most of the energy used in Canada and the United States, for instance, is for transportation that runs on oil and petroleum products. Thus, in as much as the industry uses oil and petroleum derivatives for the manufacturing of vital products, such as plastics, fertilizers, and chemical feedstock, the drifts in energy usage are not likely to decrease much in the near future. In what follows, it is a global belief that the production and consumption of oil and petroleum products might continue to increase

Consequently, a fundamental problem in environmental research in recent time has been identified in the literature to how to properly assess and control the spatial structure of pollution fields at various scales, and several studies showed that mathematical models were the only available tools for rapid computations and determinations of spilled oil fate and for

to public attention with the *Torrey Canyon* disaster in 1967.

130 Robust Control - Theoretical Models and Case Studies

regional crisis in the Nigeria Niger-Delta region.

the simulation of the various clean-up operations.

Now, consider the introduction of an optimal control theory into spill modeling to develop an optimization process that will aid effective decision-making in marine oil spill management. The purpose of the optimal control theory is to determine the control policy that will optimize (maximize or minimize) a specific performance criterion subject to the constraints imposed by the physical nature of the problem. A fundamental theorem of the calculus of variations is applied to problems with unconstrained states and controls, whereas a consideration of the effect of control constraints leads to the application of Markovian decision processes.

The optimization objectives are expressed as a performance index (value function or reward) to be optimized, whereas the optimization models are formulated to adequately describe the marine oil spill control starting from the transportation process. These models consist of conservation relations needed to specify the dynamic state of the process given by the chemical compositions and movements of crude oil in water.

### **2.1. Mathematical preliminaries and definition of terms**

In our basic optimal control problem, *u*(*t*) is used for the control and *x*(*t*) is used for the state variables. The state variable satisfies a differential equation that depends on the control variable:

$$\mathbf{x}'(t) = \mathbf{g}\left(t, \mathbf{x}(t), \boldsymbol{\mu}(t)\right) \tag{1}$$

where *x* ′ (*t*) is the state differential defining the performance index. This implies that, as a control function changes, the solution to the differential equation will also change. In other words, one can view the control-to-state relationship as a map *u*(*t*)↦ *x* = *x*(*u*) [we wrote *x*(*u*) just to remind us of the dependence on *u*]. Our basic optimal control problem therefore consisted of finding, in mathematical terms, a piecewise continuous control *u*(*t*) and the associated state variable *x*(*t*) to optimize a given objective function. That is to say,

$$\max\_{u} \int\_{t\_0}^{t\_1} f\left(t, \mathbf{x}\left(t\right), u\left(t\right)\right) dt \tag{2}$$

$$\mathbf{x}'(t) = \mathbf{g}\left(t, \mathbf{x}(t), \boldsymbol{\mu}(t)\right)$$
 
$$\text{Subject to }$$
 
$$\mathbf{x}(t\_0) = 0 \quad \text{and} \quad \mathbf{x}(t\_1) \, free$$

Such a maximizing control is called an optimal control. By "*x*(*t*1) free", it means that the value of *x*(*t*1) is unrestricted. Here, the functions *f* and *g* are continuously differentiable functions in all arguments. Thus, whereas the control(s) is piecewise continuous, the associated states are piecewise differentiable. This implies that, depending on the scale of the spatial resolution (like the case of oil spill), an introduction of space variables could alter the basic model from ordinary differential equations (with just time as the underlying variable) to partial differential equations (PDEs). Let us focus our attention to the consideration of optimal control of PDEs. Our solution to the control problem will then depend on the existence of an optimal control in the PDE.

