The Constants in A Posteriori Error Indicator for State-Constrained Optimal Control Problems with Spectral Methods

Abstract

We employ Legendre-Galerkin spectral methods to solve state-constrained optimal control problems. The constraint on the state variable is an integration form. We choose one-dimensional case to illustrate the techniques. Meanwhile, we investigate the explicit formulae of constants within a posteriori error indicator.

1. Introduction

Spectral methods provide higher accurate approximations with a relatively small number of unknowns and play increasingly important roles in design optimization, engineering design, and other scientific and engineering computations. Gottlieb and Orszag [1] summarized the theories and applications of spectral methods. There have been extensive researches on finite element methods for optimal control problems, most of which focus on control-constrained problems; see [2–5]. The authors studied the optimal control problems with the control constraint with spectral methods in [6]. In applications of engineering, one cares more about how to control the average value or -norms of the state variable. The authors [7] discussed state-constrained optimal control problems with finite element methods. However, there are few work on the state-constrained optimal control problems with spectral methods.

In order to get a numerical solution with acceptable accuracy, spectral methods only increase the degree of basis when the error indicator is larger than the a posteriori error indicator, while the finite element methods refine meshes (see [8, 9]). There have been lots of papers on the a posteriori error estimates for h-version finite element methods but not for spectral methods. Guo [10] got a reliable and efficient error indicator for -version finite element method in one dimension with a certain weight. The authors [11] deduced a simple error indicator for spectral Galerkin methods. In [12], the authors investigated Legendre-Galerkin spectral method for optimal control problems with integral constraint on state. It is difficult to obtain optimal a posteriori error indicators. Thus, if one gets the constants within upper bound a posteriori error estimates, it is easy to ensure the degree of polynomials to get an acceptable accuracy.

In this paper, we employ Legendre-Galerkin spectral methods to solve optimal control problems with state-constrained case and calculate constants in upper bound of the a posteriori error indicator, which can be used to decide the least unknowns for acceptable accuracy. With the help of auxiliary systems, we investigate explicit formulae of the constants in the a posteriori error indicator.

The outline of this paper is as follows. In Section 2, the model problem and its Legendre-Galerkin spectral approximations are listed. In Section 3, the constants within the a posteriori error indicator are investigated in detail and the explicit formulae are obtained. The conclusions are given in Section 4.

2. A Model Problem and Its Legendre-Galerkin Spectral Approximations

Throughout this paper we adopt the standard notations of Sobolev spaces [13]. Let be a Sobolev space on ,   and , and the corresponding norms are denoted by , and , respectively. This work focuses on the Legendre polynomials, which are orthogonal polynomials on [−1,1].

We concern the following distributed convex optimal control problems with integral constraint on state: where is the control variable, is the state, and is the observation.

In order to assure the existence and regularity of the solution, we assume that is a given positive constant and is an infinitely smooth function. It is well known that the problem (OCP) has a unique solution (see [3]).

We give some basic notations which will be used in the sequel. Let Hence, the state equation reduces to Then can be rewritten as finding such that We recall the following optimal conditions of (for details please refer to [7]).

Lemma 1. The pair is the optimal solution of if and only if there exists a unique pair such that

Let and let . One prefers to choose appropriate bases of such that the resulting linear system is as simple as possible. We denote by the Legendre polynomial and employ the following basis functions (see [14]): where For , we denote and . By simple calculations, these coefficients satisfy Then Legendre-Galerkin spectral approximations of can be read as finding such that The Legendre-Galerkin spectral approximations of (5) can be read as follows.

Theorem 2. The pair is the optimal solution of if and only if there exists a unique pair such that

3. Constants within the A Posteriori Error Estimates

In this section, we calculate all constants within the a posteriori error estimates. Here, we analyze the constant in the Poincaré inequality.

For all , , we recall the Poincaré inequality with -norm as (see [15])

Now, we are at the point to investigate all constants in detail. We introduce an auxiliary state , which satisfies Subtracting (12) from (3), we get Let . It is clear that Then which means Hence Then We denote by the constant in (18); that is,

Similarly, we introduce an auxiliary state , which satisfies Subtracting (20) from the continuous systems (5), we get

We select which satisfies , where denotes the integral average on of the function and . Obviously, . In fact, . Then there hold where we used the generalized Schwarz inequality, continuous systems (5), and auxiliary equation (20). Let Then It is clear that (22) reduces to Then where we used .

Hence which means that Denote by the constant in (28). With simple calculation, we have

We select which satisfies and . For instance, , which satisfies Meanwhile, Hence where .

Thus With the constant , we infer that Then We denote by the constant in (35); that is,

We calculate the error of in -norm as follows: Provided that , we get Considering the item with the constraint , we get In fact, for , the derivation of the following function is Then we have and Now, we are at the point to investigate If , we get If , , we obtain Then If ,  , we infer that Then there hold Combining the above discussions, we deduce that Obviously, We denote by the constant in (50); that is,

For any , we define a projection operator , which satisfies

Lemma 3. For all , one has where .

Proof. Firstly, assuming that is integer, we define a differential operator as From the fact that it is easy to get By iterations, we obtain Secondly, for all , we note that and Hence Finally, there hold which means that Let . It is clear that This completes the proof.

Now, we are at the point to calculate the constant for . Similarly, let and let . Then which means that Likewise, let and let . Then which is equivalent to Hence where Combining the above analyses, we get that which means that where

4. Conclusions

This paper discusses the explicit formulae of constants within upper bound of the a posteriori error estimate for optimal control problems with Legendre-Galerkin spectral methods in one dimension. Thus, with those formulae, it is easy to choose a suitable degree of polynomials to obtain an acceptable accuracy. In the future, we will study the corresponding constants in lower bound of the a posteriori error indicator. Meanwhile, the corresponding constants in a two-dimensional domain will be investigated.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is partially supported by National Natural Science Foundation of China (no. 11201212), Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (no. BS2012DX004), AMEP, and the Special Funds for Doctoral Authorities of Linyi University.