Discretization of Optimal Control Problems Governed by p-Laplacian Elliptic Equations

Abstract

In this paper, a state-constrained optimal control problem governed by p-Laplacian elliptic equations is studied. The feasible control set or the cost functional may be nonconvex, and the purpose is to obtain the convergence of a solution of the discretized control problem to an optimal control of the relaxed continuous problem.

1. Introduction and the Optimal Control Problem

Let be a bounded open convex domain of , , with a Lipschitz continuous boundary . Let be a compact subset of , and we denote by the set of measurable functions . For each , we consider the following state equation

where .

We first make the following assumptions on :

(S1) The function is measurable in is in , are continuous in . Moreover,

and for any , there exists a constant such that

The next theorem claims the well-posedness of the state equation.

Proposition 1. Suppose that (S1) holds. Then for any , there exists a unique weak solution of (1). Moreover, there exists a constant , independent of , such that

The estimate of can be obtained by the same arguments in the proof of Theorem 6.11 in [Chapter 2, 9] and the remained results of this theorem can be deduced from Lemma 3.1 in [1].

Remark 2. Since can be compactly embedded into , which shows that there exists a constant , independent of , such thatwhere is the solution of (1) corresponding to.

Let us consider another function that satisfies the following properties:

(S2) is a Carathéodry function which satisfies that for any , there exists a nonnegative function such that

Now our optimal control problem can be stated as follows.

where and is a continuous function.

In the case of no convexity assumption, optimal control problems do not have classical solutions generally, whereas the corresponding relaxed problems have solutions if some reasonable assumptions are made. To deal with these problems numerically, one needs to discretize them in some way, and then by applying some optimization method to the discrete problems to find some discrete optimal solution. Since the structures of the continuous problems are basically different from the discrete ones, it is necessary to know whether discrete optimality converges to continuous optimality.

Similar problems were considered by Casas [2] and Chryssoverghi and Kokkinis [3]. In the field of finite element approximations for optimal controls governed by PDEs, we refer the readers to the papers [4–10] and the references therein. This present paper is mainly motivated by the work of [2] where the author considered the following state equation

with

Our main goal is to generalize the results in [2] to the case of p-Laplacian. Such models arise from fluid mechanics, nonlinear diffusion and nonlinear elasticity (see [11]).

Now, we first introduce the stability concept of with respect to perturbations of the set of feasible states.

Definition 3 [1, Definition 1]. We will say that is stable to the right ifAnalogously, is stable to the left if is said stable if it is stable to the left and to the right simultaneously.

The following result shows that problem is stable under what cases, and which can be proved by the same arguments as that in the proof of Theorem 2 [2]. However, we still present the details for readers’ convenience.

Lemma 4. Suppose that (S1) and (S2) hold. There exists such that has no feasible control for . For every , except at most a countable number of them, problem is stable.

Proof. From (5), there exists a constant such that for all and . The minimum and maximum of over are denoted by and , respectively. Then we can claim that for , admits no a feasible control, while every element of is a feasible control for any . Let , and then we have that
Next we show that, for almost all , is stable. We consider a function defined by . Then except for at most a countable number of , we find that is monotone, nonincreasing and continuous. Moreover, it is easy to see that the continuity of in is equivalent to the stability of . Thus the lemma is proved.

2. The Relaxed Control Problem

In this section, we would like to apply the relaxation theory. That is the control set can be extended to a bigger space such that the new control problem has at least one solution. For this reason, we recall the concept of relaxed controls and the relations between classical controls and relaxed controls given by Warga [12] first.

Let denote the space of continuous functions endowed with the maximum norm and is the space of Radon measures in . Let be the subset of formed by the probability measures in , and be the subset of the Banach space formed by all -valued -weakly measurable functions in . That is, if and only if

and

As usual we call each member of a relaxed control and an element of a classical control, respectively.

It is known that is convex and compact, moreover, is dense in with the weak star topology of (see Warga [12, Theorem IV.2.1, p. 272 and Theorem IV.2.6, p. 275]).

We now define the relaxed control problem in the following way

where being the solution of the problem

Let us remark that can be considered as a subset of by identifying with Dirac measure-valued function . Moreover, with this identification we have and . On the other hand since is dense in , problem can be considered as an extension of . Furthermore, we will see below that has at least one solution. However we must be concerned whether . The following theorem gives the answer.

Theorem 5. Suppose that (S1) and (S2) hold. Let be as in Lemma 4. has at least one solution for every. Moreover if and only if is stable to the right.

