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Hybrid systems are systems of continuous plants, subject to disturbances, interacting with sequential automata in a network. By the synthesis problem for hybrid systems we mean extracting a finite state digital controller automaton from the system equations, constraints, and cost function which define the hybrid system. This automaton senses system state, and on the basis of its state, changes state and issues a chattering control to the actuators to control the system with epsilon optimal control for a fixed epsilon in real time. We address this problem by extracting a local cost function for the control system which transforms the infinite-dimensional optimization problem into a finite-dimensional problem.