Differential inclusions with free time

Abstract

We prove Hamiltonian necessary conditions for stateconstrained differential inclusion problems in which the basic time interval is one of the unknowns. Previous approaches typically reduce this problem to one based on a fixed time interval by transforming the time variable into an auxiliary state – a device which unfortunately requires that the data exhibit rather smooth $t$-dependence. Here we use proximal analysis to avoid this transformation, and offer the first complete treatment of free-time problems whose dynamics are assumed to be merely measurable in $t$.

Résumé

Nous démontrons des conditions nécessaires d’optimalité sous forme hamiltonienne pour des problèmes de contrôle avec contraintes sur l’état lorsque l’horizon fait partie des inconnues. Notre étude utilise l’analyse proximale et s’applique des situations où la dépendance en $t$ est simplement mesurable.