The minimum principle of hybrid optimal control theory

The hybrid minimum principle (HMP) is established for the optimal control of deterministic hybrid systems with both autonomous and controlled switchings and jumps where state jumps at the switching instants are permitted to be accompanied by changes in the dimension of the state space and where the dynamics, the running and switching costs as well as the switching manifolds and the jump maps are permitted to be time varying. First-order variational analysis is performed via the needle variation methodology and the necessary optimality conditions are established in the form of the HMP. A feature of special interest in this work is the explicit presentations of boundary conditions on the Hamiltonians and the adjoint processes before and after switchings and jumps. Analytic and numerical examples are provided to illustrate the results.