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Existence and stability of limit cycles in hybrid dynamical systems with constant derivatives. 1. General theory

We consider a class of systems whose state is described by both continuous and discrete variables. The continuous state evolves with a constant velocity determined by the discrete one. As soon as the continuous state hits the boundary of a certain polyhedron depending on the current discrete state the latter evolves instantaneously, which causes a change of the above velocity. The class under consideration encompasses, in particular, fluid models of flexible manufacturing systems and queueing networks. A criterion for existence of a finite number of limit cycles attracting all the trajectories of the system is obtained. A method to determine the number of the limit cycles, as well as the region of attraction for each of them is proposed.