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
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 . Finite element methods for quasiparabolic Integrodifferential equations problems with a smooth kernel have been discussed in Cui . 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 . 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:
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:
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., ) 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
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 ), 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
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 ) 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 ). Let be the standard Lagrange interpolation operator. For or , and as
Lemma 2.4 (see ). Let be the average interpolation operator defined in (2.25). For or , and as
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
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
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
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 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 .
and the average interpolation operator defined as in (2.25) and . Then, it follows from (2.23) and (3.5) that
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
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
By setting and Swartz inequality, we have
Integrating time from 0 to in (3.38), we obtain