Functional role of synchronization: A mean-field control perspective

Our friend and mentor Peter Caines has, together with his colleagues, created new foundations for studying collective dynamics in complex systems. Of particular inspiration to us has been his pioneering work in mean-field games (MFGs) launched two decades ago [10, 24, 25], and the related field of mean-field control. Peter pointed the way to both formulate and solve the problem of collective dynamics arising in a large population of heterogeneous dynamical systems. In this paper we survey some elements of MFGs within the context of controlled coupled oscillators. We begin by introducing a model for a single oscillator: dθ(t) = (ω + u(t)) dt + σ dξ(t), mod 2π (1) where θ(t) [0, 2π) is the phase of the oscillator at time t, ω is the nominal frequency with units of radiansper-second, {ξ(t): t 0} is a standard Wiener process, and u(t) is a control signal whose interpretation depends on the context. Unless otherwise noted, the SDEs are interpreted in their Itô form.