Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation
The problem of minimizing a given function over the attainable set (at time \(t=1)\) for non-autonomous, finite-dimensional differential inclusion is studied. The Hamilton-Jacobi method is used without assumption of differentiability of the value function. It is assumed that the right- hand-side of differential inclusion is continuous in time, locally lipschitzian in state and has non-empty, compact values. At first, two differential inequalities for the corresponding value function are introduced and fundamental properties of their solutions are studied in connections with viscosity solutions of the corresponding Hamilton-Jacobi equation. The monotonicity of solutions of these inequalities is studied. Some other different relations in the case of a locally lipschitzian value function and superdifferentiability are studied. Some sufficient conditions for the solutions of differential inequalities to be the solutions of optimal control problem under consideration are presented. The connections with Pontryagin's maximum principle are studied. The methods of generalized differentiation are used.
scientific article; zbMATH DE number 4100037
scientific article