# Stability Theory via Vector Lyapunov Functions

# Stability Theory via Vector Lyapunov Functions

This chapter describes a fundamental stability theory for nonlinear dynamical systems using vector Lyapunov functions. It first introduces the notation and definitions before developing stability theorems via vector Lyapunov functions for continuous-time and discrete-time nonlinear dynamical systems. It then extends the theory of vector Lyapunov functions by constructing a generalized comparison system whose vector field can be a function of the comparison system states as well as the nonlinear dynamical system states. It also presents a generalized convergence result which, in the case of a scalar comparison system, specializes to the classical Krasovskii–LaSalle theorem. In the analysis of large-scale nonlinear interconnected dynamical systems, several Lyapunov functions arise naturally from the stability properties of each individual subsystem.

*Keywords:*
stability theory, nonlinear dynamical system, vector Lyapunov function, comparison system, vector field, convergence, Krasovskii–LaSalle theorem, interconnected dynamical system

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