The general idea of the optimal control of PDEs here starts with a PDE with a state solution *x* and control *u*. Set ∂ to denote a partial differential operator with appropriate initial and boundary conditions:

$$\hat{\mathbf{x}} \cdot \hat{\mathbf{x}} = f\left(\mathbf{x}, \mu\right) \text{in } \Omega \times \left[0, T\right] \tag{4}$$

This implies that we are considering a problem with space *x* and time *t* within a territorial boundary, *Ω* × 0, *T* . The objective functional in this problem represents the goal of the problem, and we seek to find an optimal control *u* \* in an appropriate control set such that

$$J\left(\boldsymbol{u}^\*\right) = \min\_{\boldsymbol{u}} J\left(\boldsymbol{u}\right) \tag{5}$$

When the control cost is considered, with an objective functional

$$J(\boldsymbol{u}) = \int\_{0}^{T} \int\_{\Omega} \mathbf{g}\left(\mathbf{x}, t, \mathbf{x}\left(t\right), \boldsymbol{u}\left(\mathbf{x}, t\right)\right) d\mathbf{x} dt \tag{6}$$

To consider the properties of the functional, it is important to note the following fundamentals:


$$
\Delta J\left(\mathbf{x}, \delta\left(\mathbf{x}\right)\right) = \delta\left(J\right)\left[\mathbf{x}, \delta\left(\mathbf{x}\right)\right] + \mathbf{g}\left(\mathbf{x}, \delta\left(\mathbf{x}\right)\right)\left\|\delta\left(\mathbf{x}\right)\right\|\tag{7}
$$

where *δ*(*J*) is also linear in *δ*(*x*). Suppose that lim *δ*(*x*) →0 *g*(*x*, *δ*(*x*))=0; then, *J* is said to be differentiable on *x*, whereas *δ*(*J*) is the first variation of *J* evaluated for *x*(*t*) [5].


$$
\Delta J = J\left(\mathbf{x} + \mathcal{S}\left(\mathbf{x}\right)\right) - J\left(\mathbf{x}\right) \tag{8}
$$

**v.** A differential equation whose solutions are the functions for which a given functional is stationary is known as an Euler-Lagrange equation (Euler's equation or Lagrange's equation).

**Fundamental theorem of variational calculus [5]:** This theorem states that "if *x* \* is optimum, then it is a necessary condition that the first variation of *J* must vanish on *x*. That is to say, *δ*(*J*) *x* \* , *δ*(*x*) =0 for all admissible *δ*(*x*)".

### **2.2. Model conceptualization**

The fundamental principle upon which the pollutant fate and transport models are based is the law of conservation of mass [6]:

$$\begin{cases} \frac{\partial h}{\partial t} + \overline{\nabla} \left( h \overline{\nu} \right) - \overline{\nabla} \left( D \overline{\nabla} h \right) = R\_h\\ \frac{\partial \mathbf{C}}{\partial t} + \overline{\nabla} \left( \mathbf{C} \, \overline{\mathbf{u}} \right) - \overline{\nabla} \left( \vec{E} \, \overline{\nabla} \mathbf{C} \right) = \mathbf{R} \end{cases} \tag{9}$$

where

all arguments. Thus, whereas the control(s) is piecewise continuous, the associated states are piecewise differentiable. This implies that, depending on the scale of the spatial resolution (like the case of oil spill), an introduction of space variables could alter the basic model from ordinary differential equations (with just time as the underlying variable) to partial differential equations (PDEs). Let us focus our attention to the consideration of optimal control of PDEs. Our solution to the control problem will then depend on the existence of an optimal control in

The general idea of the optimal control of PDEs here starts with a PDE with a state solution *x* and control *u*. Set ∂ to denote a partial differential operator with appropriate initial and

This implies that we are considering a problem with space *x* and time *t* within a territorial boundary, *Ω* × 0, *T* . The objective functional in this problem represents the goal of the

> ( ) ( ) \* min *u*

( ) ( () ( )) <sup>0</sup> ,, , , *<sup>T</sup> J u g x t x t u x t dxdt*

To consider the properties of the functional, it is important to note the following fundamentals: **i.** A functional *J* is "a rule of correspondence that assigns to each function, say *x*(*t*),

with the functions in the domain is called the range of the functional" [5].

