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On the stochastic Minimum Principle for hybrid systems

A class of stochastic hybrid systems with both autonomous and controlled switchings and jumps is considered where autonomous and controlled state jumps at the switching instants are accompanied by changes in the dimension of the state space. Optimal control problems associated with this class of stochastic hybrid systems are studied where in addition to running and terminal costs, switching between discrete states incurs costs. Necessary optimality conditions are established in the form of the Stochastic Hybrid Minimum Principle. A feature of special importance is the effect of hard constraints imposed by switching manifolds on diffusion-driven state trajectories which influence the boundary conditions for the stochastic Hamiltonian and adjoint processes.