Local Optimality Conditions and Lipschitzian Solutions to the Hamilton–Jacobi Equation

We consider an optimal control problem with end-constraints formulated in terms of a differential inclusion. A sufficient condition for local optimality of a trajectory is given, involving a Lipschitzian function φ which is a generalized solution to the Hamilton–Jacobi equation. It is shown that the weakest hypothesis under which the condition is also necessary is that the problem be locally calm. It is further proved that local calmness is implied by strong normality. We thereby establish that the Carathéodory approach, modified to permit Lipschitzian functions φ, is applicable in principle when the first order optimality conditions yield nontrivial information.