**ii.** Let *δ*(*J*) be the first variation of the functional; thus, *δ*(*J*) is the part of the increment

differentiable on *x*, whereas *δ*(*J*) is the first variation of *J* evaluated for *x*(*t*) [5].

D= + *Jx x J x x gx x x* ( ) , ,,

 d

constrained in a certain set of functions, say *X*, a unique real number. The set of functions is called the domain of the functional, and the set of real numbers associated

> dd( ) ( ) () () é ù ( ) ( ) ë û (7)

> > *δ*(*x*) →0

*g*(*x*, *δ*(*x*))=0; then, *J* is said to be

W

of Δ*J*, which is linear in the variation *δ*(*x*) such that

 d

where *δ*(*J*) is also linear in *δ*(*x*). Suppose that lim

d

¶ = W´ *x f xu T* ( ) , in 0, é ù ë û (4)

*Ju Ju* = (5)

<sup>=</sup> ò ò (6)

in an appropriate control set such that

the PDE.

boundary conditions:

132 Robust Control - Theoretical Models and Case Studies

problem, and we seek to find an optimal control *u* \*

When the control cost is considered, with an objective functional

*h* = oil slick thickness,


*D* = oil fluid velocity,


*Rh* and **R** = physical chemical kinetic terms,

**u <sup>→</sup>** = grid size,

<sup>∇</sup>¯ = Cartesian coordinate, and

*t* = time.

Eq. (9) can be modified as

$$\frac{\partial \mathcal{V}\_l}{\partial i} d\mathbf{x} d\mathbf{y} dz,$$

where *dxdydz* denotes the differential volume of the state variable assuming a net chemical contaminant flux in each axial direction such that *γ <sup>i</sup>* = contaminant movement in each axial direction (*i* = *x*, *y*, *z*) and *dx*, *dy*, *dz* = differential distances in the *x, y*, and *z* directions.

The fluidity of oil in water contains the advection due to current and wind as well as the dispersive instability due to weathering processes. Thus, if we set

$$
\gamma = aq - d\nabla q \tag{10}
$$

where

*γ* = movement of contaminant vector,

*ω* = contaminant discharge vector,

*q* = contaminant molar concentration,

*d* = dispersion tensor, and

∇ = gradient operator (Laplacian).

With minor mathematical regularities, Eq. (10) will become

$$-\nabla\left(\alpha q - d\nabla q\right) = \frac{\partial \tau}{\partial t} + m \tag{11}$$

where

*τ* = total concentration of contaminant in the system,

*m* = decay rate of contaminant, and

*t* = time.

A two-dimensional differential representation of Eq. (11) is given as

#### Sequential Optimization Model for Marine Oil Spill Control http://dx.doi.org/10.5772/63050 135

$$\begin{split} \frac{\partial \sigma}{\partial t} &= \left[ c \frac{\partial \mathbf{v}\_{\mathbf{x}}}{\partial \mathbf{x}} + \mathbf{v}\_{\mathbf{x}} \frac{\partial q}{\partial \mathbf{x}} + \frac{\partial \mathbf{v}\_{\mathbf{x}}}{\partial \mathbf{x}} \frac{\partial q}{\partial \mathbf{x}} + \mathbf{v}\_{\mathbf{x}} \frac{\partial^{2} q}{\partial \mathbf{x}^{2}} - q \frac{\partial \mathbf{v}\_{\mathbf{y}}}{\partial \mathbf{y}} - \mathbf{v}\_{\mathbf{y}} \frac{\partial q}{\partial \mathbf{y}} + \frac{\partial \mathbf{v}\_{\mathbf{y}}}{\partial \mathbf{y}} \frac{\partial q}{\partial \mathbf{y}} + \mathbf{v}\_{\mathbf{y}} \frac{\partial^{2} q}{\partial \mathbf{y}^{2}} \right] - m \\ &= \frac{\partial \left( q \mathbf{v}\_{\mathbf{x}} \right)}{\partial \mathbf{x}} + \frac{\partial \left[ \mathbf{v}\_{\mathbf{x}} \frac{\partial q}{\partial \mathbf{x}} \right]}{\partial \mathbf{x}} - \frac{\partial \left( q \mathbf{v}\_{\mathbf{y}} \right)}{\partial \mathbf{y}} + \frac{\partial \left[ \mathbf{v}\_{\mathbf{y}} \frac{\partial q}{\partial \mathbf{y}} \right]}{\partial \mathbf{y}} - m, \end{split} \tag{12}$$

