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A Priori Error Estimates of Mixed Finite Element Methods for General Linear Hyperbolic Convex Optimal Control Problems
The aim of this work is to investigate the discretization of general linear hyperbolic convex optimal control problems by using the mixed finite element methods. The state and costate are approximated by the order () Raviart-Thomas mixed finite elements and the control is approximated by piecewise polynomials of order . By applying the elliptic projection operators and Gronwall’s lemma, we derive a priori error estimates of optimal order for both the coupled state and the control approximation.
With the advances of scientific computing, optimal control problems are now widely used in multidisciplinary applications such as physics, biology, medicine, engineering design, fluid mechanics, and social-economic systems. The finite element method is undoubtedly the most widely used numerical method in computing optimal control problems. Finite element approximation of a class of elliptic optimal control problems has been studied by Falk in . Then, Alt and Mackenroth in  established a priori error estimates for the finite element approximations to state constrained convex parabolic boundary control problems. Finite element approximation of optimal control problems was developed in [3–16], but there are very less published results on this topic for hyperbolic optimal control problems.
Since the pioneering work of Brezzi and Fortin , the mixed finite element methods to second order elliptic problems have drawn the attention of many specialists in partial differential equations. Mixed finite elements are appropriate for the state equations in such cases since both the scalar variable and its flux variable can be approximated to the same accuracy. In finite element methods, mixed finite element methods were widely used to approximate flux variables, although there was only very limited research work on analyzing such elements for optimal control problems. More recently, in , the authors derived a priori error estimates and superconvergence for bilinear quadratic optimal control problems using mixed finite element methods. A posteriori error analysis of mixed finite element methods for some optimal control problems was addressed in [18, 19]. In , the author discussed the semidiscrete mixed finite element methods for quadratic hyperbolic optimal control problems. By using mixed elliptic reconstruction methods, he obtained a posteriori -error estimates for both the state and the control approximation.
The purpose of this work is to obtain a priori error estimates of mixed finite element methods for general convex optimal control problems governed by linear hyperbolic partial differential equations. Analogous a priori error estimates of mixed finite element solutions for optimal control problems governed by linear parabolic equations can be found in . However, it does not seem to be straightforward to extend the existing techniques to general optimal control problems involving hyperbolic equations.
For and any nonnegative integer, let denote the Sobolev spaces endowed with the norm and the seminorm . We set . For , we denote , and , . We denote by the Banach space of all integrable functions from to with norm , , and the standard modification for .
In this paper, we focus our attention on the following general linear hyperbolic convex optimal control problems: subject to the state equations where the bounded open set is a convex polygon with the boundary , is a bounded open set in with the Lipschitz boundary , , , and are convex functionals, and . We assume that is a closed convex set in , , and . Furthermore, we assume the coefficient matrix is a symmetric -matrix and there is a constant satisfying for any vector , . The set of admissible controls is defined by
The remainder of the paper is organized as follows. In Section 2, we construct the order Raviart-Thomas mixed finite element approximation for general convex optimal control problems governed by linear hyperbolic equations and briefly state the definitions and properties of some interpolation operators. In Section 3, we derive a priori error estimates of the mixed finite element solutions for the general hyperbolic optimal control problems. Finally, we give the conclusion and the future work in Section 4.
2. Mixed Methods of Hyperbolic Optimal Control
We will now describe the mixed finite element discretization of general linear hyperbolic convex optimal control problems (1)–(6). Firstly, we introduce the costate hyperbolic equation, with the conditions,
We will take the state spaces and , where and are defined as follows: The Hilbert space is equipped with the following norm:
Taking into account the precious result in [20, 22], the optimal control problem (13) has a unique solution , and a triplet is the solution of (13) if and only if there is a costate such that satisfies the following optimality conditions: where is the inner product of and , , and are the derivatives of , , and . For simplification, the product will be denoted by .
For ease of exposition, we will assume that and are both polygons. Let and be regular triangulations or rectangulations of and , respectively. They are assumed to satisfy the angle condition which means that there is a positive constant such that, for all , , , where is the area of , is the area of , is the diameter of , and is the diameter of . Let . In addition, or denotes a general positive constant independent of .
