Relaxed Derivatives and Extremality Conditions in Optimal Control

Abstract

In previous work dating back to the early 1970’s F.H. Clarke and the author had independently derived necessary conditions for minimum including a maximum principle for optimal control problems defined by ordinary differential equations in which the right hand side f(t,·, r) and functions defining side conditions are Lipschitz continuous in their dependence on the state variable. Our results, though not the methods, were similar in the formulation of the maximum principle in which the nonexisting derivative fv(t, v, σ) was replaced by an unknown element of Clarke’s generalized Jacobian but differed in handling some side conditions. In the present paper we exhibit a maximum principle in which the dual variables and the related functions are limits of appropriate subsequences of computable sequences.