Convexity in Hamilton-Jacobi theory - I: Dynamics and duality

Value functions propagated from initial or terminal costs and constraints b y way of a differential inclusion, or more broadly through a Lagrangian tha t may take on infinity, are studied in the case where convexity persists in the state argument. Such value functions, themselves taking on infinity, a re shown to satisfy a subgradient form of the Hamilton Jacobi equation whic h strongly supports properties of local Lipschitz continuity, semidifferent iability and Clarke regularity. An extended method of characteristics is de veloped which determines them from the Hamiltonian dynamics underlying the given Lagrangian. Close relations with a dual value function are revealed.