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Finite time stability conditions for non-autonomous continuous systems. (English) Zbl 1152.34353
Consider the system $$ \dot{x} = f(t,x),\quad x(\tau)= x\sb{0}, \tag1 $$ where $f:\Bbb{R}\sb{+}\times\Bbb{R}\sp{n}\to \Bbb{R}\sp{n}$ is continuous, $\tau\ge 0, x\sb{0} \in \Bbb{R}\sp{n};$ $S(\tau,x\sb{0})$ is the set of all solutions of system (1). The origin is weakly finite time stable for system (1) if the origin is Lyapunov stable for system (1) and for all $\tau\in \Bbb{R}\sb{+}$ there exists $\delta(\tau) >0,$ such that if $\Vert x\sb{0}\Vert \le \delta(\tau)$ then for all $\varphi\in S(\tau,x\sb{0}): \varphi(t)$ is defined for $t\ge\tau$ and there exists $0\le T(\varphi)<\infty$ such that $\varphi(t) =0$ for all $t\ge \tau + T(\varphi).$ $T\sb{0}(\varphi)=\inf\{T(\varphi)\ge 0: \varphi(t)=0 \, \forall t\ge \tau +T(\varphi)\}$ is called the setting time of the solution $\varphi.$ If $ T\sb{0}(\tau,x\sb{0})=\sup\sb{\varphi\in S(\tau,x\sb{0})}T\sb{0}(\varphi)< +\infty $ then the origin is finite time stable for system (1). One of the theorems is based on a Lyapunov function $V:\Bbb{R}\sb{+}\times\Bbb{R}\sp{n}\to \Bbb{R}$ such that $$ \frac{\partial V}{\partial t}(t,x)+\sum\sb{i=1}\sp{n} \frac{\partial\sp{i}V}{\partial t\sb{i}}f\sb{i}(t,x)\le r(t,x) \tag2 $$ for all $(t,x)\in\Bbb{R}\sb{+}\times\Bbb{R}\sp{n},$ where $r:\Bbb{R}\sb{+}\to \Bbb{R}\sb{+}$ is a continuous definite function, $r(0)=0, r(t)>0 \, \forall t>0.$ \newline Theorem. Let the origin be an equilibrium point for system (1). If there exists a continuously differentiable Lyapunov function satisfying condition (2) such that for some $\varepsilon >0$ $$ \int\sb{0}\sp{\varepsilon}\frac{dz}{r(z)} <+\infty, $$ then the origin (1) is finite time stable for system (1).

34D20Stability of ODE
93D05Lyapunov and other classical stabilities of control systems
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