Proof of the maximum principle for a problem with state constraints by the v-change of time variable

Abstract

We give a new proof of the maximum principle for optimal control problems with running state constraints. The proof uses the so-called method of $ v- $change of the time variable introduced by Dubovitskii and Milyutin. In this method, the time $ t $ is considered as a new state variable satisfying the equation $ {\rm d} t/ {\rm d} \tau = v, $ where $ v(\tau)\ge0 $ is a new control and $ \tau $ a new time. Unlike the general $ v- $change with an arbitrary $ v(\tau), $ we use a piecewise constant $ v. $ Every such $ v- $change reduces the original problem to a problem in a finite dimensional space, with a continuum number of inequality constrains corresponding to the state constraints. The stationarity conditions in every new problem, being written in terms of the original time $ t, $ give a weak* compact set of normalized tuples of Lagrange multipliers. The family of these compacta is centered and thus has a nonempty intersection. An arbitrary tuple of Lagrange multipliers belonging to the latter ensures the maximum principle.