On the Relation between the Hybrid Minimum Principle and Hybrid Dynamic Programming: a Linear Quadratic Example

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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. Key aspects of the analysis are the relationship between the Hamiltonian and the adjoint process in the Hybrid Minimum Principle before and after the switching instants, the boundary conditions on the value function in Hybrid Dynamic Programming at these switching times, as well as the relationship between the adjoint process in the Hybrid Minimum Principle and the gradient process of the value function in Hybrid Dynamic Programming. The results are illustrated through an analytic example with linear dynamics and quadratic costs.