On the Hybrid Minimum Principle: The Hamiltonian and Adjoint Boundary Conditions

The Hybrid Minimum Principle (HMP) is presented 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. A feature of particular importance is the explicit presentation of the boundary conditions on the Hamiltonians and the adjoint processes before and after switchings and jumps. The numerical benefit of these expressions are demonstrated on a modified version of the Multiple Autonomous Switchings (MAS) algorithm. The results are illustrated for the hybrid model of an electric vehicle powertrain with a two-speed transmission.