The Hamiltonian–Jacobi–Bellman Equation for Time-Optimal Control

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Abstract
In this paper several assertions concerning viscosity solutions of the Hamilton–Jacobi–Bellman equation for the optimal control problem of steering a system to zero in minimal time are proved. First two rather general uniqueness theorems are established, asserting that any positive viscosity solution of the HJB equation must, in fact, agree with the minimal time function near zero; if also a boundary condition introduced by Bardi [SIAM J Control Optim., 27 (1988), pp. 776–785] is satisfied, then the agreement is global. Additionally, the Hölder continuity of any subsolution of the HJB equation is proved in the case where the related dynamics satisfy a Hörmander-type hypothesis. This last assertion amounts to a “half-derivative” analogue of a theorem of Crandall and Lions [Traps. Amer. Math. Soc., 277 (1.983), pp. 1–42] concerning Lipschitz viscosity solutions.

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