The Optimization of Trajectories of Linear Functional Differential Equations

Abstract

Our aim in this paper is to examine a number of fundamental questions in the theory of optimal control of processes monitored by certain general systems of linear functional differential equations with finite memories. In our model the controls may appear in a very general nonlinear functional manner which permits us to consider retardations of a rather general character in the control variables. In particular, we prove a maximal principle for such systems. We consider existence questions in the class of admissible Borel measurable (respectively piecewise continuous, almost piecewise continuous) initial functions and controls. We also show that certain solutions of an uncontrolled linear functional differential equation are piecewise analytic or quasi-piecewise analytic.