## 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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