Variations of the $v$-change of time in problems with state constraints

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 $t\mapsto \tau,$ under which the original time becomes yet another state variable subject to the equation $dt/d\tau = v(\tau),$ while the additional control $v(\tau)\ge 0$ 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 by 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.