- Open Access
Optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element analysis
Boundary Value Problems volume 2014, Article number: 173 (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.
We investigate the following optimal control problem governed by a hyperbolic integro-differential equation:
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 order to give the weak formulation of problem mentioned above and study the existence and regularity of the solution, we introduce the -inner products
and the bilinear forms
In the case that , , the dual pair is understood as .
We shall assume the convexity conditions
that is to say, is uniformly convex. Noting that is convex, it is easy to see that
Also, we have
because is a bounded linear operator.
Then a possible weak formulation for the state equation reads
Therefore the control problem (2.1)-(2.2) can be restated as (OCP):
Next, we will analyze the existence, uniqueness, and regularity of the solution of (2.7). Assume that there are constants and , such that for all and in :
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, , , .
Let be a minimization sequence for the system (2.7), then it is clear that are bounded in . Thus there is a subsequence of (still denoted by ) such that converges to weakly in . For the subsequence , we have
Taking in (2.9), we have
Integrating time from 0 to in (2.10), we obtain
From (2.11) and the Gronwall lemmas, we have
So we get
Then by (2.14) and (2.11)
Taking the supermaximum in (2.15), we obtain
Then from (2.14) and (2.16), we also have
Then we have , and . Thus
Integrating time from 0 to in (2.9), we obtain
Taking the limits in (2.18) as , we have
So we have
Further, from (2.9), we obtain
This means .
Since is a convex function on space and is a strictly convex function on , we have
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, , .
Differentiating (2.2) with respect to , we have
and we obtain
Taking in (2.22), we have
Integrating time from 0 to in (2.23), in the same way as getting (2.16) and (2.17), we can deduce
Then and . Further we have
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.
For simplicity, we will only consider the case of quadratic objective functionals as follows:
where is the observation.
3.1 The optimality conditions of model problem
The following theorem states the optimality conditions of the problem (2.7).
A pairis the solution of the optimal control problem (2.7), if and only there exists a co-state, such that the triplesatisfies the following optimality conditions:
whereis independent with. is the adjoint operator of.
Let , where
By the standard method in , the optimal conditions read
Next, we compute . Let us differentiate the state equation (2.7) at in the direction . By (2.7), we have
Taking the limits in (3.9) as , we obtain
where we used the equality that for any ,
Then (3.10) is equivalent to
Define the co-state satisfying
Since , (3.13) is equivalent to
Letting in (3.12), we have
By (3.8) and (3.15), we have
By (3.6)-(3.8), and (3.16), the optimality conditions read
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 .
Then the finite element approximation of is thus defined by :
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).
It is well known that a pair is a solution of (3.18)-(3.19), if and only there exists a co-state such that the triple satisfies the following optimality conditions:
The optimality conditions in (3.20)-(3.22) are the semi-discrete approximation to the problem (3.3)-(3.5). Let be the local averaging operator given by
It is an obvious fact that for any . By the operator , (3.22) is equivalent to
In the next sections, we will analyze the a priori error estimates of the approximation solution.
4 A priori error analysis
For simplicity, we consider the zero obstacle problem:
or the integration obstacle problem:
In the case of (4.1), (3.5) and (3.22) yield
In the case of (4.2), (3.5) and (3.22) yield
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.
Let us introduce the auxiliary problem
Since is the standard finite element of , from , we get the following results.
Letbe the solutions of the systems (4.5)-(4.6). Then we have the a priori error estimates
Letandbe the solutions of the systems (4.5)-(4.6) and (3.20)-(3.22). Then we have the a priori error estimate
From (4.5) and (3.20), we obtain
Similarly, from (4.6) and (3.21), we have
Taking in (4.10), we obtain
Integrating time from 0 to in (4.12) and noting that , , we have
Letting be small enough, we get
By the Gronwall lemma, we have
Similarly letting in (4.11), we also have
From (4.14), (4.15), and Lemma 4.2, we only need to estimate .
we need the estimate .
