Recently, hybrid control has become increasingly relevant thanks to modern application domains involving complex systems, sub-system interconnections, mode-transitions, and heterogeneous computational devices. Hybrid models, in which continuous and discrete dynamics interact, have proved useful for capturing these types of phenomena. In this book, we consider hybrid systems with regional dynamics where transitions between different regimes occur as the state of the system crosses given switching surfaces. In particular, we focus on the associated optimal control problem and present a hybrid Bellman equation that provides a characterization of global optimality. The main benefit of the proposed approach lies in the fact that it presents both a theoretical representation of the solution's structure, and, from a more application-driven point of view, a practical algorithm that yields globally optimal solutions in a regional dynamics framework. The recursive algorithm is explored through numerous examples including a heterogeneous multi-agent problem. The results should be of interest to anyone working in: computer science, control theory, system dynamics, and operations research.

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