^{1}

^{2}

^{2}

^{2}

The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physics of plasma and other real life problems. In this paper, we deal with a class of the constrained OCP for parabolic systems. It is converted to new unconstrained OCP by adding a penalty function to the cost functional. The existence solution of the considering system of parabolic optimal control problem (POCP) is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved.

Many researches in recent years have been devoted to the studies of optimal control problems for a distributed parameter system. Optimal control is widely applied in aerospace, physics, chemistry, biology, engineering, economics and other areas of science and has received considerable attention of researchers.

The optimal boundary control problem for parabolic systems is relevant in mathematical description of several physical processes including chemical reactions, semiconductor theory, nuclear reactor dynamics, population dynamics [

Optimization can be of constrained or unconstrained problems. The presence of constraints in a nonlinear programming creates more problems while finding the minimum as compared to unconstrained ones. Several situations can be identified depending on the effect of constraints on the objective function. The simplest situation is when the constraints do not have any influence on the minimum point. Here the constrained minimum of the problem is the same as the unconstrained minimum, i.e., the constraints do not have any influence on the objective function. For simple optimization problems it may be possible to determine, beforehand, whether or not the constraints have any influence on the minimum point. However, in most of the practical problems, it will be extremely difficult to identify it. Thus one has to proceed with general assumption that the constraints will have some influence on the optimum point. The minimum of a nonlinear programming problem will not be, in general, an extreme point of the feasible region and may not even be on the boundary. Also the problem may have local minima even if the corresponding unconstrained problem is not having local minima. Furthermore, none of the local minima may correspond to the global minimum of the unconstrained problem. All these characteristics are direct consequences of the introduction of constraints and hence we should to have general algorithms to overcome these kinds of minimization problems [

The algorithms for minimization are iterative procedures that require starting values of the design variable x. If the objective function has several local minima, the initial choice of x determines which of these will be computed. There is no guaranteed way of finding the global optimal point. One suggested procedure is to make several computer runs using different starting points and pick the best Rao [

All of the many methods available for the solution of a constrained nonlinear programming problem can be classified into two broad categories, namely, the direct methods and the indirect methods approach. In the direct methods the constraints are handled in an explicit manner whereas in the most of the indirect methods, the constrained problem is solved as a sequence of unconstrained minimization problems or as a single unconstrained minimization problem. Here we are concerned on the indirect methods of solving constrained optimization problems. A large number of methods and their variations are available in the literature for solving constrained optimization problems using indirect methods. As is frequently the case with nonlinear problems, there is no single method that is clearly better than the others. Each method has its own strengths and weaknesses. The quest for a general method that works effectively for all types of problems continues. Sequential transformation methods are the oldest methods also known as Sequential Un-Constrained Minimization Techniques (SUMT) based upon the work of Fiacco and McCormick, 1968. They are still among the most popular ones for some cases of problems, although there are some modifications that are more often used. These methods help us to remove a set of complicating constraints of an optimization problem and give us a frame work to exploit any available methods for unconstrained optimization problems to solve, perhaps, approximately. [

In the meanings of constrained conditions, these optimal control problems can be divided into control con-strained problems and state constrained problems. In each of the branches referred above, there are many excellent works and also many difficulties to be solved.

The rest of this paper is organized as follows. In Section 2, the proposed system of optimal control problem with respect to a parabolic equation is offered. Section 3 describes the analysis of existence and uniqueness of the solution of the POCP. In Section 4, the variation of the functional and its gradient is presented. Section 5 describes Lipschitz continuity of the gradient cost functional. Finally, conclusions are presented in Section 6.

Consider the following POCP process be described in:

with the initial and the boundary conditions:

where the solution of the problem (1-3) is

Many physical and engineering settings have the mathematical model (1-3), in particular in hydrology, material sciences, heat transfer and transport problems [

The purpose is to find the optimal control

and

where

where

This section present the concept of the weak solution of the system (1-3) and the existence solution. Let a function

The weak solution

Theorem 1:

Under the above conditions, the optimal control problem has an optimal solution

Proof: when

verges to the function

tional

The main objective here, the proof of Theorem 2 (found in tail of this section) which requires the following two lemmas; lemma 1 and lemma 2. Let the first variation of the cost functional

therefore,

where

Therefore the function

Lemma 1:

If the direct system (1-3) have the corresponding solution

then the following integral identity holds for all elements

Proof: At

At the boundary conditions in (13) and (14) for the functions

Using the definition of the Fréchet-differential and the above the scalar product definition in V, transform the right-hand side of (15) need into the following expression:

Now we need to show that the last two terms on the right-hand side of (15) are of order

Lemma 2:

If the parabolic problem (12) have the solution

where

Proof:

Multiplying the Equation (12) by

We obtain energy identity after applying the initial and boundary conditions as the following:

We use the

Applying the Cauchy inequality to estimate the term

By integrating the both sides of above inequality on

and use this estimate on the right-hand side of (20):

From (19) with above inequality, we obtain:

where

Hence, the last integral (15) is bounded by

we obtain the following theorem:

Theorem 2:

The cost functional

In this section, by helping the gradient of cost functional

In many situation estimations of determine the parameter

Lemma 3:

The functional

where

where

Proof: Let the following backward parabolic problem

has the solution

implies the following two inequalities:

and

Multiplying the first and the second inequality by

Computing of the second integral on the right-hand side of (26) by the same term. From the energy identity (30) we can obtain the following:

This, with the last estimate, concludes

where

In this paper, we studied a class of the constrained OCP for parabolic systems. The existence and uniqueness of the system is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved. By using the exterior penalty function method, the constrained problem is converted to new unconstrained OCP. The common techniques of constructing the gradient of the cost functional using the solving of the adjoint problem is investigated.

El-Sayed, M.A., Salama, M.M., Farag, M.H. and Al-Thobai- ti, F.B. (2017) The Cost Functional and Its Gradient in Optimal Boundary Control Pro- blem for Parabolic Systems. Open Journal of Optimization, 6, 26-37. https://doi.org/10.4236/ojop.2017.61003