From (3.5), (3.22), we have
On the one hand, we take in (4.10), and in (4.11), and integrate time from 0 to , to have
On the other hand
Applying the above two estimates, from Lemma 4.2, we can get
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.
Letandbe the solutions of the systems (3.3)-(3.5) and (3.20)-(3.22). Then we have the a priori error estimate:
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.
The numerical example is the following control problem:
The solutions of (5.1)-(5.2) are
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.
Alt W: On the approximation of infinite optimisation problems with an application to optimal control problems. Appl. Math. Optim. 1984, 12: 15-27. 10.1007/BF01449031
Falk FS: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 1973, 44: 28-47. 10.1016/0022-247X(73)90022-X
French DA, King JT: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim. 1991, 12: 299-315. 10.1080/01630569108816430
Malanowski K: Convergence of approximations vs. regularity of solutions for convex, control constrained, optimal control systems. Appl. Math. Optim. 1982, 8: 69-95. 10.1007/BF01447752
Neittaanmäki P, Tiba D: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. Dekker, New York; 1994.
Optimal Shape Design for Elliptic Systems. Springer, Berlin; 1984.
Tiba D: Lectures on the Optimal Control of Elliptic Equations. University of Jyväskylä Press, Jyväskylä; 1995.
Tiba D: Optimal Control of Nonsmooth Distributed Parameter Systems. Springer, Berlin; 1990.
Tiba D, Tröltzsch F: Error estimates for the discretization of state constrained convex control problems. Numer. Funct. Anal. Optim. 1996, 17: 1005-1028. 10.1080/01630569608816739
Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin; 1971.
Hermann B, Yan NN: Finite element methods for optimal control problems governed by integral equations and integro-differential equations. Numer. Math. 2005, 101: 1-27. 10.1007/s00211-005-0608-3
Shen W, Ge L, Yang D: Finite element methods for optimal control problems governed by linear quasi-parabolic integro-differential equations. Int. J. Numer. Anal. Model. 2013, 10: 536-550.
Millor RK: An integro-differential equation for rigid heat conductions with memory. J. Math. Anal. Appl. 1978, 66: 313-332. 10.1016/0022-247X(78)90234-2
London S, Staffans O: Volterra Equations. Springer, Berlin; 1979.
Staffans O: On a nonlinear hyperbolic Volterra equation. SIAM J. Math. Anal. 1980, 11: 793-812. 10.1137/0511071
Dafermos C, Nohel J: Energy methods for nonlinear hyperbolic Volterra integro-differential equations. Commun. Partial Differ. Equ. 1979, 41: 219-278. 10.1080/03605307908820094
Ismatov M: A mixed problem for the equation describing sound propagation in a viscous gas. J. Differ. Equ. 1984, 20: 1023-1035.
Sun P: The finite element methods and application of interpolated techniques for hyperbolic integro-differential equations. Math. Appl. 1996, 9: 433-440.
Zhang T, Li C: Superconvergence of finite element approximations to parabolic and hyperbolic integro-differential equations. Northeast. Math. J. 2001, 17(3):279-288.
Lions JL, Magenes E: Non Homogeneous Boundary Value Problems and Applications. Springer, Berlin; 1972.
Adaptive Finite Elements Methods for Optimal Control Problem Governed by PDEs. Sciences Press, Beijing; 2008.
Li R:On multi-mesh -adaptive algorithm. J. Sci. Comput. 2005, 24: 321-341. 10.1007/s10915-004-4793-5
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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
About this article
Cite this article
Shen, W., Yang, D. & Liu, W. Optimal control problem governed by a linear hyperbolic integro-differential equation and its finite element analysis. Bound Value Probl 2014, 173 (2014). https://doi.org/10.1186/s13661-014-0173-8
- optimal control problem
- linear hyperbolic integro-differential equations
- optimality conditions
- finite element methods
- a priori error estimate