Regularizing properties of Pontryagin’s maximum principle

Abstract

This work is dedicated to the necessary and sufficient conditions for minimizing sequences in problems with inexact initial data. These conditions are tightly bound with the classical Pontryagin’s maximum principle. The paper also covers regularizing properties of these sequences and those of the maximum principle itself, considering a minimizing sequence (rather than the classical optimal control) as the central theoretical notion. It is well-known that Pontryagin’s maximum principle [1] results from real needs, first of all, applied studies ([2], P. 7). However, in most papers which deal with the theory of the necessary conditions in the optimal control, the initial data in problems under consideration are assumed to be known exactly. The papers on optimal control problems, where studying the necessary and sufficient conditions, one takes into account (somehow or other) the possibility of inexact definition of input data, are relatively few [3, 4]. At the same time, it seems natural to develop the theory of the necessary and sufficient conditions in order to tolerate inexact definition of initial data. Consider, for comparison, the development of solution methods for optimization and optimal control problems [5], the theory of ill-posed problems [6]. In favor of this observation, let us adduce the following arguments. First, in numerous applications one inevitably encounters the necessity to use inexact initial data. Second, in the analysis of solution algorithms for optimization and optimal control problems, the necessary and sufficient optimality conditions play the most important role. Finally, third, generally speaking, optimal control problems represent a class of mathematical problems, where the instability of the initial data with respect to a perturbation is anticipated.