On the relation between the Minimum Principle and Dynamic Programming for Hybrid systems

Hybrid optimal control problems are studied for systems where autonomous and controlled state jumps are allowed at the switching instants and in addition to running costs, switching between discrete states incurs costs. A key aspect of the analysis is the relationship between the Hamiltonian and the adjoint process in the Minimum Principle before and after the switching instants as well as the relationship between adjoint processes in the Minimum Principle and the gradient of the value function. In this paper we prove that under certain assumptions the adjoint process in the Hybrid Minimum Principle and the gradient of the value function in Hybrid Dynamic Programming are governed by the same dynamic equation and have the same boundary conditions and hence are identical to each other.