Optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element analysis
© Shen et al.; licensee Springer. 2014
Received: 2 April 2014
Accepted: 27 June 2014
Published: 24 September 2014
In this paper, the mathematical formulation for a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation is established. We first show the existence and regularity for the solution of the optimal control problem. The finite element approximation is based on the optimality conditions, which are also derived. Then the a priori error estimates for its finite element approximation are obtained with the optimal convergence order. Furthermore some numerical tests are presented to verify the theoretical results.
The distributed optimal control problem has been a classic research topic in the discipline of applied mathematics. Since it is normally difficult to obtain a closed form solution, finite element approximations of optimal control problems governed by partial differential equations have been extensively studied in the literature. In particular, there have been extensive studies in convergence and a priori error estimates of the standard finite element approximation of optimal control problems; see for instance, –, although it is impossible to give even a very brief review here.
For optimal control problems governed by classic linear PDEs such as elliptic, parabolic and hyperbolic equations, the existence and the optimality conditions are well known, see . Furthermore their finite element approximation and a priori error estimates were established long ago, for example, see –, . Recently research has been carried out for the control governed by the integro-differential equations such as elliptic and parabolic integro-differential equations; see , . However, there exists little research on the optimal control problem governed by hyperbolic integro-differential equations, in spite of the fact that such control problems are widely encountered in practical engineering applications and scientific computations. Integro-differential equations and their control of this nature appear in applications such as heat conduction in materials with memory, population dynamics, and visco-elasticity; cf., e.g., –. The physical backgrounds and the existence and uniqueness of the solution of the hyperbolic integro-differential equations have been studied in –. One very important characteristic of all these models is that they all express conservation of a certain quantity; mass, momentum, heat etc. in any moment for any subdomain.
Furthermore the finite element approximation of optimal control problem governed by hyperbolic integro-differential equations has not been studied yet, although there exists much research on the finite element approximation of hyperbolic integro-differential equations, see, e.g., .
The purpose of this paper is to investigate the weak formulation of the optimal control problem governed by integro-differential equations of hyperbolic type, and then its finite element approximation. Furthermore we derive the optimality conditions and establish the a priori error estimates for the constrained optimal control problems. Finally we present some numerical tests to verify the theoretical results.
The outline of the paper is as follows. In Section 2, we present the weak formulation and prove the existence of the solution for the optimal control problem. In Section 3, we present the optimality conditions and the finite element approximation. In Section 4, we establish the optimal a priori error estimates for the finite element approximation of the control problem. Finally, we present some numerical tests, which illustrate the theoretical results.
2 Model problem and its weak formulation
Let , with the Lipschitz boundary , and be bounded open sets in , , and . We introduce some Sobolev spaces. Throughout the paper, we adopt the standard notation for Sobolev spaces on with norm , and semi-norm . Set . Also denote by , with norm , and semi-norm . 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 unknowns and the mesh parameters introduced later.
To fix ideas, we will take the state space with and the control space with . Let the observation space be with . Let be a convex subset.
where is the control, is the state, is a closed convex subset with the respect to the control, , , and are some suitable functions to be specified later. is a linear strongly elliptic self-adjoint partial differential operator of second order with coefficients depending smoothly on the spatial variables, and is an arbitrary second-order linear partial differential operator, with coefficients depending smoothly on both time and spatial variables in the closure of their respective domains; is a suitable continuous operator. A precise formulation of this problem is given later.
Here we assume is a convex functional which is continuously differentiable on , and is a strictly convex continuously differentiable functional on . We further assume that as and that is bounded below. Details will be specified later.
In the case that , , the dual pair is understood as .
because is a bounded linear operator.
In the following, we will give the existence and uniqueness of the solution of the system (2.7).
Assume that the above conditions (a)-(d) hold. There exists a unique solutionfor the minimization problem (2.7) such that, , , .
This means .
So is one solution of (2.7). Since is a strictly convex function on , hence the solution of the minimization problem (2.7) is unique. □
The following theorem states the regularity of the solution of (2.7).
Assume that the above condition (a)-(e) holds andis an-regularity elliptic operator of second order and, . Then the solution of (2.7) is regular in the sense that, , .
Thus by the Gronwall lemmas, . This completes the proof of Theorem 2.2. □
In this paper, we suppose that is independent of . The above results also hold for the case provided suitable smoothness of the operator is assumed.
3 The optimality conditions and its finite element approximation
In this section, we study the optimality conditions and the finite element approximation for the optimal control problem governed by hyperbolic integro-differential equation.
where is the observation.
3.1 The optimality conditions of model problem
The following theorem states the optimality conditions of the problem (2.7).
whereis independent with. is the adjoint operator of.
where is defined in (3.14). This completes the proof of Theorem 3.1. □
3.2 Finite element approximation
In the following, we discuss the finite element approximation of the control problem (2.7). 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 , . 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 of continuity. Let . It is easy to see that . Let denote the maximum diameter of the element in . To simplify our presentation we here only consider the piecewise constant finite element space for the approximation of the control. Let denote all the zeroth-order polynomial over . Therefore we always take . is a closed convex set in . For ease of exposition, in this paper we assume that .
where , , and are the approximations of and .
Since (3.19) is a linear functional equation, and (3.18) is a strictly convex and finite dimensional optimal problem, we can prove that the problem (3.18)-(3.19) has a unique solution in the same way as proving the uniqueness of the solution of (2.1)-(2.2).
In the next sections, we will analyze the a priori error estimates of the approximation solution.
4 A priori error analysis
In the following, we will give the a priori error estimates in -norm. We first present some lemmas.
Letbe given by (4.1) or (4.2). Thenfor any.
Since is the standard finite element of , from , we get the following results.
From (4.14), (4.15), and Lemma 4.2, we only need to estimate .
we need the estimate .
Thus we complete the proof of Lemma 4.3. □
Then from Lemma 4.1, Lemma 4.2, and the triangle inequality, we have the following.
5 Numerical experiment
In this section, we carry out a numerical experiment to verify the a priori error estimates derived in Section 4. The numerical tests were done by using AFEpack software package (see ).
In the numerical example, we take . We use linear finite element spaces to approximate the state and co-state, and the piecewise constant finite element spaces to approximate the control. For the time variable, a Euler backward-difference procedure is used to solve the discrete system. Here the time step size is controlled to demonstrate the relation between the error function and the spatial sizes.
Numerical result: for adaptive time steps 50
From Table 1, we see that the -norm convergent rate of the control variable is , i.e., we have first-order accuracy with respect to the spatial size; the -norm convergent rate of the state and co-state variables and also are ; and the -norm convergent rate of the state and co-state approximation errors and are , consistent with our theoretical analysis.
In this paper, a quadratic optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element approximation are investigated for the first time. By selecting suitable state and control spaces, and defining the bilinear forms, the mathematical formulation is established. Then a priori estimates have been carried out using the standard functional analysis techniques, and the existence and regularity of the solution are provided by using these estimates. We then approximate the optimal control using the standard finite element method and study the approximation errors. Based on these studies, a priori error estimates with the optimal convergence rates are derived. Finally numerical results are presented. Through our investigation, it is clear the standard finite element method works well, both from the point of view of theory and practice, for the quadratic optimal control governed by a linear hyperbolic integro-differential equation when there is no convection term present. However, when there exists strong convection, it is very likely that very different finite element approximation schemes need to be used.
The authors express their thanks to the referees for their valuable comments and suggestions. This work is supported by National Natural Science Foundation of China (Grant: 11326226), Science and Technology Development Planning Project of Shandong Province (No. 2012G0022206) and Nature Science Foundation of Shandong Province (No. ZR2012GM018).
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