Global stabilization of nonlinear systems based on vector control lyapunov functions

TL;DR

This paper demonstrates that the existence of a vector control Lyapunov function is both necessary and sufficient for designing smooth, globally stabilizing feedback laws for nonlinear systems, with practical applications to reaction networks.

Contribution

It establishes a fundamental equivalence between vector control Lyapunov functions and the existence of smooth global stabilizers for nonlinear systems.

Findings

Necessary and sufficient condition for smooth global stabilization.

Proposed simple, checkable conditions for feedback law existence.

Application to stabilization of reaction networks in chemical reactors.

Abstract

This paper studies the use of vector Lyapunov functions for the design of globally stabilizing feedback laws for nonlinear systems. Recent results on vector Lyapunov functions are utilized. The main result of the paper shows that the existence of a vector control Lyapunov function is a necessary and sufficient condition for the existence of a smooth globally stabilizing feedback. Applications to nonlinear systems are provided: simple and easily checkable sufficient conditions are proposed to guarantee the existence of a smooth globally stabilizing feedback law. The obtained results are applied to the problem of the stabilization of an equilibrium point of a reaction network taking place in a continuous stirred tank reactor.