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Finite time stability conditions for non-autonomous continuous systems. (English) Zbl 1152.34353

Consider the system

$\stackrel{˙}{x}=f\left(t,x\right),\phantom{\rule{1.em}{0ex}}x\left(\tau \right)={x}_{0},\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $f:{ℝ}_{+}×{ℝ}^{n}\to {ℝ}^{n}$ is continuous, $\tau \ge 0,{x}_{0}\in {ℝ}^{n};$ $S\left(\tau ,{x}_{0}\right)$ 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 $\tau \in {ℝ}_{+}$ there exists $\delta \left(\tau \right)>0,$ such that if $\parallel {x}_{0}\parallel \le \delta \left(\tau \right)$ then for all $\varphi \in S\left(\tau ,{x}_{0}\right):\varphi \left(t\right)$ is defined for $t\ge \tau$ and there exists $0\le T\left(\varphi \right)<\infty$ such that $\varphi \left(t\right)=0$ for all $t\ge \tau +T\left(\varphi \right)·$ ${T}_{0}\left(\varphi \right)=inf\left\{T\left(\varphi \right)\ge 0:\varphi \left(t\right)=0\phantom{\rule{0.166667em}{0ex}}\forall t\ge \tau +T\left(\varphi \right)\right\}$ is called the setting time of the solution $\varphi ·$ If ${T}_{0}\left(\tau ,{x}_{0}\right)={sup}_{\varphi \in S\left(\tau ,{x}_{0}\right)}{T}_{0}\left(\varphi \right)<+\infty$ then the origin is finite time stable for system (1). One of the theorems is based on a Lyapunov function $V:{ℝ}_{+}×{ℝ}^{n}\to ℝ$ such that

$\frac{\partial V}{\partial t}\left(t,x\right)+\sum _{i=1}^{n}\frac{{\partial }^{i}V}{\partial {t}_{i}}{f}_{i}\left(t,x\right)\le r\left(t,x\right)\phantom{\rule{2.em}{0ex}}\left(2\right)$

for all $\left(t,x\right)\in {ℝ}_{+}×{ℝ}^{n},$ where $r:{ℝ}_{+}\to {ℝ}_{+}$ is a continuous definite function, $r\left(0\right)=0,r\left(t\right)>0\phantom{\rule{0.166667em}{0ex}}\forall t>0·$

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 $\epsilon >0$

${\int }_{0}^{\epsilon }\frac{dz}{r\left(z\right)}<+\infty ,$

then the origin (1) is finite time stable for system (1).

##### MSC:
 34D20 Stability of ODE 93D05 Lyapunov and other classical stabilities of control systems