Stabilizing chaotic system on periodic orbits using multi-interval and modern optimal control strategies

In this paper, optimal approaches for controlling chaos is studied. The unstable periodic orbits (UPOs) of chaotic system are selected as desired trajectories, which the optimal control strategy should keep the system states on it. Classical gradient-based optimal control methods as well as modern optimization algorithm Particle Swarm Optimization (PSO) are utilized to force the chaotic system to follow the desired UPOs. For better performance, gradient-based is applied in multi-intervals and the results are promising. The Duffing system is selected for examining the proposed approaches. Multi-interval gradient-based approach can put the states on UPOs very fast and keep tracking UPOs with negligible control effort. The maximum control in PSO method is also low. However, due to its inherent random behavior, its control signal is oscillatory.

► We use optimal control to stabilize a chaotic system.
► The Duffing system is selected as a chaotic system for optimal control.
► The BFGS quasi-Newton gradient-based method is used for direct optimization.
► The Particle Swarm Optimization is used as an Evolutionary Algorithm optimization.
► Chaotic system is optimally stabilized in much less time than delayed feedback control.