## Asymptotic controllability implies feedback stabilization.(English)Zbl 0892.93053

The paper deals with the connection between asymptotic controllability to $$0\in\mathbb{R}^n$$ of $\dot x= f(x,u)\tag{1}$ by $$u:[0,+\infty)\to U\subset\mathbb{R}^n$$ and existence of a feedback control $$k:\mathbb{R}^n\to U$$ that makes the equilibrium at $$0\in\mathbb{R}^n$$ of $\dot x= f(x,k(x))\tag{2}$ globally asymptotically stable. These properties which are equivalent in the linear case, turn to be equivalent in the nonlinear case, with a suitable choice of the class of feedback control functions. These are the $$s$$-stabilizing functions defined as piecewise constant functions $$k(x(t))= k(x(t_i))$$, $$t_i\leq t\leq t_{i+1}$$, with $$x(t_i)$$ the endpoint of the solution on the preceding interval. The solution thus defined, of $\dot x= f(x,k(x(t_i))),\quad t_i\leq t\leq t_{i+1}\tag{3}$ is called $$\pi$$-trajectory. A function $$k(x)$$ is $$s$$-stabilizing if the equilibrium at $$0$$ of (3) is attractive, Lyapunov stable and the solutions are equi-bounded.
The main result of the paper reads simply: System (1) is asymptotically controllable iff it admits a $$s$$-stabilizing feedback.
This deep result is proved using techniques from control Lyapunov functions, nonsmooth analysis and positional differential games.
The paper gives first a selfcontained exposition of proximal subgradients for the control Lyapunov functions. This theory allows construction of the stabilizing control on compact sets: the solution is “guided” from one compact domain to another by constant controls. The domains that converge to the origin are bounded by level surfaces of the Lyapunov functions. Then the global result is proved.

### MSC:

 93D15 Stabilization of systems by feedback 93B05 Controllability 93C57 Sampled-data control/observation systems
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