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Stabilization of uncertain control systems through piecewise constant feedback. (English) Zbl 0891.93068
An uncertain system with dynamics defined by $$\dot{x}(t)=f(x(t),u(t),d(t)),\quad x(t) \in R^n,\ u(t) \in U \subset R^m,\ d(t) \in D \subset R^l, $$ where $t$, $u$, $d$, and $x$ are the time, the control, the disturbance and the state, is considered. The sets $U$ and $D$ are supposed to be closed. Let $V:R^n \to R^+$ be a given function and $a$ be a positive parameter. The following approximate stabilization problem is studied: Fix $T>0$ and $\eta>0$. The aim is to find a control law $u=u(t,x)\in U$ such that for any measurable $d(.)$ the system of ordinary differential equations $$\dot{x}(t)=f(x(t),u(t,x(t)),d(t)),\ t \in [0,T] $$ has at least one solution and any solution $x(.)$ satisfies $$ V(x(t))\le V(x(0))e^{-at}+\eta,\ t \in [0,T]. $$ Assuming that $V$ is Lipschitz continuous and satisfies $$ \inf_{u \in U} \sup_{d\in D} \ \langle f(x,u,d),p\rangle \le aqV(x) $$ for each proximal normal $(p,q)$ to the epigraph $\text{epi}V$ of $V$ at the point $(x,V(x))$, the existence of a piecewise constant feedback satisfying the approximate stabilization property is proved under natural technical assumptions on $f$. This is the main result of the paper. Its proof relies on the fact that the conditions imposed on $V$ imply viability of the set $\text{epi}V$ with respect to the following augmented system: $$ \dot{x}(t)=f(x(t),u(t),d(t)),\ u(t) \in U,\ d(t) \in D,\ \dot{w}(t)=-aw(t).$$ Thus, for any $\varepsilon > 0$ there exists a piecewise control such that for any measurable disturbance $d(.)$ the corresponding trajectory $(x(.),w(.))$ starting from $(x_0, V(x_0))$ is defined on $[0,T]$ and the distance between $(x(t),w(t)), \ t \in [0,T]$ and $\text{epi}V$ is not greater than $\varepsilon$.

MSC:
93D21Adaptive or robust stabilization
93D15Stabilization of systems by feedback
93C41Control problems with incomplete information
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References:
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