Dual Value Function Research Articles

Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations

Value functions for convex optimal control problems on infinite time intervals are studied in the framework of duality. Hamilton-Jacobi characterizations and the conjugacy of primal and dual value functions are of main interest. Close ties between the uniqueness of convex solutions to a Hamilton-Jacobi equation, the uniqueness of such solutions to a dual Hamilton-Jacobi equation, and the conjugacy of primal and dual value functions are displayed. Simultaneous approximation of primal and dual infinite horizon problems with a pair of dual problems on finite horizon, for which the value functions are conjugate, leads to sufficient conditions on the conjugacy of the infinite time horizon value functions. Consequently, uniqueness results for the Hamilton-Jacobi equation are established. Little regularity is assumed on the cost functions in the control problems, correspondingly, the Hamiltonians need not display any strict convexity and may have several saddle points.

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

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.