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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 481295, 24 pages
A Posteriori Error Estimates for a Semidiscrete Parabolic Integrodifferential Control on Multimeshes
School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China
Received 18 August 2012; Accepted 26 September 2012
Academic Editor: Hua Su
Copyright © 2012 Wanfang Shen. 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.
We extend the existing techniques to study semidiscrete adaptive finite element approximation schemes for a constrained optimal control problem governed by parabolic integrodifferential equations. The control problem involves time accumulation and the control constrain is given in an integral obstacle sense. We first prove the uniqueness and existence of the solution of this optimal control problem. We then derive the upper a posteriori error estimators for both the state and the control approximation, which are useful indicators in adaptive multimesh finite element approximation schemes.
For the last twenty years great progress has been made on standard finite element approximation of optimal control problems governed by elliptic or parabolic equations; see, for example, [1–9], although it is impossible to give even a very brief review here. More specifically, research on finite element approximation of parabolic optimal control problems can be found in, for example, [10, 11]. Systematic introduction of the finite element method for PDEs and optimal control problems can be found in, for example, [5, 12, 13].
In many real modeling applications it is important to consider time accumulation. Parabolic integrodifferential equations and their control of this nature appear in heat conduction in materials with memory, population dynamics, and viscoelasticity; see, for example, Friedman and Shinbrot , Heard , and Renardy et al. . This calls for more studies on the finite element approximation on integrodifferential equations. For instance, finite element methods for parabolic integrodifferential equations with a smooth kernel have been discussed in, for example, Cannon et al. , Sloan and Thomée , and Yanik and Fairweather .
Recently adaptive finite element method has been widely studied and applied in practical control computations. In order to obtain a numerical solution of acceptable accuracy for the optimal control problem, the finite element meshes have to be refined according to a mesh refinement scheme. Adaptive finite element approximation uses a posteriori indicators to guide the mesh refinement procedure. Only the area where the error indicator is larger will be refined so that a higher density of nodes is distributed over the area where the solution is difficult to approximate. In this sense efficiency and reliability of adaptive finite element approximation rely very much on the error indicator used.
Adaptive finite element schemes have been widely studied for some optimal control problems governed by elliptic and parabolic PDEs as well. Some of recent accounts of progress can be found, for example, in [11, 20, 21]. Although there exists so much progress in adaptive finite element elliptic and parabolic control problems, it is much more complicated to study and implement adaptive multimeshes computational schemes for parabolic integrodifferential control problems. More specificity, there has been a lack of a posteriori error estimates for any parabolic integrodifferential control problem, in spite of the fact that such control problems are widely encountered in practical engineering applications and scientific computations as we discussed above.
In this work, we extend the results and the techniques used in [20–22] to a quadratic optimal control problem governed by a linear parabolic integrodifferential equation, which can be generalized to the control problems with more general objective functions. The semidiscrete finite element scheme for this problem is presented. We derive the upper a posteriori error estimates for the semidiscrete finite element approximation for the case where the control constraints are given in an integral sense: .
We are interested in the following optimal control problems: subject to where and are bounded open sets in with the Lipschitz boundary and , , , , , is a closed convex subset; is a linear strongly elliptic self-adjoint partial differential operator of second order with coefficients depending smoothly on spacial variables, is an arbitrary second-order linear partial differential operator, with coefficients depending smoothly on both of time and spacial variables in the closure of their respective domains, and is a suitable linear and continuous operator.
Here we assume is a convex functional which is continuously differential on , and is a strictly convex continuously differential functional on . We further assume that as and that is bounded below. Details will be specified later.
The plan of the paper is as follows: in Section 2, we give the weak formulation and prove the existence and uniqueness of the solution for this optimal control problem. In Section 3, we will give a brief review on the finite element methods and optimality conditions and construct the semidiscrete finite element approximation schemes for the optimal control problem. In Section 4, the upper a posteriori error bounds in -norm are derived for the control problem. In Section 5, we obtain the a posteriori error bounds in -norm for the control problem.
2. Existence and Uniqueness of the Solution of the Model Problem
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 . In addition or denotes a general positive constant independent of .
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 . To fix idea, we will take the state space with and the control space . Let the observation space be with .
Assume that there are constants and , such that for all , :
We will assume the following convexity conditions: that is to say is uniformly convex.
Noting that is convex, it is easy to see that
Also, we have that because is a bounded linear operator.
Lemma 2.1. For the optimal control problem (2.6), there exists the unique solution , such that and and .
Proof. Assume that is a minimization sequence for the problem (2.6). It follows that are bounded in . Therefore there is a subsequence of (still denoted by ) such that converges to weakly in . It is clear that for the subsequence
By taking and integrating time from to in (2.7), and applying Gronwall's inequality, we have
This infers that and , and
Now let us integrate time from 0 to in (2.7) and take limits as . Clearly we have
Furthermore, we have
It follows that .
Since is a convex function on space and is a strictly convex function on , we have
It follows that is one solution of (2.6). Because is a strictly convex function on , we have that the solution for the minimization problem (2.6) is unique. The proof of Lemma 2.1 is completed.
