A Systematic Approach for Multi-switching Compound Synchronization of Nonidentical Chaotic Systems Using Optimal Control

Multi-switching compound synchronization involves creating a new master system by combining various chaotic systems where state errors are defined in different ways resulting in multi-switching behavior. While some successful instances are documented in the literature, a generic solution to this complex problem remains elusive. This research aims to systematically explore multi-switching compound synchronization among nonidentical chaotic systems using optimal control. A crucial prerequisite for achieving synchronization with optimal control is the complete controllability of the slave chaotic systems. We propose a methodology to confirm it, which involves linearizing the slave system using the Jacobian matrix. Then, through the application of optimal control, we propose a compound multi-switching synchronization strategy to ensure the convergence of the synchronization error to zero. We illustrate our approach using four well-known and nonidentical chaotic systems. The numerical results are consistent with the theory. Additionally, we compare our method with active control, a commonly used approach in this context, and find that optimal control leads to faster synchronization error convergence. Our findings suggest no theoretical limitations to adopting optimal control in this type of synchronization. This systematic approach can be applied to other sets of nonidentical chaotic systems with minor changes.