Variations of the v-Change of Time in Problems with State Constraints

Abstract

For a general optimal control problem with a state constraint, we propose a proof of the maximum principle based on a v-change of the time variable t ↦ τ, under which the original time becomes yet another state variable subject to the equation dt/dτ = v(τ), while the additional control v(τ) ≥ 0 is piecewise constant and its values are arguments of the new problem. Since the state constraint generates a continuum of inequality constraints in this problem, the necessary optimality conditions involve a measure. Rewriting these conditions in terms of the original problem, we get a nonempty compact set of collections of Lagrange multipliers that fulfil the maximum principle on a finite set of values of the control and time variables corresponding to the v-change. The compact sets generated by all possible piecewise constant v-changes are partially ordered with respect to inclusion, thus forming a centered family. Taking any element of their intersection, we obtain a universal optimality condition, in which the maximum principle holds for all values of the control and time.