Convexity in Hamilton--Jacobi Theory I: Dynamics and Duality

Abstract

Value functions propagated from initial or terminal costs and constraints by way of a differential inclusion, or more broadly through a Lagrangian that may take on $\infty$, are studied in the case where convexity persists in the state argument. Such value functions, themselves taking on $\infty$, are shown to satisfy a subgradient form of the Hamilton--Jacobi equation which strongly supports properties of local Lipschitz continuity, semidifferentiability and Clarke regularity. An extended method of characteristics is developed which determines them from the Hamiltonian dynamics underlying the given Lagrangian. Close relations with a dual value function are revealed.