Convexity and Duality in Hamilton-Jacobi Theory

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

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