Abstract and Applied Analysis

Volume 2012 (2012), Article ID 808514, 26 pages

doi:10.1155/2012/808514

Research Article

Adaptive Finite Element Method for Optimal Control Problem Governed by Linear Quasiparabolic Integrodifferential Equations

Wanfang Shen1,2 and Hua Su²

¹School of Mathematics, Shandong University, Jinan 250100, China

²School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China

Received 21 June 2012; Accepted 10 July 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Wanfang Shen and Hua Su. 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.

The mathematical formulation for a quadratic optimal control problem governed by a linear quasiparabolic integrodifferential equation is studied. The control constrains are given in an integral sense: . Then the a posteriori error estimates in -norm and -norm for both the state and the control approximation are given.

1. Introduction

Integrodifferential equations of quasiparabolic and their control of this nature appear in applications such as biology mechanics, nuclear reaction dynamics, heat conduction in materials with memory, and viscoelasticity. All these models express a conservation of a certain quantity in any moment for any subdomain and the historical accumulation feature in the physical models. This in many applications is the most desirable feature of the approximation method when it comes to numerical solution of the corresponding initial boundary value problem. The existence and uniqueness of the solution of the quasiparabolic Integrodifferential equations has been studied in [1]. Finite element methods for quasiparabolic Integrodifferential equations problems with a smooth kernel have been discussed in Cui [2]. Although there is so much work for the finite element approximation of this problem, to our knowledge, there has been a lack of a posteriori error estimates for finite element approximation of any quasiparabolic Integrodifferential optimal control problem.

The finite element approximation of optimal control problems has been an important topic in engineering design works. There have been extensive theoretical and numerical studies for various optimal control problems, see, for instance, [3–11], although it is impossible to give even a very brief review here. And research on finite element approximation of parabolic optimal control problems can be found in, for example, [12, 13].

Among many finite element methods, the adaptive finite element method based on a posteriori error estimates has become a central theme in scientific and engineering computations for its high efficiency. In order to obtain a numerical solution of acceptable accuracy, it is essential for the adaptive finite element method to use a posteriori error estimate indicators to guide the mesh refinement procedure. We only need refine the area where the error indicators are larger, so that a higher density of nodes are distributed over the area where the solution is difficult to approximate. In this sense, adaptive finite element approximation relies very much on the error indicators used, which are often based on a posteriori error estimates of the solutions.

The purpose of this paper is to derive the a posteriori error estimates for the semidiscrete finite element approximation of a quadratic optimal control problem governed by a linear quasiparabolic Integrodifferential equation, which paves a way to derive the a posteriori error estimates for the full discrete finite element approximation for this control problem and thus to develop its adaptive finite element schemes. We extend the existing techniques and results in [14–16] to the optimal control problem governed by the Integrodifferential equation of quasiparabolic type.

The outline of the paper is as follows. In Section 2, we first briefly introduce the optimal control problem and give the optimality conditions, then construct the finite element approximation schemes for the optimal control problem. In Section 3, we give the a posteriori error bounds in -norm for the control problem. And the a posteriori error bounds in -norm for the control problem are derived in Section 4.

2. Optimal Control Problem and Its Finite Element Approximation

Let and be bounded convex polygon domains in with Lipschitz boundary and . In this paper, we adopt the standard notation for Sobolev spaces on with norm , and seminorm  . We set . We denote by , with norm  , and seminorm  .

We denote by the Banach space of all integrable functions from into with norm for and the standard modification for . Similarly, one can define the spaces and . The details can be found in [17]. In addition, or denotes a general positive constant independent of the mesh size .

In the following, we will give semi-discrete finite element approximation schemes for the optimal control problem governed by a linear quasiparabolic Integrodifferential equation.

2.1. Model Problem and Its Weak Formulation

We will take the state space with and the control space with . Let the observation space with and a convex subset.

We are interested in the following optimal control problem:

subject to

where is control, is state, is the observation, is a closed convex subset, , and and are some suitable functions to be specified later. is a linear bounded operator from to independent of . And

such that there is a constant satisfying that for any vector as follows:

.

Let

In the case that and , the dual pair is understood as .

Assume that there are constants and , such that for all and in as follows:

for any and in .

Then the weak form of the state equation reads as

It is well known (see, e.g., [1]) that the above weak formulation has at least one solution in  .

Therefore, the weak form of the control problem (2.1) and (2.2) reads as (OCP)

In the following, we first give the existence and uniqueness of the solution of the system (2.8).

