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A class of optimal control problems of hybrid nature governed by semilinearparabolic equations is considered. These problems involve the optimization ofswitching times at which the dynamics, the integral cost, and the bounds on thecontrol may change. First- and second-order optimality conditions are derived.The analysis is based on a reformulation involving a judiciously chosentransformation of the time domains. For autonomous systems and time-independentintegral cost, we prove that the Hamiltonian is constant in time when evaluatedalong the optimal controls and trajectories. A numerical example is provided.

Hybrid Optimal Control Problems for a Class of Semilinear Parabolic Equations