Geometric Theory of Linear Controlled Systems

Abstract

Geometric techniques are widely used in the study of optimal control problems. In the proof of the maximum principle [1] the essential arguments involved geometric ideas such as the cone of attainability and the existence of a separating hyperplane in a certain space. The generation of this cone depended on linearization and the proof of the maximum principle then followed from the linear theory by showing that the linearized cone gave a good approximation (essentially a fixed point result of the type given by BROUWER was all that was needed). In this paper geometric ideas will be exploited to obtain results from which necessary and sufficient conditions for optimal control follow for linear models and to establish an existence theorem for optimal control. The necessary condition, the maximum principle, can be established for certain nonlinear models using the geometric results and a fixed point argument.

Research sponsored by Air Force Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-571-66.