Theorem 2.1. Assume that the condition (2.6) (a)–(c) holds. There exists the unique solution for the minimization problem (2.8) such that , , and .

Proof. Let be a minimization sequence for the system (2.8), then the sequence is bounded in . Thus there is a subsequence of (still denote by ) such that converges to weakly in . For the subsequence , we have

By setting and integrating from to in (2.9), we give

Applying Gronwall's inequality to (2.10) yields

So is a bounded set in and is a bounded set in . Thus

Let .

By integrating time from to in (2.9) and taking limit as , we obtain

Then,

Furthermore, we have

Then, we get

This means . So is one solution of (2.8).

Since is a convex function on space and is a strictly convex function on , hence is a strictly convex function on , so the minimization problem (2.8) has one unique solution.

2.2. Optimality Conditions and Their Finite Element Approximation

By the theory of optimal control problem (see [18]), we can similarly deduce the following optimality conditions of the problem (2.8).

Theorem 2.2. A pair is the solution of the optimal control problem (2.8), if and only if there exists a costate such that the triple satisfies the following optimality conditions:

where is the adjoint operator of .

Let us consider the semi-discrete finite element approximation of the control problem (2.8). Here, we only consider triangular and conforming elements.

Let be a polygonal approximation to with boundary  . Let be a partitioning of into disjoint regular -simplices , so that . Each element has at most one face on , and and have either only one common vertex or a whole edge or face if and . We further require that   where is the vertex set associated with the triangulation . As usual, denotes the diameter of the triangulation . For simplicity, we assume that is a convex polygon so that .

Associated with is a finite-dimensional subspace of , such that are polynomials of order for all and . Let . Note that we do not impose a continuity requirement. It is easy to see that .

Let be a partitioning of into disjoint regular -simplices , so that . and have either only one common vertex or a whole edge or face if and . We further require that , where is the vertex set associated with the triangulation . For simplicity, we again assume that is a convex polygon so that .

Associated with is another finite-dimensional subspace of , such that are polynomials of order for all and . Here there is no requirement for the continuity. Let . It is easy to see that . Let denote the maximum diameter of the element in .

Due to the limited regularity of the optimal control in general, there will be no advantage in considering higher-order finite element spaces than the piecewise constant space for the control. We therefore only consider the piecewise constant finite element space for the approximation of the control, though higher-order finite element spaces will be used to approximate the state and the co-state. Let denote all the 0-order polynomial over . Therefore, we always take . is a closed convex set in . For ease of exposition, in this paper, we assume that .

Then a possible semi-discrete finite element approximation of is as follows

such that

where and is the approximation of .

In the same way of proving Theorem 2.1, we can easily prove that the problem (2.20)-(2.21) has a unique solution .

It is well known (see [18]) that a pair is a solution of (2.20)-(2.21), if and only if there exists a co-state such that the triple satisfies the following optimality conditions:

The optimality conditions in (2.22)–(2.24) are the semi-discrete approximation to the problem (2.17)–(2.19).

Introduce the local averaging operator given by

Then, we have for any and (2.24) is equivalent to

In the following, we derive the a posteriori error estimates for semi-discrete finite element approximation (2.22)–(2.24), allowing different meshes to be used for the state and the control.

The following lemmas are important in deriving the a posteriori error estimates of residual type.

Lemma 2.3 (see [19]). Let be the standard Lagrange interpolation operator. For or , and as

Lemma 2.4 (see [20]). Let be the average interpolation operator defined in (2.25). For or , and as

Lemma 2.5 (see [21]). For all as

3. A Posteriori Error Estimates in -Norm

In this paper, the control constraints are given in an integral sense as follows:

The following lemma is the first step to derive the a posteriori error estimates of residual type.

Lemma 3.1. Let and be the solutions of (2.17)–(2.19) and (2.22)–(2.24). Then, we have

where

is the -projection from to , and is defined by the following system:

Proof. From (2.19), we have

Then, by (2.24) and (3.6), we have

Next, we will estimate , respectively.

(1) We first estimate . Let be the -projection from to .

We have

Since , so , then . So that we can take in .

For given , let

We have . We will show that is the solution of the variational inequality in (2.24) assuming is known.

Since , we have

Thus, , we have . Note that for all , we have

If , then

If , since

we have

From (3.11)–(3.14), we obtain

So is the solution of the variational inequality in (2.24) assuming is known.