Remark 2.2. Here we suppose that the operator is independent of time variable . The above results can also be applied to the case provided that suitable conditions for the operator are to be imposed.
3. Finite Element Approximation of Control
In this section, we firstly state the optimality conditions and set up the finite element approximation for optimal control problems governed by parabolic integrodifferential equation.
Theorem 3.1. A pair is the solution of the optimal control problem (2.6), if and only if there exists a costate such that the triple satisfies the following optimality conditions: where is the adjoint operator of .
In the following, we construct the semidiscrete finite element approximation of the control problem (2.6) by approximating the optimality conditions.
Let be a polygonal approximation to with boundary . For simplicity, we assume that is a convex polygon so that . 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 . As usual, denotes the diameter of the triangulation .
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 . For simplicity, we again assume that is a convex polygon so that . and have either only one common vertex or a whole edge or face if and . Let denote the maximum diameter of the element in .
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 .
In this paper we only consider the piecewise constant finite element space for the approximation of the control for the reason of the limited regularity of the optimal control in general. For ease of exposition, in this paper we assume that .
In order to derive a posteriori error estimates of residual type, we need the following important lemmas.
Lemma 3.2 (see ). Let be the standard Lagrange interpolation operator. For or , and ,
Lemma 3.4 (see ). For all ,
Then a possible semidiscrete finite element approximation of (OCP) is thus defined by : subject to where , is the approximation of .
By applying  again we can show that a pair is a solution of (3.7)-(3.8), if and only if there exists a costate such that the triple satisfies the following optimality conditions, which we will label :
4. A Posteriori Error Estimates for First-Order Derivatives
Adaptive finite element approximation has been found very useful in computing optimal control, as mentioned in Introduction. It uses an a posteriori error indicator to guide the mesh refinement procedure. Furthermore it has been recently found that for constrained control problems, different adaptive meshes are often needed for the control and the states; see . Using different adaptive meshes for the control and the state allows very coarse meshes to be used in solving the state and costate equations. Thus much computational work can be saved since one of the major computational loads is to solve the state and costate equations repeatedly. In this section, we derive the upper a posteriori error estimates for the optimal control problem allowing different meshes to be used for the states and the control.
For simplicity, we will only consider the case of quadratic objective functionals as follows:
Here where is a positive regularity constant. By differential theory, we have
In this paper, we consider the integration obstacle type control constraint:
And we will consider the special case of the differential operator of and : where , such that there are constants satisfying and .
In this case, we have the following bilinear form
4.1. Main Results
We first state the main results of this section.
Theorem 4.1. Let and be the solutions of and , respectively. Let and be the solution of the costate equations (3.2) and (3.10). Then there hold the a posteriori error estimates where is defined as follows: where is a face of an element , is the maximum diameter of , and are the normal derivative jumps over the interior face , defined by where is the unit normal vector on outwards . For later convenience, one defines and when .
4.2. Some Lemmas
We have the following.
Proof. It follows from (4.4) that we have
Then by (4.5) and (4.18)
Since is the -projection, then, for any , we have
Note that and . Then we have , thus . So that we can take in and by (4.7), we have
From (4.15) and (3.1), we have for and from (4.17) and (3.2)
Then from (4.23), (4.24) and integrating by part
Following from (4.21)–(4.25), let be small enough:
The proof of Lemma 4.2 is completed.
and be the average interpolation operator defined as in  and . Then from (3.10) and (4.17)
So we have
By letting in (4.30) and from (2.2), we have
Integrating time from to in (4.31) and by Schwartz inequality, Lemmas 3.3 and 3.4, we have
Letting be small enough, we obtain
By Gronwall inequality and (4.30)–(4.33)
Similarly, we have
Then from (4.30), (4.34), and (4.35)
Similarly by analysis of , we let
By (3.9) and (4.15)
By letting and applying Swartz inequality, we have
By integrating time from 0 to in (4.40)
Since is small, then from (4.41) and Gronwall inequality we have and we obtain
In the same way of getting (4.36), we also have
Then the desired results (4.27) follow from (4.34)–(4.36) and (4.43)–(4.44).
4.3. Proof of Theorem 4.1
Proof. By using the triangle inequality, Lemmas 4.2 and 4.3, and from (4.29) and (4.38), [17–19], using the following stability results
we can easily obtain (4.11). The proof of Theorem 4.1 is completed.
5. A Posteriori Error Estimates in Integral
Often we need sharper a posteriori error estimates. Then we need deriving the a posteriori error estimates in -norm. To this end we first state the main results in this paper.
5.1. Main Results
We have the following results.
Theorem 5.1. Let and be the solutions of and , respectively. Let and be the solution of the costate equations (3.2) and (3.10). Then there hold the a posteriori error estimates where is defined in Theorem 4.1 and
In order to prove Theorem 5.1, we need the following dual equations and lemmas.
5.2. Dual Equations and Some Lemmas
For given and the following equation:
we have its dual equation: