PROOF OF THE MAXIMUM PRINCIPLE FOR A PROBLEM WITH STATE CONSTRAINTS BY THE V-CHANGE OF TIME VARIABLE

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 dt/d tau = v; where v(tau) >= 0 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.