A Universal Construction of Artstein's Theorem on Nonlinear Stabilization

Authors

Eduardo D. Sontag

Year

1989

Journal Paper

Algorithm

Sontag Universal Formula

Source

Systems and Control Letters

Identifiers

DOI 10.1016/0167-6911(89)90028-5

Abstract¶

This note presents an explicit proof of the theorem - due to Artstein - which states that the existence of a smooth control-Lyapunov function implies smooth stabilizability. Moreover, the result is extended to the real-analytic and rational cases as well. The proof uses a 'universal' formula given by an algebraic function of Lie derivatives; this formula originates in the solution of a simple Riccati equation.

Summary¶

Sontag gives a universal feedback construction that turns suitable control Lyapunov functions into stabilizing nonlinear controllers. The formula made Artstein's existence theorem constructive and became a standard tool in nonlinear stabilization and control-Lyapunov-function design.