Monotonicity and differential properties of the value functions in optimal control

Using the “basic monotonicity property” along locally admissible trajectories, we extend to very general problems certain existing results concerning the differential inequalities verified by the value function of an optimal control problem; these differential inequalities are expressed in terms of its contingent, quasitangent, and peritangent (Clarke’s) directional derivatives and in terms of certain sets of “generalized tangent directions” to the “locally admissible trajectories.” Under additional, rather restrictive hypotheses on the data, which allow suitable estimates (and even exact characterizations) of the sets of generalized tangent directions to the trajectories, the differential inequalities are shown to imply previous results according to which the value function is a “generalized solution” (in the “contingent,” “viscosity,” or “Clarke” sense) of the associated Hamilton-Jacobi-Bellman (HJB) equation.