Adaptive alternatives in the velocity control of Mean-Field Kuramoto Models

This paper presents an alternative control design for the macroscopic model of Kuramoto oscillators. The resulting partial differential equation describing the density of the collective dynamics is a nonlinear version of the continuity equation in which the control signals are coupled to the state. An optimal control for the bilinear system provides optimality of the controllers but it also results in an open-loop policy due to the backward in time integration of adjoint states. To tackle this, an adaptive alternative is considered and which views the control signals corresponding to a target density, as unknown spatiotemporally varying parameters. An adaptive observer is proposed along with the Lyapunov-redesign of the parameter adaptive laws. The stability of the closed-loop density equation using adaptive estimates of the controllers along with tracking convergence are examined.