Pseudo-Spectral Method for Finite and Infinite Time Synchronization Problems

This paper presents a unified pseudo-spectral framework for solving both finite-time and infinite-time synchronization problems in nonlinear dynamical systems. The core of our approach is the formulation of two novel optimal control problems, where the achievement of synchronization is established as the optimal solution. We rigorously prove that the solution to these optimal control problems guarantees synchronization. The proposed method leverages a Legendre pseudo-spectral technique to transcribe the continuous-time optimal control problems into nonlinear programming problems. The unknown coefficients of the approximating polynomials are then efficiently determined by solving the resulting algebraic system derived from the Karush-Kuhn-Tucker optimality conditions. The efficacy and superiority of the method are demonstrated through several numerical examples, where it is compared against other well-known synchronization techniques. The results confirm that our method offers significant advantages in accuracy and convergence. Finally, we discuss the potential for extending this approach to a broader class of complex synchronization problems.