Abstract
In this paper, we consider semidiscrete splitting positive definite mixed finite element methods for optimal control problems governed by hyperbolic equations with integral constraints. The state and costate are approximated by the lowest order Raviart-Thomas mixed rectangular finite element, and the control is approximated by piecewise constant functions. We derive some convergence and superconvergence results for the control, the state and the adjoint state. A numerical example is provided to demonstrate our theoretical results.
1. Introduction
There have been extensive studies in error estimates of standard finite element methods (FEMs) for optimal control problems (OCPs). The convergence or superconvergence results of standard FEMs for elliptic and parabolic OCPs can be found in [1–13], respectively.
Because large temperature gradients during cooling or heating may lead to its destruction in temperature control problems, the gradient stands for Darcy velocity in flow control problems, stiffness optimization in nonlinear pantographic structures [14], and topology optimization of a cycloidal metamaterial [15]; their objective functionals contain not only the primal state variable but also its gradient. At this time, mixed finite element methods (MFEMs) will be a very good choice for solving this kind of OCPs. The convergence or superconvergence results of MFEMs for elliptic and parabolic OCPs can be found in [16–20], respectively. However, mixed finite element spaces have to satisfy the Ladyženskaja-Babuška-Brezzi (LBB) condition, which brings very little available approximation spaces and expensive computing costs.
In order to avoid the limitation of LBB condition, a splitting positive definite MFEM was first proposed for solving miscible displacement of compressible flow in a porous medium [21]. Compared with the classic MFEMs, the main advantages of this method are that the original problems can be split into two independent symmetric positive definite subschemes and that the LBB condition is not necessary. Recently, splitting positive definite MFEMs have been used to solve hyperbolic equations [22, 23], elliptic OCPs [24], and parabolic OCPs [25]. To our best knowledge, most of the published papers on different FEMs or MFEMs for OCPs are focused on elliptic or parabolic cases. Although Xu in [26] established a priori error estimates and superconvergence results of splitting positive definite MFEM for pseudohyperbolic integrodifferential OCPs and Lu et al. in [27] derived the convergence of finite volume element method for nonlinear hyperbolic OCPs, there are very little studies on hyperbolic OCPs, .
The goal of this paper is to investigate splitting positive definite MFEMs for hyperbolic OCPs and derive the convergence and superconvergence.
We are interested in the following hyperbolic OCPs: where is a rectangle domain, , , , with and . The coefficient matrix is a symmetric matrix, and there are constants satisfying any vector , . is a set defined by
In this paper, we adopt the standard notation for Sobolev spaces on with a norm given by , a seminorm given by . For , we set , and . We denote by the Banach space of all integrable functions from into with norm , and the standard modification for . For simplicity of presentation, we denote by . Similarly, one can define the spaces and . In addition, denotes a general positive constant independent of , where is the spatial mesh-size for the control and state discretization.
The plan of this paper is as follows. In Section 2, we give an equivalent optimality conditions for the OCP (1)–(5) and construct its splitting positive definite mixed finite element approximation scheme. In Section 3, we derive the convergence for the control variable, the state variables, and the adjoint state variables. In Section 4, we derive the superconvergence properties between the projections and the approximation solutions of the control and the state variables. In the last section, we present a numerical example to illustrate our theoretical results.
2. Splitting Positive Definite MFEMs for OCPs
In this section, we shall construct a splitting positive definite mixed finite element approximation of the control problems (1)–(5). To fix the idea, we shall take the state spaces and , where and are defined as follows.
Let and the inner products
Let , a mixed weak form of (2), and (3) can be given by
As in [24], taking in (10), (9) differentiating twice with respect to , and then substituting the two resulting equations, we derive
By using (10) and (11), we get the following new mixed variational form:
It is easily seen that (12) is separated from (13) so that can be solved independently from (12).
We recast (1)–(5) as the following weak form: find such that
It then follows from [28] that the optimal control problems (14)–(18) have a unique solution , and that a triplet is the solution of (14)–(18) if and only if there is a costate such that satisfies the following optimality conditions:
The inequality (27) can be expressed as where and .
Let be a uniform rectangulation of the domain , and denotes the diameter of element and . Let denote the lowest order Raviart-Thomas mixed finite element space [29, 30], namely where indicates the space of polynomials of degree no more than and in and on , respectively. And the approximated space of control is given by
We introduce two projection operators. First, we define the standard -projection [29] , which satisfies the following: for any
Second, we recall the Fortin projection (see [29, 31]) , which satisfies the following: for any
Then the splitting positive definite mixed finite element discretization of (14)–(18) is as follows: find such that
The optimal control problems (36)–(40) again have a unique solution , and a triplet is the solution of (36)–(40) if and only if there is a costate such that satisfies the following optimality conditions:
Similar to (28), we have where and .
