In this talk, I will provide a historical survey of the small-gain theory for the stability analysis and control synthesis of dynamical networks. Recent developments in cyclic-small-gain theorems will be reviewed and applied to answer the first fundamental question: When is a dynamical network comprised of more than two subsystems stable? Constructive methods for the generation of Lyapunov functions for the total network on the basis of Lyapunov functions for the subsystems will be presented. Tools such as gain assignment techniques will be discussed to address the second fundamental question: When can a dynamical network be made stable by feedback? Throughout the talk, applications of the nonlinear/network small-gain theory will be shown by means of classical problems in nonlinear control design and modern problems arising from quantized and event-based control of nonlinear systems. Finally, if time permits, I will present our current work in integrating small-gain theory and adaptive dynamic programming for the learning-based adaptive optimal control of connected and autonomous vehicles.

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