Numerical Analysis of a Distributed Optimal Control Problem Governed by an Elliptic Variational Inequality

Abstract

The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energy . The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positive (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter goes to zero.

1. Introduction

We consider a bounded domain whose regular boundary consists of the union of two disjoint portions and with . We consider the following free boundary problem :where the function in (1) can be considered as the internal energy in , is the constant temperature on , and is the heat flux on . The variational formulation of the above problem is given as follows: find such that whereWe note that is bilinear, continuous, and symmetric on and a coercive form on [1]; that is to say, there exists a constant such thatIn [2], the following continuous distributed optimal control problem associated with or the elliptic variational inequality (3) was considered as follows.

Problem . Find the continuous distributed optimal control such that [3–5]where the quadratic cost functional is defined bywith , a given constant, and is the corresponding solution of the elliptic variational inequality (3) associated with the control .

Several continuous optimal control problems are governed by elliptic variational inequalities, for example, the process of biological waste-water treatment; reorientation of a satellite by propellers; and economics: the problem of consumer regulation of a monopoly and so forth. There exists an abundant literature for optimal control problems governed by elliptic variational equalities or inequalities [6–12], for numerical analysis of variational inequalities or optimal control problems [13–16].

The objective of this work is to make the numerical analysis of the optimal control problem which is governed by the elliptic variational inequality (3) by proving the convergence of a discrete solution to the continuous optimal control problems.

In Section 2, we establish the discrete elliptic variational inequality (10) which is the discrete formulation of the continuous elliptic variational inequality (3), and we obtain that these discrete problems have unique solutions for all positive . Moreover, on the adequate functional spaces these solutions are convergent when to the solutions of the continuous elliptic variational inequality (3).

In Section 3, we define the discrete optimal control problem (31) corresponding to continuous optimal control problem (6). We prove the existence of a discrete solution for the optimal control problem () for each parameter and we obtain the convergence of this family with its corresponding discrete state system to the continuous optimal control with the corresponding continuous state system of the problem ().

2. Discretization of the Problem

Let be a bounded polygonal domain; a positive constant; and a regular triangulation with Lagrange triangles of type 1, constituted by affine-equivalent finite elements of class over , being the parameter of the finite element approximation which goes to zero [17, 18]. We take equal to the longest side of the triangles and we can approximate the sets and bywhere is the set of the polynomials of degree less than or equal to in the triangle . Let be the corresponding linear interpolation operator and a constant (independent of the parameter ) such that, , [17]:The discrete variational inequality formulation of system is defined as follows: find such that

Theorem 1. Let , , and ; then there exists unique solution of the problem given by the elliptic variational inequality (10).

Proof. It follows from the application of Lax-Milgram Theorem [1].

Lemma 2. Let and , be the solutions of for and , respectively; then one has that(a)there exists a constant independent of such that(b)(c)if in weak, then in strong for each fixed .

Proof. (a) If we consider in the discrete elliptic variational inequality (10) we havewhere is the trace operator and therefore (11) holds.
(b) As and are, respectively, the solutions of discrete elliptic variational inequalities (10) for and  , we havefor . By coerciveness of we deducethus (12) holds.
(c) Let . From item (a) we have that ; then there exist such that in weak (in strong). If we consider the discrete elliptic inequality (10) we haveand using the fact that is a lower weak semicontinuous application then, when goes to infinity, we obtain thatand from uniqueness of the solution of problem , we deduce that .
Now, it is easily to see thatand from the coerciveness of we obtainAs in and in , by passing to the limit when in the previous inequality, we obtain

Henceforth we will consider the following definitions [2]: given and , we have the convex combinations of two data itemsthe convex combination of two discrete solutionsand we define as the associated state system which is the solution of the discrete elliptic variational inequality (10) for the control .

Then, we have the following properties.

Lemma 3. Given the controls , one has that(a) (b) 

Proof. (a) From definition (22) we getand then we conclude (23).
(b) It follows from a similar method to part (a).

Theorem 4. If and are the solutions of the elliptic variational inequalities (3) and (10), respectively, for the control , then in strong when .

