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We present in this paper a new technique based on Gelfand’s triplet [1] and include differential theory to make a theoretical analysis of an optimal control problem with constraints governed by coupled partial differential equations. This technique allowed us to give some theoretical results of existence and uniqueness of the solution of constraints and characterize the optimal control.

The objective of this paper is to make a theoretical analysis of an optimal control problem with constraints governed by coupled partial differential equations.

The interest of this work is two-fold. Indeed firstly the resolution of optimal control problem with partial differential equations constraints is a challenge for the current research in the field both theoretically and numerically. Secondly, the mathematical model studied is of utmost importance in practice since the physical phenomenon is modeled through this paper concerning the pollution of surface water. The studied problem is very complex since the objective function is not explicitly defined in term of control directly. Hence there is a need to solve equations that govern constraints that are also complex as we couple the model which describes the pollutants dissolution with Navier-Stokes equations.

To overcome these difficulties, we built a new technique based on the Gelfand’s triplet and theory of differential solutions to transform the partial differential equations into ordinary differential equations [

The outline of this paper is as follows: in this second section we present the mathematical model; the third section is devoted to the main results of existence uniqueness and also estimation of the system; in the fourth section we conduct the characterization of the optimal control and we end this work with a conclusion and perspectives.

Let

We consider the following model:

subject to

where:

・ z_{d} is an observation function; D is a given linear continuous operator of observation; N is a given positif real;

・ c denotes the concentration of pollutants, g is a source term, d is a diffusion coefficient,

・

・ Equations (2.a)-(2.c) modelling the transport and dissolution of pollutants;

・ Systems (2.d)-(2.g) is Navier-Stokes equations.

We rewrite problems (1)-(2) in the following compact form:

where U_{ad} denotes set of admissible controls. In this form it is easier to establish results of existence and uniqueness concerning the control when knowing some properties of J and U_{ad} [

To make the theoretical study of system (2), we define the following Gelfand triplet:

We choose in this paper

We assume that there exist two positive constants A and A* such that the following inequality hold:

We also recall the following result concerning the Navier-Stokes equations whose proof can be found in [

Theorem 1. Let

and

This first result concerning existence and uniqueness of solution of the model that governs the dissolution of pollutants:

For this we introduce Gelfand’s triplet for the following operators:

and

We consider a function

and

With these notations Equation (6) becomes:

where

We define the space of solutions by:

where

and

Theorem 2. If

solution of the problem (11).

Proof. To proof uniqueness we first establish the following intermediate result:

Indeed,

Assuming existence of two solutions

by multiplying the first equation of (18) by

using the monotony of

since

To prove existence of the solution we first prove that the problem approached of the problem model has a unique solution and that this approximate solution converges to the exact solution. Let consider the following approached problem

where

We assume that this approximated problem is posed on the subspace

・

・ The set of linear combinations of elements of this sequence is dense in

Proposition 3. Under the assumptions of theorem (2),

Proof. The proof is based on the Carathéodory’s theorem which we apply to the following function:

・

・ the condition of minoration on any compact of

Then, for every

which is therefore the solution of (22).

In order to demonstrate the convergence of the approximated solution to the exact solution, we establish the following priori estimation results.

Proposition 4. If

Proof. To prove relation (25) we form the scalar product of the first equation of Equation (11) with

Integrate over

then

We have:

and we deduce the following successive inequalities

To establish relation (26) we integrate the scalar product over

Using (25) we have:

then we have

To proof relation (27) we use the fact that the Laplacian is a bounded linear operator and the following lemma:

Lemma 3.1.

Proof. For all

According to Poincaré inequality there exist

Norms

then we have

Lemma 3.2. If

Proof. For all

According to Lemma (3.1) we have:

Consider

Finally we get

Using estimates (25) and (26), we obtain:

To achieve the proof of Theorem 2 it must be shown that the approximate solution

Using the relation (25), we deduce that the terms of the sequence

Let denote by

From relation (26) we deduce that the sequence

Using relation (27) we obtain the weak convergence

According to [

On the other hand let

and

Since the family

Also to [_{n} such as

Let

Effect the scalar product of (40) by

By Lemma 3.2, we have:

Let:

then

According to Willet-Wong’s inequality we deduce:

Using relation (25), one can have a priori estimation of

Let

The space of admissible controls considered in this paper is

Theorem 5 The optimal control problem (3) has a unique solution

Proof. to proof this result, one can proof that the functional J is inferior semi- continuous, strongly convex and differentiable.

Let us proof that

Let denote by

Apply Fatou’s lemma to the sequence

according to (45) we have:

So that

To proof the strong convexity, we choose

Applying the first unequally Clarkson to the first term in right member, we obtain:

By applying the equality of the parallelogram to the second term the right member, we obtain:

And the differentiability of

Let us denoted by

Let us designed by

then

Note that if

According to these properties of

From relation (47) we have the inequality:

As

search

by tender

since

Theorem 6. If D is a surjective operator of the space

where P is the solution of the adjoint equation.

First Case: We assume that D is an injective operator define from the space in

Seconde Case: D is the identity operator define from

when

Proof. The derivative given by (51) is unusable since for each test W we must solving the tangent model (53). The introduction of the adjoint state allows us to obtain a explicit expression of

We first perform the scalar product of the first equation of (53) with P and then integrate by parts we obtain:

with additional conditions on

we obtain:

By identifying this result with the first term of

To characterize the control we consider the two cases given by relations (56), and (57). Then if the adjoint equation is given by the proof is the same for both cases

In this case relation (51) becomes:

We form the scalar product between the first equation of the adjoint equation with

We have therefore:

So that thus one can write

His equation had to be satisfied whatever the disturbance on

The aim of this paper is a mathematical analysis of an optimal control problem of surface water pollution by using a triplet of evolution adapted to order of derivations. This technique allowed us to propose theorems not only of existence and uniqueness of the control but also of the solution of the equation that governs the constraints which is nonlinear. Then we solved the control problem directly and gave an optimal control characterization by using the adjoint equation.

It is essential to remember that this study only allowed us to characterize the optimal control without giving an analytical expression of it. In a future work we will proceed to a numerical approximation of this model in order to propose solution to the decision makers.

Moustapha, D., Haoua, H. and Bisso, S. (2017) Mathematical Analysis of an Optimal Control Problem of Surface Water Pollution. Applied Mathematics, 8, 164-179. https://doi.org/10.4236/am.2017.82014