so that we have *vx* and *vy*, which represented the fluid velocities in the *x* and *y* directions. By applying the principle of the conservation of mass, the steady-state equation of spill transpor‐ tation is given as

$$\begin{split} \frac{\partial VST}{\partial \mathbf{x} \partial \mathbf{y} \partial \mathbf{b}} &= \frac{\partial h\_{\mathbf{x}}}{\partial \mathbf{x}} \frac{\partial p^{2}}{\partial \mathbf{x}} + h\_{\mathbf{x}} \frac{\partial^{2} p^{2}}{\partial \mathbf{x}^{2}} + \frac{\partial h\_{\mathbf{y}}}{\partial \mathbf{y}} \frac{\partial p^{2}}{\partial \mathbf{y}} + h\_{\mathbf{y}} \frac{\partial^{2} p^{2}}{\partial \mathbf{y}^{2}} \\ &= \frac{\partial}{\partial \mathbf{x}} \bigg[h\_{\mathbf{x}} \frac{\partial p^{2}}{\partial \mathbf{x}}\bigg] + \bigg[h\_{\mathbf{y}} \frac{\partial p^{2}}{\partial \mathbf{y}}\bigg] \end{split} \tag{13}$$

where

**u**

**<sup>→</sup>** = grid size,

*t* = time.

where

where

*t* = time.

<sup>∇</sup>¯ = Cartesian coordinate, and

134 Robust Control - Theoretical Models and Case Studies

Eq. (9) can be modified as

contaminant flux in each axial direction such that *γ <sup>i</sup>*

*γ* = movement of contaminant vector,

*q* = contaminant molar concentration,

*ω* = contaminant discharge vector,

∇ = gradient operator (Laplacian).

*m* = decay rate of contaminant, and

*d* = dispersion tensor, and

dispersive instability due to weathering processes. Thus, if we set

With minor mathematical regularities, Eq. (10) will become

*τ* = total concentration of contaminant in the system,

g w , *<sup>i</sup> dxdydz <sup>i</sup>*

where *dxdydz* denotes the differential volume of the state variable assuming a net chemical

The fluidity of oil in water contains the advection due to current and wind as well as the

( ) *q dq m*

¶ -Ñ - Ñ = +

w

A two-dimensional differential representation of Eq. (11) is given as

*t* t = contaminant movement in each axial

= -Ñ *q dq* (10)

¶ (11)

¶g

¶

direction (*i* = *x*, *y*, *z*) and *dx*, *dy*, *dz* = differential distances in the *x, y*, and *z* directions.

*h* = oil penetrability trajectory,

*p* = oil stress,

*V* = oil viscidness,

*S* = source of oil mass fluidity,

*T* = temperature,

*b* = molecular weight of oil, and

*l* = a fixed length of the *z* direction.

According to Refs. [5–7], "the transport and fate of the spilled oil is governed by the advection due to current and wind, horizontal spreading of the surface slick due to turbulent diffusion, gravitational force, force of inertia, viscous and surface tension forces, emulsification, mass transfer of heat, and changes in the physiochemical properties of oil due to weathering processes (evaporation, dispersion, dissolution, oxidation, etc.)". Thus, Eq. (13) can be transformed to

$$\frac{\partial q(\mathbf{x},t)}{\partial t} = -h\_{\mathbf{x}} \frac{\partial}{\partial \mathbf{x}} q(\mathbf{x},t) + D \frac{\partial^2}{\partial \mathbf{x}^2} q(\mathbf{x},t) + R + S \tag{14}$$

where *q* ={*qe*, *qd* , *qp*} denotes the oil spill concentration in emulsified, dissolved, and particulate phases, respectively, at state *x* and time *t; h* is the fluid velocity; *D* is the spreading function, and *R* and *S* denote the environmental factors and the spill source term, respectively.

