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.

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 [

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 [

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

Throughout this paper we adopt the standard notations of Sobolev spaces [

We concern the following distributed convex optimal control problems with integral constraint on state:

In order to assure the existence and regularity of the solution, we assume that

We give some basic notations which will be used in the sequel. Let

The pair

Let

The pair

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

For all

Now, we are at the point to investigate all constants in detail. We introduce an auxiliary state

Similarly, we introduce an auxiliary state

We select

Hence

We select

Thus

We calculate the error of

For any

For all

Firstly, assuming that

Now, we are at the point to calculate the constant for

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.

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

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.