#### Abstract

The contribution of this thesis is the somewhat general analysis, of singular solutions which arise in problems of optimal control and the development of certain analytical procedures for detecting and calculating singular solutions. The basic optimal control problem considered in this study is the task of choosing a control u(t) which will a) transfer the state of a system, described by the a first order ordinary differential equations, from some prescribed initial state to some prescribed final (terminal) state and b) simultaneously minimize (with respect to the control u) an index of performance J of the form. It is assumed that the allowable values, of the control .u may be constrained to lie in some set U. The conventional mathematical techniques presently being used in optimal control theory are discussed. It is shown that for a certain class of optimal control problems, which are characterized by the control u appearing linearly in the system state equations (l) and the integrand of the index of performance (2), the optimal control u*(t) is found (formally) to the of the "bang-bang" type In (3), A and B are, respectively, the upper and lower hounds on the admissible control u and F(t) is a certain function of time which is called the switching function. When the switching function becomes identically zero over a finite time interval the conventional mathematical 'techniques fail to yield any information about the desired optimal control. The solution in this ease is said to he "singular" and the corresponding control is termed "singular control". The nature of singular solutions is being investigated in detail and the apparent failure of the conventional mathematical techniques has been shown to be due to the fact that singular optimal controls lie in the interior (rather than on the boundary) of the admissible set U. The concept of a singular control surface in the system state space was introduced and is used to examine the geometry of singular solutions. Some mathematical properties of the singular control surface are being derived and a backward tracing scheme is used to aid in establishing the role of singular sub-arcs in the solution of optimal control problems. It is being shown that the singular control u*(t) can be obtained from the condition F(t) O and in some cases can be synthesized as a function of the system state variables. The proposed techniques for solving optimal control problems with singular solutions can be illustrated by means of four examples which are worked out in detail.

#### Date of this Version

8-1-1963

## Comments

Thesis National Science Foundation Contract G-16460. PRF 2743