A posteriori error estimates of spectral method for nonlinear parabolic optimal control problem

In this paper, we investigate the spectral approximation of optimal control problem governed by nonlinear parabolic equations. A spectral approximation scheme for the nonlinear parabolic optimal control problem is presented. We construct a fully discrete spectral approximation scheme by using the backward Euler scheme in time. Moreover, by using an orthogonal projection operator, we obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}(H^{1})-L^{2}(L ^{2})$\end{document}L2(H1)−L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control. Finally, by introducing two auxiliary equations, we also obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{2}(L^{2})-L^{2}(L^{2})$\end{document}L2(L2)−L2(L2) a posteriori error estimates of the approximation solutions for both the state and the control.


Introduction
Optimal control problems appear frequently in the operation of physical, social, economic processes, and other fields, and the numerical solutions of optimal control problems are extremely important for better performance of those fields. Therefore, one needs some efficient numerical methods to approximate the solutions of optimal control problems. Finite element methods seem to be the most popular used numerical methods in solving optimal control problems. Meanwhile, other numerical methods, such as the spectral method, the mixed finite element method and the finite volume method have also been applied to approximate some optimal control problems. For example, there has been done much work on the finite element method for optimal control problems [11,[13][14][15]17], the spectral method for optimal control problems [4,8,9], the mixed finite element method for optimal control problems [3, 5-7, 18-20, 23, 24], and the finite volume method for optimal control problems [21,22].
The spectral method has two important features: it enjoys the great superiority of fast convergence rate and provides very accurate approximations with a relatively small number of unknowns when the solutions are smooth. Let us briefly review the current literature. In [9], Ghanem and Sissaoui derived a posteriori error estimates by a spectral method of a linear-quadratic elliptic optimal control problem without inequality constraints. In [8], the authors studied the Legendre Galerkin spectral approximation of optimal control problems governed by elliptic equations and obtained a priori and a posteriori error estimates. A posteriori error estimates of a Legendre Galerkin spectral approximation of optimal control problems governed by parabolic equations were derived in [4]. To the best of our knowledge, a posteriori error estimates of the spectral method for nonlinear optimal control problems have never been studied. Nonlinear optimal control problems appear frequently in real life such as economics, chemical engineering, robotics and aeronautics, and the spectral method has several attractive features. Therefore, it is necessary to study a posteriori error estimates of the spectral method for nonlinear parabolic optimal control problems.
The purpose of this work is to derive a posteriori error estimates for the spectral approximation of an optimal control problem governed by nonlinear parabolic equations. We present a fully discrete scheme which uses the backward Euler scheme in time and uses the spectral approximation in space, and we obtain a posteriori error estimates of the spectral approximation solution for both the state and the control.
The outline of this paper is as follows. In Sect. 2, we shall construct spectral approximation scheme for nonlinear parabolic optimal control problem. In Sect. 3, by using orthogonal projection operator, L 2 (H 1 ) -L 2 (L 2 ) a posteriori error estimates of the spectral approximation solutions for optimal control problem solution are derived. In Sect. 4, by using two auxiliary equations, L 2 (L 2 ) -L 2 (L 2 ) a posteriori error estimates of the spectral approximation solutions for optimal control problem are derived. In the last section, we briefly give conclusions and some possible future work.

Spectral approximation of nonlinear parabolic optimal control
In this section, we shall state the spectral approximation scheme and its optimality conditions for the optimal control problem governed by nonlinear parabolic equations. Now, we set the state space W = L 2 (0, T; H 1 0 ( )), the control space X = L 2 (0, T; L 2 ( )), and V = H 1 0 ( ). We will study the following nonlinear parabolic optimal control problem: where K is a set defined by and f , y d ∈ L 2 (0, T; L 2 ( )), y 0 ∈ H 1 0 ( ), let B be a linear continuous operator from X to L 2 (0, T; V ), φ(·) ∈ W 2,∞ (-R, -R) for any R > 0, φ (y) ∈ L 2 ( ) for any y ∈ H 1 ( ), φ ≥ 0, and the matrix A(·) = (a i,j (·)) 2×2 ∈ (C ∞ (¯ )) 2×2 , such that there is a constant c > 0 satisfying Let a(y, w) = (A∇y) · ∇w dx, ∀ y, w ∈ H 1 0 ( ), It is well known that there are constants c and C > 0 such that Then a weak formula of the optimal control problem can be obtained: It is well known (see, e.g., [16]) that the optimal control problem (2.5)-(2.7) has at least one solution (y, p, u), and that if a triplet (y, p, u) is the solution of (2.5)-(2.7), then there is a co-state p ∈ W such that (y, p, u) satisfies the following optimality conditions: where B * is the adjoint operator of B. Proof For any function p ∈ W , we have which satisfies the following variational inequality: If B * p > 0, then it is easy to see u = B * p -B * p and So we have u = max{0, B * p} -B * p.
Next, we will use the Legendre Galerkin spectral method to investigate the spectral approximation of the nonlinear parabolic optimal control problem (2.8)-(2.10). From now on, we assume that = (-1, 1) 2 . Firstly, let us introduce some basic notations which will be used in the sequel. For x i , i = 1, 2, we denote by L r (x i ) the rth degree Legendre polynomial in the variable x i , and we set where N ≥ 0 is an integer. We define a product space such as We introduce the finite dimensional spaces V N = X N V and K N = X N K . In addition, C denotes a general positive constant independent of N , the order of the spectral approximation.
Then the Legendre Galerkin spectral approximation for the nonlinear parabolic optimal control problem is 12) y N (x, 0) = y N 0 (x), x ∈ , (2.13) where y N ∈ H 1 (0, T; V N ) and y N 0 ∈ V N is an approximation of y 0 . It follows that the optimal control problem (2.11)-(2.13) has at least one solution (y N , u N ), and that if a pair (y N , u N ) is the solution of (2.11)-(2.13), then there is a co-state p N such that (y N , p N , u N ) satisfies the following optimality conditions: (2.14) Now, we shall construct the fully discrete approximation scheme for the above semidiscrete problem.
Similarly, construct the approximation spaces K N i ⊂ L 2 ( ) (similar to K N ) on the ith time step. The fully discrete approximation scheme of (2.11)- It follows that the optimal control problem (2.17)-(2.19) has at least one solution . . , M, satisfies the following optimality conditions: For any function w ∈ C(0, T; Then the optimality conditions (2.20)-(2.24) can be restated: For any u ∈ L 2 ( ), we define the orthogonal projection operator P N : It can be shown (see [2]) that The following lemma will play a very important role in a posteriori error estimates. It can be found in the reference book of Ref. [2].
are bounded functions in¯ , more details can be found in [7].

