A considerable number of real-world processes like automotive systems, chemical processes, manufacturing plants, etc. consist of multiple subsystems which are activated by sophisticated rules. Such dynamical systems, involving continuous, real-valued states and discrete, finite-valued states, are considered as hybrid systems. The continuous, time-driven values are described by differential equations, while the discrete, event-driven dynamics are described by finite state automata.

Optimal control for hybrid systems is challenging since it requires both, the solution of the optimal activation sequence, as well as the optimal continuous control inputs to the corresponding subsystems. Several solution methods like Pontryagin's minimum (maximum) principle have been extended to the hybrid minimum (maximum) principle providing necessary optimality conditions. 

A topic of current research is the extension of the hybrid minimum principle to the case of intersecting switching manifolds for hybrid systems with partitioned state space and autonomous switching. Based on the hybrid minimum principle, efficient algorithms are developed for the solution of hybrid optimal control problems.

Optimization algorithms for hybrid optimal control are e.g. used for gait synthesis of legged robots. Mathematical models of legged locomotion are hybrid because of the strong coupling between the continuous dynamics of the robot joints and the discrete dynamics of variable ground contact. An additional challenge for optimal control is the requirement of orbital stability of periodic gaits.