The two principal and most commonly used approaches in optimal control, namely the Minimum Principle and Dynamic Programming, are studied for the general case of hybrid systems where the system state space dimension and dynamics are permitted to undergo changes at autonomous and controlled switching instants and, further, the system performance function is permitted to include running and terminal costs as well as costs associated with switches between discrete states. Subject to technical conditions, 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 times, the boundary conditions on the value function in Hybrid Dynamic Programming at such instants, as well as the identity of the adjoint process in the Hybrid Minimum Principle and the gradient process of the value function in Hybrid Dynamic Programming.
In addition to analytical examples, the particular problem of gear changing for electric vehicles is studied in this framework where the special structure of the transmission augments an additional degree of freedom to the system during the transition period. The results of time optimality and energy optimality are presented for an electric vehicle with a two speed transmission.
This is joint work with Peter E. Caines.