### **2.3. Optimality problem**

When hydrocarbons enter an aquatic environment, their concentrations tend to decrease with time due to the evaporation, oxidation, and other weathering processes. This could be described as a death process and could be modeled as a first-order reaction [7]. Having known this, the optimal control problem can then be formulated by setting *R* in Eq. (14) to be

$$R = -kC(\mathbf{x}, t) \tag{15}$$

so that *k* denotes a kinetic constant of the environmental factors that influenced the concen‐ tration of oil in water. Here, it is assumed that the source term is not known so that' *S* = 0.

Then, Eq. (14) can be expressed as

$$\frac{\partial q\left(\mathbf{x},t\right)}{\partial t} = -\nabla \left(Vq\left(\mathbf{x},t\right)\right) + \nabla \cdot \left(D\nabla q\left(\mathbf{x},t\right)\right) - kq\left(\mathbf{x},t\right) \tag{16}$$

which is called "oil spill dynamical (or transport) problem". To solve this problem, a mecha‐ nism for controlling the system in marine environment can be set up as follows:

Let Ω be an open, connected subset of ℝ*<sup>n</sup>*, where ℝ*<sup>n</sup>*i is the Euclidean *n*-dimensional space. We defined the spatial boundary of the problem as Ω. The unit variable is *t* and is contained in the interval 0, *T* , where *T* <*∞*. Let *x* be the space variable associated with Ω, and let ∂ be a partial differential operator with appropriate initial and boundary conditions, where ∂Ω is the differential boundary of Ω; then,

$$q\_t(\mathbf{x}, t) - a\Delta q(\mathbf{x}, t) = q\left(\mathbf{x}, t\right) \left(1 - q\left(\mathbf{x}, t\right)\right) - u\left(\mathbf{x}, t\right)q\left(\mathbf{x}, t\right) \quad \text{in} \qquad \Omega \times \left[0, T\right]$$

$$q\left(\mathbf{x}, 0\right) = q\_0\left(\mathbf{x}\right) \ge 0 \quad \text{on} \qquad \Omega, t = 0 \qquad \text{(seebed boundary)}\tag{17}$$

$$q\left(\mathbf{x}, t\right) = 0 \qquad \text{on} \qquad \partial\Omega \times \left[0, T\right] \quad \text{(seea-side boundary)}$$

where ∂*Ω* × 0, *T* mathematically defined an operation with a PDE operator ∂ in the spatial boundary of the problem Ω within a specified upper and lower horizons 0, *T* .

Eq. (17) is defined as the state equation with a logistic growth *q*(1−*q*) and a constant diffusion coefficient *α* due to weathering processes. The symbol Δ represents the Laplacian. The state *q*(*x*, *t*) denotes the volume or concentration of the crude oil and *u*(*x*, *t*)is the control that entered the problem over the volumetric domain. The zero boundary conditions imply the limitation of the slick at the surrounding environment.

The reward or value objective functional can be obtained as

where *q* ={*qe*, *qd* , *qp*} denotes the oil spill concentration in emulsified, dissolved, and particulate phases, respectively, at state *x* and time *t; h* is the fluid velocity; *D* is the spreading function,

When hydrocarbons enter an aquatic environment, their concentrations tend to decrease with time due to the evaporation, oxidation, and other weathering processes. This could be described as a death process and could be modeled as a first-order reaction [7]. Having known

so that *k* denotes a kinetic constant of the environmental factors that influenced the concen‐ tration of oil in water. Here, it is assumed that the source term is not known so that' *S* = 0.

