Adaptive control of recurrent trajectories based on linearization of Poincare map

The problem of adaptive output feedback control aimed at stabilization of a (periodic or chaotic) goal trajectory is considered. Advantages and drawbacks of chaos control method based on linearization of Poincaré map (first proposed by Ott, Grebogi and Yorke in 1990) are discussed. It is suggested that the recurrence of the goal trajectory is the key property for applicability of approach. Algorithms of adaptive control based on linearization of controlled Poincaré map and method of goal inequalities are proposed. It is shown that stabilization of recurrent trajectories is possible under additional controllability-like and observability-like conditions. Examples of stabilization of periodic and chaotic trajectories for forced brusselator and Rössler systems are studied by computer simulations.