Proof. From Lemma 2 we have that there exists a constant independent of such that , and then we conclude that there exists so that in weak as and . On the other hand, given there exist such that for each and in strong when goes to zero. Now, by considering in the discrete elliptic variational inequality (10) we getand when we pass to the limit as in (26) by using the fact that bilinear form is lower weak semicontinuous in we obtain that it is to say, and, from the uniqueness of the solution of the discrete elliptic variational inequality (3), we obtain that .
Now, we will prove the strong convergence. If we consider in the elliptic variational inequality (3) and in (10), from the coerciveness of and by some mathematical computation, we obtain thatthen by passing to the limit when it results in .

3. Discretization of the Optimal Control Problem

Now, we consider the continuous optimal control problem which was established in (6). The associated discrete cost functional is defined by the following expression:and we establish the discrete optimal control problem as follows: find such thatwhere is the associated state system solution of the problem which was described for the discrete elliptic variational inequality (10) for a given control .

Theorem 5. Given the control , one has(a) (b) for some constant independent of ;(c)the functional is a lower weakly semicontinuous application in ;(d)there exists a solution of the discrete optimal control problem (31) for all .

Proof. (a) From the definition of we obtain (a) and (b).
(c) Let in weak; then by using the equality we obtain that . Therefore, we have(d) It follows from [4].

Lemma 6. If the continuous state system has the regularity then one has the following estimations :(a) (b) where ’s are constants independent of .

Proof. (a) As , we have that . If we consider in (10), by using the inequalities (29), we obtainand then (34) holds.
(b) From the definitions of and , it results inand therefore

Following the idea given in [2] we define an open problem: given the controls and ,

Remark 7. We have that (39)⇒(40).

Remark 8. The equivalent inequality (39) for the continuous optimal control problem is true; that is [2], for all , and ,where , is the unique solution of the elliptic variational inequality (3) when we consider instead of and is the unique solution of the elliptic variational inequality (3) when we consider instead of .

Remark 9. If (40) (or (39)) is true, then the functional is -elliptic and a strictly convex application because we haveand therefore, the uniqueness for the discrete optimal control problem () holds in Theorem 5.

Now, we will show the convergence result for optimal control problems governed by elliptic variational inequalities in order to generalize the result for optimal control problems governed by elliptic variational equalities [19]. We remark that there exist a few numbers of papers for the numerical analysis of optimal control problems governed by elliptic variational inequalities, for example [20–22].

Theorem 10. Let be the continuous state system associated with the optimal control which is the solution of the continuous distributed optimal control problem (6). If, for each , one chooses an optimal control which is the solution of the discrete distributed optimal control problem (31) and its corresponding discrete state system , one obtains that

Proof. Let and let be a solution of (31), and let be its associated discrete optimal state system which is the solution of the discrete elliptic variational inequality (10) for each . From (30) we have that for all Then, if we consider and its corresponding associated state system, it results in the following:From Lemma 2 we have that ; then we can obtainIf we consider in inequality (10) for , we obtainthereforeand from the coerciveness of the application we have that and in consequence .
Now we can say that there exist and such that in weak (in strong), and in weak when . Then, and in ; that is, .
Letting , there exist such that in strong when . Then, if we consider the variational elliptic inequality (10) for we haveTaking into account that the application is a lower weak semicontinuous application in and by passing to the limit when goes to zero in (49) we obtain thatand by the uniqueness of the solution of the problem given by the elliptic variational inequality (3), we deduce that .
Finally, the norm on is a lower semicontinuous application in the weak topology; then we can prove thatand because of the uniqueness of the optimal problem (6), it results in and .
Now, if we consider in the elliptic variational inequality (3) for the control and we define , we have thatand by considering for in inequality (10) we obtainand then by the coerciveness of we getWhen we pass to the limit as in (54) and by using the strong convergence of to on and the weak convergence of to on , we haveThe strong convergence of the optimal controls to is obtained by using Theorem 5 and weakly on ; that is,then and therefore .

4. Conclusions

We have proved the convergence of a discrete optimal control and its corresponding discrete state system governed by a discrete elliptic variational inequality to the continuous optimal control and its corresponding continuous state system which is also governed by a continuous elliptic variational inequality by using the finite element method with Lagrange’s triangles of type 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper has been partially sponsored by Project PIP no. 0534 from CONICET-UA, Rosario, Argentina, and AFOSR-SOARD Grant FA9550-14-1-0122.