Article Contents
Article Contents

# 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 ${\rm d} t/ {\rm d} \tau = v,$ 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.

Mathematics Subject Classification: Primary: 49K15; Secondary: 49K27, 46N10.

 Citation:

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