3. Convergence Analysis
In this section, we will derive the convergence of splitting positive definite MFEMs for hyperbolic OCPs. For , we define the discrete state solution associated with which satisfies
It is clear that the exact solution and its approximation can be written in the following way:
Lemma 1. Let be the solution of (19)–(26) and be the solution of (51)–(58) with , respectively. If the solution satisfies then we have
Proof. For ease of presentation, we set , , , . From Equations (19)–(26) and (51)–(58) and the fact that , with the definition of and , we can easily obtain the following error equations:
Setting in (62), we have
Notice that
Substitute (67) into (66), integrating the resulting equation from to , and using Hölder’s inequality, Young’s inequality, Gronwall’s inequality, and the assumption on and (34) and (35), we have
Letting in (63) and in (65), respectively. Then integrating the resulting equations from to and to , respectively, we get
where we also used Hölder’s inequality, Young’s inequality, and Gronwall’s inequality.
At last, setting as the test function in (64) and integrating the resulting equation from to , similar to (68), we arrive at
Note that , and we have
Combining (68)–(74), (32), (34), and (35) and the triangle inequality, we complete the proof.
Using the same estimates as in Lemma 1, we get the following.
Lemma 2. Let and be the solutions of (51)–(58) with and , respectively. Then we have
Now, from the above Lemmas 1 and 2, we can derive the following convergence results.
Theorem 3. Let be the solution of (19)–(27) and be the solution of (41)–(49), respectively. Assume that all the assumptions in Lemma 1 are valid. Then we have
Proof. Let in (31), and we have
By (80), we have
Thus, we know that .
It follows from (27) and (49) that
Next, we estimate (82) term by term. For , using Hölder’s inequality, Young’s inequality, and Lemma 1, we have
From (28), we find that
Thus
Set , , , and ; then from (51)–(58), we have the following error equations:
choosing in (86), in (87), in (88), and in (89), respectively. Since , integrating the resulting equations from to , we can see that
For and , using Hölder’s inequality, Young’s inequality, Lemma 2, and (32), we get
For , by Hölder’s inequality, the triangle inequality, Lemma 1, and (32), we arrive at
Combining (82), (83), (85), and (90)–(92), we derive (76). Using (76), Lemmas 1 and 2, and the triangle inequality, we complete the proof.
4. Superconvergence Properties
In this section, we will derive some superconvergence properties for the optimal control problems. In order to derive the main results, we need the following lemmas.
Lemma 4. Let be the solution of (19)–(27) and be the solution of (51)–(58) with , respectively. If the solution satisfies then we have
Proof. At first, for any and , by applying the proof of Theorems 4.1, 5.1, and Example 6.2 in [32], we can prove Moreover, using (32) and (35), we have Similar to Lemma 1, using (98), we can prove (95)–(97). We omit the proof here.
Lemma 5. Let and be the solutions of (51)–(58) with and , respectively. Then we have
Proof. First, set , , , and , and we choose and in (51)–(58), respectively; then we obtain the following error equations: Noting from the fact that and (31) then, as in Lemma 1, using the stability estimates, we complete the proof.
Lemma 6. Let and be the solutions of (51)–(58) with and , respectively. Then we have
Proof. Set , , , and , similar to (101)–(104), and we obtain the following error equations: choosing in (107), in (108), in (109), and in (110), respectively. Integrating the resulting equations from to , we derive which yields to (106).
Theorem 7. Let be the solution of (19)–(27) and be the solution of (41)–(49), respectively. Assume that all the assumptions in Lemma 1 are valid. Then, we have
Proof. We choose in (23) and in (44) to get the following two inequalities: Note that . Adding the above two inequalities, we get Thus, by (114) and (26), we find that At first, from Cauchy inequality, (26), (28), and (24), it is easy to get Using Hölder’s inequality, Young’s inequality, and Lemmas 4 and 5, we arrive at Combining (84), (85), and (115)–(118) and Lemma 6, we derive (112).
Using Theorem 7 and the stability estimates as in Lemma 1, we can arrive at the following.
Lemma 8. Let and be the solutions of (51)–(58) with and , respectively. Assume that all the assumptions in Theorem 7 are valid. Then we have
Combining Lemma 4–8 and using the triangle inequality, we derive the following superconvergence results.
Theorem 9. Let and be the solutions of (19)–(27) and (41)–(49), respectively. Assume that all the assumptions in Theorem 7 are valid. Then we have
5. Numerical Experiments
In this section, we present a numerical example to validate our convergence and superconvergence results. The hyperbolic OCP was dealt numerically with codes developed based on AFEPack. The package is freely available and the details can be found in [33].
Let , and . Set
Then, a fully discrete splitting positive definite mixed finite element solution of (1)–(6) satisfies the following system:
Example 10. Let , where denotes identity matrix. The data under testing are as follows:
Let and . We set and . The numerical results of convergence and superconvergence based on a sequence of uniformly refined meshes are reported in Tables 1–4, respectively. They are consistent with our theoretical results.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The first author is supported by the Scientific Research Project of Hunan Provincial Department of Education (20C0854) and the Scientific Research Program in Hunan University of Science and Engineering (20XKY059). The second author is supported by the National Natural Science Foundation of China (11401201), the Natural Science Foundation of Hunan Province (2020JJ4323), the Scientific Research Project of Hunan Provincial Department of Education (20A211), and the Construct Program of Applied Characteristic Discipline in Hunan University of Science and Engineering.