Stabilizing the unstable periodic orbits of a hybrid chaotic system using optimal control

In this paper, we are interested in the control of a chaotic hybrid system with an application to Chua’s system. It is known that chaotic attractors contain an infinite number of unstable periodic orbits (UPO) with different lengths, our idea consists in stabilizing a predetermined orbit of a given length by using an optimal control method. Our approach is to determine the switching instants from one subsystem to the other while minimizing the difference between two successive orbits. Should the switchings be state dependent, as is the case for the well known Chua’s circuit, then our approach consists in perturbing the switching boundaries such that the system trajectory hits those boundaries at the specified instants. Numerical simulations illustrating the efficiency of the proposed method are presented.