Adaptive Dynamic Programming for Optimal Synchronization of Kuramoto Oscillator

This paper addresses the problem of optimal synchronization of the Kuramoto oscillator model with the Ott-Antonsen framework. The Ott-Antonsen ansatz is used to analyze the dynamics of the large networks of interactive units. Besides, this approach reduces the infinite-dimensional dynamics to phase space flow, i.e., low dimensional dynamics for certain systems of globally coupled phase oscillators. For a collection of non-homogeneous oscillators, the states are elucidated as phase angles, which is the modification of the model for a coupled Kuramoto oscillator. In order to achieve optimal synchronization of the Kuramoto oscillator model, the Hamiltonian-Jacobi-Bellman (HJB) expression obtained from the Ott-Antonsen framework, which is extremely difficult to solve in general is solved using adaptive dynamic programming (ADP). This paper develops an ADP algorithm for learning approximate optimal control laws in terms of coefficient of coupling and order parameter to address the synchronism of the Kuramoto oscillator model. ADP has been contemplated as one of the efficient methods to solve optimal control of nonlinear systems. Finally, local synchronism of the coupled Kuramoto oscillator model is shown with the help of simulation analysis for the order parameter as a function of time.