Symplectic Geometry in Hybrid and Impulsive Optimal Control

Abstract: Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory is applied to these systems and a hybrid version of Pontryagin's maximum principle is presented. This hybrid maximum principle is presented to emphasize its geometric nature which makes its study amenable to the tools of geometric mechanics and symplectic geometry. One explicit benefit of this geometric approach is that the symplectic structure (and hence the induced volume) is preserved. This allows for a hybrid analog of caustics and conjugate points. Additionally, an introductory analysis of singular solutions (beating and Zeno) is discussed geometrically. This work concludes on a biological example where beating can occur.