( ) ( ) ( ) ( ) ( ) ( ) , , ,, *q xt Vq x t D q x t kq x t*

which is called "oil spill dynamical (or transport) problem". To solve this problem, a mecha‐

Let Ω be an open, connected subset of ℝ*<sup>n</sup>*, where ℝ*<sup>n</sup>*i is the Euclidean *n*-dimensional space. We defined the spatial boundary of the problem as Ω. The unit variable is *t* and is contained in the interval 0, *T* , where *T* <*∞*. Let *x* be the space variable associated with Ω, and let ∂ be a partial differential operator with appropriate initial and boundary conditions, where ∂Ω is the

= -Ñ + Ñ × Ñ -

nism for controlling the system in marine environment can be set up as follows:

() () () () ( ) ( )( )

, , , 1 , , , 0,

= ¶W ´é ù - ë û

*<sup>t</sup> q x t q x t q x t q x t u x t q x t in T*

= ³ W=

boundary of the problem Ω within a specified upper and lower horizons 0, *T* .

,0 0 , 0 ( ) , 0 0, ( )

*q x q x on t seabed boundary q x t on T sea sideboundary*

where ∂*Ω* × 0, *T* mathematically defined an operation with a PDE operator ∂ in the spatial

Eq. (17) is defined as the state equation with a logistic growth *q*(1−*q*) and a constant diffusion coefficient *α* due to weathering processes. The symbol Δ represents the Laplacian. The state *q*(*x*, *t*) denotes the volume or concentration of the crude oil and *u*(*x*, *t*)is the control that entered the problem over the volumetric domain. The zero boundary conditions imply the limitation

W ´é ù ë û

( ) ()

0


( )

of the slick at the surrounding environment.

*R kC x t* = - ( ) , (15)

(17)

¶ (16)

and *R* and *S* denote the environmental factors and the spill source term, respectively.

this, the optimal control problem can then be formulated by setting *R* in Eq. (14) to be

**2.3. Optimality problem**

136 Robust Control - Theoretical Models and Case Studies

Then, Eq. (14) can be expressed as

differential boundary of Ω; then,

a

*t*

¶

$$J\left(u\right) = \int\_{0}^{T} \int\_{\Omega} e^{-\theta t} \left(\tilde{\xi}u\left(\mathbf{x},t\right)q\left(\mathbf{x},t\right) - Au\left(\mathbf{x},t\right)^{2}\right)d\mathbf{x}dt\tag{18}$$

Here, *ξ* denotes the price of spilled oil, so that *ξuq* represents the reward from the control amount *uq*. Note that a quadratic cost for the clean-up effort with a weighted coefficient *A*, where *A* is assumed to be a positive constant, is applied. The term *e* <sup>−</sup>*θ<sup>t</sup>* is introduced to denote a discounted value of the accrued future costs with0≤*θ* <1. By setting *ξ* =1 (for convenience), an optimal control *u* \* is needed to optimize a control strategy focusing on the actual detected spill point, such that application of any control on a no-spill region (look-alike) would be minimized [i.e., *u* \* (*x*, *t*)=0] and the value of all future earnings would be maximized. In other words, we seek for *u* \* such that

$$J\left(\boldsymbol{u}^{\bullet}\right) = \max\_{\boldsymbol{u}\in U} J\left(\boldsymbol{u}\right) \tag{19}$$

where *U* denotes a set of allowable control, and the maximization is over all measurable controls with 0≤*u*(*x*, *t*)≤*m*<1 a.e. Under this set-up, it follows that, within the context of optimal control, the state solution satisfies *q*(*x*, *t*)≥0 on *Ω* ×(0, *T* ) by the maximum principle for parabolic equations.

**Lemma 1 [8]:** Let *U* be a convex set and *J* be strictly convex on *U*. Then, there exists at most one *u* \* ∈*U* such that *J* has a minimum at *u* \* . This implies that, by the maximum principle for parabolic equations, the necessary conditions for optimality are satisfied whenever the state solution satisfies *q*(*x*, *t*)≥0 on *Ω* ×(0, *T* ).