Let denote the order Raviart-Thomas space  associated with the triangulations or rectangulations of . denotes the space of polynomials of total degree at most and indicates the space of polynomials of degree no more than and in and , respectively. If is a triangle, , and if is a rectangle, , . We define
By the definition of finite element subspace, the mixed finite element discretization of (13) is as follows: compute such that where and and are two finite element approximations of and .
It is well known (see, e.g., [7, 20]) that the optimal control problem (24) again has a unique solution and that a triplet is the solution of (24) if and only if there is a costate such that satisfies the following optimality conditions:
Let be the orthogonal -projection into defined by which satisfies
Let be the Raviart-Thomas projection , which satisfies We have the commuting diagram property where denotes identity matrix. We point out and .
In the rest of the paper, we will use some intermediate variables. For any control function , we first define the state solution associated with that satisfies
Correspondingly, we define the discrete state solution associated with that satisfies
Thus, as we defined, the exact solution and its approximation can be written in the following way:
In the following, we further assume that , , and are locally Lipschitz continuous, that and are bound functions on and , and that there is a such that For , we will write where is bounded function in .
3. A Priori Error Estimates
Now we will construct an analogue of the family of elliptic projection operators defined by Wheeler  in her thesis. Let be the solution of (14)–(18). Then, define the elliptic projection of to be by the following relations: where we assume that .
By using Lemma 3 in , we can obtain the following technical results.
Lemma 2. For and for sufficiently small, if , there is a positive constant independent of such that
By Theorem 3 in , we can establish the following useful result.
Lemma 3. Suppose . Then,
Now, we investigate the intermediate error estimates between and the intermediate solution . Benefit from the previous results in this section, we only need to estimate , and , .
Lemma 4. Assume that the optimal control problems (1)–(5) have a unique solution and that is 2-regular. Assume that the regularity assumptions (10) are valid. There is a positive constant , independent of , such that
Proof. Firstly, we prove the first inequality (71). From (61)-(62) and (50)-(51), we can derive the following error equations:
Differentiating (73) with respect to , we obtain
Taking and in (75) and choosing and , we can derive that
Next, taking and in (75) and choosing and , we also find that . Now, choosing and as test functions in (73) and (74), we have
From (76), we find that , and then we have
Then we obtain (71) from (77), (78), and the triangle inequality.
Furthermore, we prove the second inequality (72). By using (34), subtract (18)-(19) and (46)-(47) to get the following error equations: Noting that and taking in (79), we find that By using Lemma 1 and (77), we can obtain that Taking and in (74), since , we have Differentiating (75) and (74) with respect to , we obtain Selecting and as test functions in (84) and (85), respectively, we get Integrating (86) from to , using (83) and the Gronwall’s Lemma, we obtain Differentiating (79) with respect to , we obtain Now we choose and as test function in (80) and (88), and we have Then, integrating (89) from into , using (83) and (87), we obtain Note that ; then . Since , we have Then we complete the proof by combining (90), (91), and the triangle inequality.
Theorem 5. Assume that the optimal control problems (1)–(5) have a unique solution and that is 2-regular. Assume that the regularity assumptions (10) are valid. There is a positive constant , independent of , such that
By applying the results we have proved above, we only need to estimate , and , . For convenience, let
Proof. From (25)-(26) and (50)-(51), we obtain the following error equations:
Let and in (98); since , we have . We differentiate (98) with respect to , and we derive
Choose and as test functions and add the two relations of (99) and (100); using the Cauchy inequality, we obtain
Integrating (101) with respect to time from to , we derive
By using Gronwall’s lemma to (102), we obtain
Since , using (103), we have
Then we derive (96).
From (29)-(30) and (54)-(55), we obtain the following error equations: Let and in (105); since , we have Introduce the symbol , let , and integrate (105) with respect to time from to , and we obtain Set in (108) and in (105), note that , and then add those equations to derive Integrating (109) with respect to time from to , using (107) and Yong’s inequalities, we get Choosing and as test functions in (105) and (106), it is easy to get where is an arbitrary small positive constant. Namely,