The minimum principle for time-varying hybrid systems with state switching and jumps

The hybrid minimum principle (HMP) is extended to hybrid systems with autonomous (internally forced) switching on switching manifolds, controlled (externally forced) switching, jumps of the continuous state when switching, and time-varying functions for specifying the continuous and discrete dynamics. The formulation of the hybrid optimal control problem (HOCP) includes running, switching, and terminal costs. The HMP provides necessary optimality conditions for a solution of the HOCP and can be used as basis for developing numerical optimal control algorithms.