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Hybrid control systems and viscosity solutions. (English) Zbl 1108.49024

Summary: We investigate a model of hybrid control system in which both discrete and continuous controls are involved. In this general model, discrete controls act on the system at a given set interface. The state of the system is changed discontinuously when the trajectory hits predefined sets, namely, an autonomous jump set \(A\) or a controlled jump set \(C\) where the controller can choose to jump or not. At each jump, the trajectory can move to a different Euclidean space. We prove the continuity of the associated value function \(V\) with respect to the initial point. Using the dynamic programming principle satisfied by \(V\), we derive a quasi-variational inequality satisfied by \(V\) in the viscosity sense. We characterize the value function V as the unique viscosity solution of the quasi-variational inequality by the comparison principle method.

MSC:

49L20 Dynamic programming in optimal control and differential games
34H05 Control problems involving ordinary differential equations
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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