The Spectral Methods For Optimal Control Problems With Constraints On State

There is an active and attractive area of the research about theoretical analysis and numerical methods for the optimal control problems governed by partial differential equa-tions. Although there are lots of research about the control constrained optimal control problems with finite element methods, not with the spectral methods. In [32], the authors investigated the control integral constrained optimal control problems. Recently, many re-searchers concern about the state constrained optimal control problems, which are met in applications, by the finite element methods. However, to the author's knowledge, it seems that there are few works have been made to systematically discuss spectral methods for these problems.This paper investigates the state constrained optimal control problems with spec-tral methods. For simplicity, we choose the Poisson equation and the first bi-harmonic equation as the state equations. Obviously, the same techniques and conclusions can be expanded to general model problems. We focus on the a priori error estimates and a posteriori error estimates, specially, a gradient projection algorithm and its convergent property are investigated.With the orthogonal property of the Legendre polynomials, we get an improved a posteriori error estimator to [38], which is given with explicit formulations. Then, the ex-plicit error estimator is convenient for engineering applications. Especially, the improved a posteriori error estimator only depends on the right hand side item in the state equation. In fact, there are the two coefficients of the Legendre expansion. Similarly, we discuss the p-version finite element methods and its a posteriori error estimator.According to the property of the basis for two dimension and the orthogonal property of Legendre polynomials, we study the Poisson equation in two-dimension and deduce the explicit a posteriori error estimator, which depends on four items of the coefficients for the Legendre expansion(with the symmetric property, there are only three items). Easily, we get the discrete formulation for p-version finite element methods and the explicit formulation for the a posteriori error estimator in two-dimensional domain.There are lots of works applying finite element methods to approximate the fashion-able optimal control problems. Obviously, if the initial data are sufficient smooth, we can choose suitable spectral methods to obtain "spectral accuracy". So, we investigate the optimal control problems with spectral methods. Firstly, we systematically analyze state integral constrained optimal control problems in one-dimension. We deduce the optimal conditions, the a priori error estimates and the a posteriori error estimator with explicit formulations.In engineering applications, the state equations can be described by the first bi-harmonic equations. So, it is the point for us to investigate state integral constrained optimal control problems subjecting to the first bi-harmonic equations. With the help of KKT conditions, we declare the optimal conditions easily. Moreover, we investigate the a priori error estimates and derive an efficient projection gradient algorithm, specially, we discuss the convergence of the algorithm.In many optimal control problems, the objective functions include two-order deriva-tive of the state variables. Then, how to ensure a high accuracy of this item is the key point in numerical approximation for the state equations. As we all known, the mixed finite element methods introduce an intervening variable to enhance the accuracy of ap-proximation for gradient item. Following the same ideals, we introduce an auxiliary vari-able to construct mixed spectral methods to investigate state integral constrained optimal control problems with the first bi-harmonic equation. We derive the optimal conditions, a priori error estimates and an efficient gradient projection algorithm, especially, we obtain the convergent rate of the algorithm.We perform numerical examples to confirm our theoretical results.