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A Lyapunov characterization of robust stabilization. (English) Zbl 0947.34054

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.

MSC:

34H05 Control problems involving ordinary differential equations
93D09 Robust stability
93D15 Stabilization of systems by feedback
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