Ledyaev, Yuri S.; Sontag, Eduardo D. A Lyapunov characterization of robust stabilization. (English) Zbl 0947.34054 Nonlinear Anal., Theory Methods Appl. 37, No. 7, A, 813-840 (1999). Consider control systems of the type \((1)\) \(dx/dt = f(x,u)\), \(x \in \mathbb{R}^n\), \(u \in U\) where \(U\) is a locally compact space and \(f: \mathbb{R}^n \times U \rightarrow \mathbb{R}^n\) is continuous. The goal is to find a feedback control \(k: \mathbb{R}^n \rightarrow U\) such that the origin in \(\mathbb{R}^n\) is asymptotically stable with respect to the closed-loop system \(dx/dt = f (x,k(x))\). The feedback \(k\) is called robust when \(k\) drives the state of the system to a small neighborhood of the origin even in the presence of (small enough) external disturbance \(w\) and measurement error \(e\;(dx/dt =f(x,k(x+e(t)))+w(t))\). The authors prove the important result: For the control system (1) there exists a smooth control Lyapunov function if and only if there exists a robustly stabilizing feedback. This feedback is in general discontinuous. Reviewer: Klaus R.Schneider (Berlin) Cited in 1 ReviewCited in 38 Documents MSC: 34H05 Control problems involving ordinary differential equations 93D09 Robust stability 93D15 Stabilization of systems by feedback Keywords:robust stabilization; discontinuous feedback; control Lyapunov function PDFBibTeX XMLCite \textit{Y. S. Ledyaev} and \textit{E. D. Sontag}, Nonlinear Anal., Theory Methods Appl. 37, No. 7, 813--840 (1999; Zbl 0947.34054) Full Text: DOI