Coextremals and the value function for control problems with data measurable in time

Abstract

Suppose x ∗(·) is a solution to an optimal control problem formulated in terms of a differential inclusion. Known first-order necessary conditions of optimality assert existence of a coextremal, or adjoint function, p(·), which together with x ∗(·) satisfies the Hamiltonian inclusion and associated transversality condition. In this paper we interpret extremals in terms of generalized gradients of the value function V by demonstrating that p(·) can in addition be chosen to satisfy (p(t) · x ̇ ∗(t), −p(t)) ϵ ∂V(t, x ∗(t)) , a.e. The hypothesis imposed are more or less the weakest under which the Hamiltonian inclusion condition is known to apply and permit, in particular, measurable time dependence of the data. The proof of the results relies on recent developments in Hamilton Jacobi theory applicable in such circumstances. An analogous result is proved for problems where the dynamics are modelled by a differential equation with control term.