Proof. Step 1. We would like to prove the existence of one solution of for . Indeed, similar to (5), there exists a constant , independent of , such thatIt follows from (16) and (S2) thatTherefore, there exists a minimizing sequence with the property ofSince is convex and compact, there exists a such that (as )Moreover, without losing generality, we can suppose that there exists a function such thatwhen , where is the solution of (15) corresponding to , respectively. Finally the continuity of and shows that is a solution of .
Step 2. We deal with the remainder part of the theorem. To do this, we first sate the following inequalitiesThe first and the last inequalities can be deduced from the identification of every feasible control for (resp. ) with a feasible control for (resp. ). We only need prove the second inequality. Since is dence in , if is a feasible control for , then there exists such that (as )That is,Let be the solution of (15) corresponding to . From (23), we have thatwhich means that uniformly in , thentherefore for any and bigger than some , only depending on . Thus are feasible solutions for problem andthe desired inequality is obtained.
Next, we only need to prove thatLet be a solution of for every . Since is compact, one can take a sequence , with , such that weakly for some . By the uniform convergence in , for every , we have thatthis shows that is a feasible control for . Hence we obtain thatFinally, it follows from (22) and (27) the proof can be deduced. In fact, if , then we have thatthat is is stable to the right. On the other hand, if , thenand the proof is completed.

Corollary 6. Suppose that (S1) and (S2) hold. If problem is stable to the right and it has a solution , then is also a solution of .

3. Numerical Approximation of the Control Problem

In this section the numerical discretization of problem will be considered, and the convergence of optimal discrete controls to optimal relaxed controls in some topology will be proved.

We first give some standard notations to use the finite element method (see Ciarlet [13] or Casas [2]). Let be a regular family of triangulations in satisfying the inverse assumption. Let us take with the interior and the boundary . Then we assume that is convex and the vertices of placed on the boundary are points of . To any boundary triangle of we associate another triangle with two interior sides to coincident with two sides of and the third side is the curvilinear arc of limited by the other two sides. Denote by the family formed by these boundary triangles with a curvilinear side and the interior triangles to of , and thus . We now consider the spaces

where denotes the space of the polynomials of degree less than or equal to 1. It is noticed that since we assume the set is convex, the inclusion holds. For any we denote by the unique element of that satisfies (for any ):

Now we state the finite dimensional optimal control problem as follows:

where is the set of vertices of .

From Theorem 5.3.2 in [13], we can prove the discrete solution converges to the solution of (1), as we now show.

Lemma 7. Suppose that (S1) holds. Let there be given a family of finite element spaces as previously described. If, are the solutions of (1), (34), respectively. Then

The following result shows that problem has at least one solution.

Lemma 8. Suppose that (S1) and (S2) hold. For everythere existssuch that has at least one solution for all.

Proof. By the compactness of and the continuity of , we can claim that has one solution if one can show the set of feasible controls is nonempty. In fact, let be a feasible control for and we take such that for almost all as . Then it follows from (36) that uniformly in . Using this uniform convergence, we have that for any . Thus for , there exists a constant such that holds for all and each . That is to say that is feasible for and thus the proof is over.

Finally, we will prove the main result in this paper.

Theorem 9. Suppose that (S1) and (S2) hold. Let us assume that is stable and let be as in Lemma 8. Given a family of controls being a solution of , there exist subsequences , with as , and elements such that in the weakly topology of . Each one of these limit points is a solution of . Moreover we have that

Proof. Let be the state corresponding to and we set . Since is a weakly compact subset of the space and , there exists a subsequence such that and weakly in for some . Now we show that is a solution of . Let be the state associated to . Similar to the proof of Lemma 8, since converges to uniformly in and for any , therefore , which shows that is feasible for the problem .
For , with fixed, and we let be a solution of . Since is dense in , there exists sequence such that weakly in . By the uniform convergence , one can claim that there exists such that for every and . For any fixed we can take a sequence such that for almost all . It follows from the uniform convergence and (for each ) that is a feasible control for . Hence, we have that whenever . Therefore, we getNow passing to the limit as , we gain that . Finally, from the feasibility of for and the stability condition (Definition 3), we conclude thatwhich shows that is a solution of . The rest of this theorem is obvious.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Funding

This work was partially supported by the National Natural Science Foundation of China under Grants 11726619, 11726620, 11601213, the Natural Science Foundation of Guangdong Province under Grant 2018A0303070012, and the Key Subject Program of Lingnan Normal University (No. 1171518004).