SINGULAR CONTROLS IN LINEAR SYSTEMS WITH QUADRATIC COSTS

Abstract

The aim of this thesis is to study some approaches used to solve singular optimal control problems of linear systems with quadratic costs. Two main approaches will be discussed here. In Chapter 4, we present an analytical approach using the concept of jump fields. While in Chapter 5, a numerical approach known as the Continuation Method for Optimal Control Synthesis is given. The first two chapters explain some basic concepts in control theory and the notations used subsequently. In Chapter 3, we will consider some properties of non-singular optimal control problems. A brief description of each chapter is given as follows: Chapter 1 gives a brief introduction to linear system. The main point in this chapter is the introduction of the transition matrix and its properties. The concept of adjoint equation is also introduced. In Chapter 2, we consider the general optimal control problems. We present a reasonably broad survey of sufficient conditions for the existence of at least one optimal control. The main aim of this chapter is to describe a set of conditions that any optimal control must necessarily satisfy. This set of necessary conditions is collectively known as the Pontryagin Maximum Principle. Chapter 3 is devoted to the study of non-singular optimal control problems of linear system with quadratic costs. We will show that optimals and extremals are equivalent for this case. We will first show that extremals intersect at finite number of partitions. We then prncccd to show that the set of branching points of an extremal is a finite set depending on the equation of the control system only. The remaining chapter explores the situation when the time interval for the control problem is infinite. In Chapter 4, we consider the singular optimal control problem of linear system with quadratic costs. We will show that for a given boundary conditions, there always exist a particular "turnpike" or sub-optimal trajectory with only two "access" points and the optimal strategy is to get to the turnpike as fast as possible, follow it most of the way and then exit to the terminal point as quickly as possible. This is an analytical approach through the application of jump fields. The method is described as optimal synthesis in the chapter. In Chapter 5, we will consider a numerical approach, known as the Continuation Method for Optimal Control Synthesis. We will use a perturbation technique to transform the singular optimal control problem to a non-singular one. We will then apply the continuation method to the perturbed problem. This continuation method produces a solution to the perturbed problem that is convergent to the solution of the original singular problem. Besides, the continuation method only requires lo solve the two-point boundary value problem once and then follows by solving a series of initial boundary value problems. which is a more efficient approach compared to many numerical procedures that are currently available.