The approximate maximum principle in constrained optimal control

This paper concerns optimal control problems for dynamical systems described by a parametric family of discrete/finite-difference approximations of continuous-time control systems. Control theory for parametric systems governed by discrete approximations plays an important role in both qualitative and numerical aspects of optimal control and occupies an intermediate position in dynamic optimization: between optimal control of discrete-time (with fixed steps) and continuous-time control systems. The central result in optimal control of discrete approximation systems is the approximate maximum principle ( AMP), which gives the necessary optimality condition in a perturbed maximum principle form with no a priori convexity assumptions and thus ensures the stability of the Pontryagin maximum principle (PMP) under discrete approximation procedures. The AMP has been justified for optimal control problems of smooth dynamical systems with endpoint constraints under some properness assumption imposed on the sequence of optimal controls. In this paper we show, by a series of counterexamples, that the properness assumption is essential for the validity of the AMP, and that the AMP does not hold, in its expected ( lower) subdifferential form, for nonsmooth problems. Moreover, a new upper subdifferential form of the AMP is established for ordinary and time-delay control systems. The results obtained surprisingly solve ( in both negative and positive directions) a long-standing and well-recognized question about the possibility of extending the AMP to nonsmooth control problems, for which the affirmative answer has been expected in the conventional lower subdifferential form.