L 2 (H 1 ) -L 2 (L 2 ) a posteriori error estimates
In this section, we shall derive a L 2 (H 1 ) -L 2 (L 2 ) posteriori error estimates for the spectral approximation of the optimal control problem governed by nonlinear parabolic equations. Set It can be shown that (see [12]) where p(U N ) is the solution of the auxiliary equations: We assume that the cost function J is strictly convex near the solution u, i.e., for the solution u there exists a neighborhood of u in L 2 such that J is convex in the sense that there is a constant c > 0 satisfying for all v in this neighborhood of u. The convexity of J(·) is closely related to the second order sufficient optimality conditions of optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many references, the authors assume the following second order sufficiently optimality condition (see [13]): there is c > 0 such that J (u)v 2 ≥ c v 2 0 . From the assumption (3.8), by the proof contained in [1], there is a c > 0 independent of N , such that where u and u N are the solutions of (2.8)-(2.10) and (2.14)-(2.16), respectively.
where p(U N ) is defined by (3.5) and Proof According to (3.9), we have Note that U N ∈ K N ⊂ K . It follows from (2.10) that we have Therefore, we can get Firstly, we can easily estimate the first term E 1 of (3.11) as follows: L 2 (0,T;L 2 ( )) . (3.12) For the second term E 2 , we can also easily obtain where L 2 (0,T;L 2 ( )) , By using Lemma 2.2, we can estimate the first term I 1 as follows: where δ is an arbitrary positive number, C(δ) is a constant dependent on δ. Note that φ (·) ≥ 0, we can obtain L 2 (0,T;H 1 ( )) . (3.21) For the third term I 3 , we can derive Similarly, for I 4 we have the following estimate: Due to φ(·) ∈ W 2,∞ (-R, -R), we can get Then we can estimate the last term I 6 of (3.19) as follows: Thus From Lemma (2.2), we have For J 2 , by using the assumption of φ, we can get L 2 (0,T;L 2 ( )) + δ e y 2 L 2 (0,T;L 2 ( )) ≤ C(δ)η 2 6 + δ e y 2 L 2 (0,T;H 1 ( )) . (3.32) For J 3 , it is clear that Proof Combining Theorem 3.1 and Theorem 3.2, we have

L 2 (L 2 ) -L 2 (L 2 ) a posteriori error estimates
In this section, we shall carry out L 2 (L 2 ) -L 2 (L 2 ) a posteriori error estimates for the spectral approximation of the optimal control problem governed by nonlinear parabolic equations. In order to estimate the error P Np(U N ) 2 L 2 (0,T;L 2 ( )) , we shall use two auxiliary equations. Now, we set the following dual auxiliary equations: and The following well-known stability results are presented in [10].

Concluding remarks and future work
In this paper, we present a fully discrete scheme in which we use the backward Euler scheme in time and use the spectral approximation in space for the nonlinear parabolic optimal control problem. By using the orthogonal projection operator and some auxiliary equations, we obtain L 2 (H 1 ) -L 2 (L 2 ) a posteriori error estimates of the spectral approximation solutions for both the state and the control, and also obtained L 2 (L 2 ) -L 2 (L 2 ) a posteriori error estimates of the spectral approximation solutions for both the state and the control. The results obtained and techniques used can be extended to such control problems with more general objective functions. Furthermore, we shall consider the spectral approximation for hyperbolic optimal control problems.