Existence of regular synthesis for general control problems

Abstract

The concept of regular synthesis has been introduced by Bolt’anski in his classical paper [I] (cf. also [2]) on the sufficiency of the Pontrjagin maximum principle for time-optimal control problems. It has been used as an assumption on a closed-loop control to generate open-loop optimal controls. Using the theory of subanalytic sets it has been proved in [3] that every normal linear system admits a regular time-optimal synthesis. Subsequently, in [S], Sussmann has been able to dispose of the normality condition and to extend the theorem to a certain class of nonlinear systems. All of the mentioned papers deal entirely with the time-optimal control problem for systems which are linear in the control, with polyhedral control domains. The present paper constitutes an extension towards optimal control problems with general performance criteria and general control domains. The abstract theorem proved in this paper is modelled after an important class of problems-linear-quadratic optimal control problems with linear control constraints. This problem will be dealt with in a forthcoming paper. An important requirement in Bolt’anski’s definition of regular synthesis which is followed in [3,4] as well as [8, 91 is that the optimal trajectories enter the switching surfaces (called cells) transversally. This requirement has to be dropped not only in the linear-quadratic optimal control problem but also in the linear time-optimal control problem with a control domain having a piecewise smooth curvilinear boundary, as the following example demonstrates: Consider the system