On the computational algorithms for optimal control problems with general constraints.

Abstract

In this thesis we used the following four types of optimal control problems: (i) Problems governed by systems of ordinary differential equations; (ii) Problems governed by systems of ordinary differential equations with time-delayed arguments appearing in both the state and the control variables; (iii) Problems governed by linear systems subject to sudden jumps in parameter values; (iv) A chemical reactor problem governed by a couple of nonlinear diffusion equations. • The aim of this thesis is to devise computational algorithms for solving the optimal control problems under consideration. However, our main emphasis are on the mathematical theory underlying the techniques, the convergence properties of the algorithms and the efficiency of the algorithms.

Chapters II and III deal with problems of the first type, Chapters IV and V deal with problems of the second type, and Chapters VI and VII deal with problems of the third and fourth type respectively. A few numerical problems have been included in each of these Chapters to demonstrate the efficiency of the algorithms involved. The class of optimal control problems considered in Chapter II consists of a nonlinear system, a nonlinear cost functional, initial equality constraints, and terminal equality constraints. A Sequential Gradient-Restoration Algorithm is used to devise an iterative algorithm for solving this class of problems. 'I'he convergence properties of the algorithm are investigated.

The class of optimal control problems considered in Chapter III consists of a nonlinear system, a nonlinear cost functional, and terminal as well as interior points equality constraints. The technique of control parameterization and Liapunov concepts are used to solve this class of problems, A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints or inequality type was developed by Rei. 103 in 1989. In Chapter IV, we extend the results of Ref. 103 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints, as well as terminal state inequality constraints and continuous state inequality constraints. In Ref. 104, a computational scheme using the technique of control parameterization was developed for solving a class of optimal control problems in which the cost functional includes the full variation of control. Chapter V is a straightforward extension of Ref. 104 to the time-delayed case. However the main contribution of this chapter is that many numerical examples have been solved. In Chapter VI, a class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek for the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. In Chapter VII, a chemical reactor problem and its control to achieve a desired output temperature is considered. A finite element Galerkin method is used to convert the original distributed optimal control problem into a quadratic programming problem with linear constraints, which can he solved by any standard quadratic programming software .