Optimality and Viability Conditions for State-Constrained Optimal Control Problems

We provide a new proof of the Pontryagin Maximum Principle for problems with state constraints of the form g(t; x(t))_ 0 using a technique of exact penalization. In particular, we show how to use two theorems of nonsmooth analysis, namely the multidirectional mean value inequality of Clarke and Ledyaev and a subgradient formula for maximum-type functions due to Ledyaev and Trieman, to derive the Pontryagin Maximum Principle for problems with state constraints and to derive viability-type results for state constrained problems. This chapter explains about state constrained optimal control problems and provides a novel approach to the study of optimality conditions and viability through nonsmooth analysis and exact penalization. It shows how to use two theorems of nonsmooth analysis, namely, the multidirectional mean value inequality of F. Clarke and Yu. Ledyaev and a subgradient formula for maximum-type functions due to Ledyaev and Jay Trieman, in order to derive the Pontryagin Maximum Principle for problems with state constraints and obtain viability results for state-constrained problems. Since the appearance of Clarke's subdifferential in 1973, the field of nonsmooth analysis has seen considerable proliferation of subdifferentials, each with particular advantages and disadvantages. The chapter outlines equipped with an elementary background in nonsmooth analysis.