Optimal Control and Stochastic Synchronization of Phase Oscillators*

Synchronization of oscillations is a phenomenon prevalent in natural, social, and engineering systems, such as neural circuitry in the brain, sleep cycles in biology, semiconductor lasers in physics, and periodic vibrations in mechanical engineering. The ability to control synchronization of oscillating systems then has important research and clinical implications, for example, for the study of brain functions. In this paper, we study optimal control and synchronization of nonlinear oscillators described by the phase model. We consider both deterministic and stochastic cases, in which we derive open-loop controls that create desired synchronization patterns and devise feedback controls that maximize the steady-state synchronization probability of coupled oscillators in the presence of shared and unshared noisy stimuli modeled by the Brownian motion.