Then,

Since , we have

Consider

By (3.4) and (2.17), we have for

and from (3.5) and (2.18), we have

Then, from (3.19), (3.20), and integrating by part we have

Following from (3.17)–(3.21), let be small enough as

This completes the proof.

Lemma 3.2. Let and be the solutions of (2.17)–(2.19), and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as

where

where is a face of an element , is the maximum diameter of ,  and   and are the normal derivative jumps over the interior face defined by

where is the unit normal vector on outwards . For later convenience, one can define and when .

Proof. Let

and the average interpolation operator defined as in (2.25) and . Then, it follows from (2.23) and (3.5) that

We have

Taking in (3.28) and from (2.6), we have

Integrating time from to in (3.29) and by Schwartz inequality, Lemmas 2.4 and 2.5, we have

Letting be small enough, we have

Then, from Gronwall inequality and (3.28)–(3.31) we have

Similarly,

In the same way of getting (3.32),by setting in (3.28), we have

Similarly analysis for , we let

From (2.22) and (3.4), we obtain

We have

By setting and Swartz inequality, we have

Integrating time from 0 to in (3.38), we obtain

Since is small enough, then from (3.39) and Gronwall inequality, we have

Then,

In the same way of getting (3.34), we can similarly obtain

Then the desired results (3.23) follow from (3.32)–(3.34) and (3.41)-(3.42).

From Lemmas 3.1 and 3.2, we have the following results.

Theorem 3.3. Let and be the solutions of(2.17)–(2.19) and (2.22)–(2.24) respectively. Then, there hold the a posteriori error estimates as

where is defined in Lemma 3.1.

Proof. First,  from (3.27) and (3.36), and [2], we have the following stability results:

Then, the desired results (3.43) follows from triangle inequality, (3.44) and Lemmas 3.1 and 3.2.

This completes the proof.

4. A Posteriori Error Estimates in -Norm

In the following, we will derive the a posteriori error estimates in -norm.

We will introduce the following dual equations.

For given , we have

and its dual equation

From [1, 2], we have the following stability results.

Lemma 4.1. Assume that is a convex domain. Let and be the solution of (4.1) and (4.2), respectively. Then,

where , and is defined similarly.

Using Lemmas 3.1 and 4.1, we have the following upperbounds.

Lemma 4.2. Let and be the solutions of (2.17)–(2.19) and (2.22)–(2.24), respectively. Then, there hold the a posteriori error estimates as

where is defined in Lemma 3.2, and

Proof. We first estimate .

Let be the solution of (4.1) with ,  and   the interpolation of in Lemma 2.3.

From (4.1), (3.27), and by integrating by parts we obtain

It follows from Lemmas 2.3, 2.5, and 4.1 that

By Schwartz inequality, we have

Letting be small enough, it follows from (4.6)–(4.8) that

Next, we estimate . Similarly let be the solution of (4.2) with ,  and   the interpolation of in Lemma 2.3.

Then, it follows from (3.36) and integrating by parts that

Similarly, it follows from Lemmas 2.3,  2.5 and 4.1 that

Letting be small enough, then from (4.10)–(4.11), we have

The desired results (4.4) follows from (4.9)–(4.12).This completes the proof.

Using Lemmas 3.1 and 4.2, we have the following upper bounds.

Theorem 4.3. Let and be the solutions of (2.17)–(2.19) and (2.22)–(2.24), respectively. Then, there hold the a posteriori error estimates as

Proof. By triangle inequality, (3.44) and (3.38), Lemmas 3.1 and 4.2, we can easily prove (4.13) in the same way of getting (3.43).

This completes the proof.

5. Conclusion

In this paper, we study the semi-discrete adaptive finite element method for optimal control problem governed by a linear quasiparabolic Integrodifferential equation. We extend the existing methods in studying adaptive finite element approximation of optimal control governed by a parabolic Integrodifferential equation to the control governed by a quasiparabolic Integrodifferential equation. After presenting the weak form and the existence and uniqueness of the solution for the optimal control problem, the a posteriori error estimates for semi-discrete finite element approximations in -norm and -norm are derived. The work will pave a way to derive the a posteriori error estimates of full discrete finite element approximations of this optimal control problem

Acknowledgments

This research was supported by Science and Technology Development Planning project of Shandong Province (no. 2011GGH20118), Shandong Province Natural Science Foundation (no. ZR2009AQ004), and National Natural Science Foundation of China (no. 11071141).

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