  Hybrid optimal control problems are studied for a general class of hybrid
systems where autonomous and controlled state jumps are allowed at the
switching instants and in addition to terminal and running costs switching
between discrete states incurs costs. The statements of the Hybrid Minimum
Principle and Hybrid Dynamic Programming are presented in this framework and it
is shown that under certain assumptions the adjoint process in the Hybrid
Minimum Principle and the gradient of the value function in Hybrid Dynamic
Programming are governed by the same set of differential equations and have the
same boundary conditions and hence are almost everywhere identical to each
other along optimal trajectories. Analytic examples are provided to illustrate
the results and, in particular, a Riccati formalism for linear quadratic hybrid
tracking problems is presented.
