Table of Contents
ISRN Applied Mathematics
Volume 2014, Article ID 579047, 12 pages
http://dx.doi.org/10.1155/2014/579047
Research Article

Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

1Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China
2School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China

Received 16 September 2013; Accepted 12 December 2013; Published 10 February 2014

Academic Editors: M. Braack, Y. M. Cheng, and W. Yeih

Copyright © 2014 Yongquan Dai and Yanping Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables.

1. Introduction

Finite element approximation of optimal control problems plays a very important role in numerical methods for these problems. There have been extensive studies on this aspect, for example, [114]. A systematic introduction of finite element method for PDEs and optimal control problems can be found in, for example, [1518]. The superconvergence of nonlinear parabolic problem was studied in [19]. In [20], superconvergence was obtained for parabolic optimal control problems with convex control constraints, where the state partial differential equations are linear.

Optimal control problems governed by nonlinear parabolic state equations, a priori error estimates of finite element approximation, were studied in, for example, [21, 22]. In this paper, we will study the superconvergence of both the control variable and the state variables for this problem.

The model optimal control problem that we shall study in detail is the following convex optimal control problem: Here, the bounded open set is a convex polygon or has the smooth boundary . Let be a linear continuous operator from to and . Assume that , with being a symmetric matrix and, for any vector , there is a constant satisfying Here, denotes the admissible set of the control variable, which is defined by

In this paper, we adopt the standard notation for Sobolev spaces on with a norm given by a semi-norm given by We set . For , we denote

We denote by the Banach space of all integrable functions from into , with norm and the standard modification for , where . Similarly, one can define the spaces and . The details can be found in [23]. In addition, and denote general positive constants independent of .

The plan of the paper is as follows. In Section 2, we shall give a brief review on the finite element method and then construct the approximation scheme for the optimal control problem. In Section 3, we shall give some preliminaries and some intermediate error estimates. In Section 4, superconvergence results for both control and state variables were derived. In Section 5, we give a numerical example to demonstrate our theoretical results. In the last section we make a conclusion and state some future works.

2. Approximation for the Optimal Control Problem

To fix idea, we shall take the state space with and the control space with . Then the problem (1) can be restated as where , , and It follows from the assumption on that there is a positive constant such that , for all .

We make the following assumptions.(1) for any , for any , and .(2)Let , where is a smooth and convex function such that and . The function has the same property as .

It is well known (see, e.g., [24]) that the control problem (8)–(10) has a solution and that if a pair is the solution of (8)–(10), then there is a costate such that the triplet satisfies the following optimality conditions for : where is the adjoint operator of .

In the following we construct the finite element approximation for the optimal control problem (8)–(10). For ease of exposition we will assume that is a convex polygon. Let be a quasi-uniform (in the sense of [25]) partition of into triangles. And let be the maximum diameter of in . Moreover, we set where is the space of polynomial of degree less than or equal to 1.

For simplicity, in this paper we shall assume that . Now, the semidiscrete finite element approximation of the problem (8)–(10) is as follows: where and is an approximation of which will be defined below. The control problem (16)–(18) has a solution , and if a pair is the solution of (16)–(18), then there is a costate such that the triplet satisfies the following optimal conditions:

3. Some Preliminaries

First, we shall use some intermediate variables. For any , let be the solution of the following equations:

Then, for any , let be the solution of the following equations: Thus, we have

We define the standard -orthogonal projection , which satisfies, for any ,

Next, let us recall the elliptic projection , which satisfies, for any , Let

We have the approximation properties:

Now, we establish the following error estimates for the intermediate variables.

Lemma 1. Let be the solution of (12)–(14); for sufficiently small, there exists a positive constant which only depends on and , such that

Proof. Let and in (22)-(23), respectively; then we have the following error equations: for any and . We shall estimate (31) and (32), respectively.
First, let in (33); we have namely, Now, we estimate the right side of (36). Using the continuity of and (29), we have Combining (36)-(37), using the -Cauchy’s inequality and the assumption of and , we have Notice that then, integrating (38) in time and using Gronwall’s lemma, we have
Then, by choosing in (34), we have namely, Now, we estimate the right side of (42). From the assumption of the convex function , we have and using the assumption of and -Cauchy’s inequality, we have where we used the embedding . Combining (42)–(44) and from the assumption of and , we have Notice that then integrating (45) in time, using Gronwall’s lemma and (31), we have which completes the proof of Lemma 1.

Lemma 2. For any , if the intermediate solution satisfies then, one has

Proof. From (22)-(23) and (24), we have the following error equations: for any and . Using the definition of , the above equation can be restated as Let in (52); we have For the first term of (54), using the definition of , we have Similarly, from the assumption of , we can obtain
Combining (54)–(56), using -Cauchy’s inequality, and from the assumption of and , we have
Notice that Then, integrating (57) in time and using Gronwall’s lemma, we have
Let in (53); we have Now we estimate the right side of (60): where we used the assumption of : where we used the definition of : where we used the embedding and the assumption of : where we used the assumption of and the definition of .
Notice that then, combining (60)–(64), using -Cauchy’s inequality, and the assumption of and , we have Integrating (66) in time and using Gronwall lemma, we have which implies (50). Thus, we complete the proof of Lemma 2.

4. Superconvergence Properties

In this section, we will discuss the superconvergence for both the control variable and the state variables by using the results we have got before. Let be the average operator defined in [26]. Let In this paper, we assume that and are regular such that meas  .

Let and be the solution of (9) and (17), respectively. Set Then the reduced problems of (8) and (16) read as respectively. It can be shown that where is the solution of (22)-(23) with .

In many applications, is uniform convex near the solution . The convexity of is closely related to the second order sufficient conditions of the control problem, which are assumed in many studies on numerical methods of the problem. For instance, in some applications, is convex; see [27] for examples. Thus if is uniform convex (e.g., ), which is frequently met, then, there is a constant , independent of , such that where and are solutions of (70) and (71), respectively, is the orthogonal projection of which is defined in (26). We shall assume the above inequality throughout this paper.

First, we are going to formulate the superconvergence result for the control variable.

Theorem 3. Let be the solution of (12)–(14) and let be the solution of (19)–(21). One assumes that the exact control and state solution satisfy Then, one has

Proof. Let in (14) and in (21) and then, add the two inequalities; we have Hence, For the second term of the right hand of (77), we divide it into four parts: then, from (77)-(78), we have
Using the definition of and the assumption of , we have
From Taylor’s expansion of the function , there exists some value such that where we used the assumption of and the approximation property (29).
Notice that
Obviously, . From (14), we have pointwise a.e. ; we choose and , so that . Hence, . Then, From the assumption of , we have Then, integrating (79) in time and combining Lemmas 1 and 2 and (79)–(84), we have where we have used -Cauchy’s inequality which implies (75). Thus, we complete the proof of Theorem 3.

In the following, we shall establish the superconvergence results for the state variable and costate variable .

Theorem 4. Let be the solution of (12)–(14) and let be the solution of (19)–(21). One assumes that the exact control and state solution satisfy Then, one has

Proof. First, we have the following error equation from (12) and (19): for any .
Using the definition of in (27), we have for any .
We take in (90), and using the assumption of and , then Now, we estimate the right hand of (91). From (29)-(30), (75), and using -Cauchy’s inequality, we have
Then, using the assumption of , we have
Therefore, inserting (92)–(94) in (91), we have
Notice that then, integrating (95) in time, using Gronwall’s lemma, and from the result of Theorem 3, we can easily obtain that which implies (87).
Then, from (13) and (20), we have the following error equation for any . Using the definition of in (27), we have We take , and using the assumption of and , then Now, we estimate the right hand of (100). From the assumption of , we have Using the definition of , we have From the assumption of and the definition of , we can obtain From the assumption of and -Cauchy’s inequality, we have
Therefore, inserting (101)–(104) in (100), we have Notice that then, integrating (105) in time, using Gronwall’s lemma and (97), we have Thus, we complete the proof of Theorem 4.

5. Numerical Example

In this section, we carry out a numerical example to demonstrate our theoretical results. The optimal problem was solved numerically by a precondition projection algorithm; see, for instance, [28], with codes developed based on AFEPack [29]. In order to validate the superconvergence results, we shall consider the following full-discrete scheme. Let , , being the time-step, and , the integral part of . In the example, we choose the domain and .

We now shall consider the fully discrete approximation for semidiscrete problem (19)–(21) by using the backward Euler scheme in time. The scheme is as follows: find such that