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Finite time stability conditions for non-autonomous continuous systems. (English) Zbl 1152.34353
Consider the system $\dot{x} = f(t,x),\quad x(\tau)= x_{0}, \tag{1}$ where $$f:\mathbb{R}_{+}\times\mathbb{R}^{n}\to \mathbb{R}^{n}$$ is continuous, $$\tau\geq 0, x_{0} \in \mathbb{R}^{n};$$ $$S(\tau,x_{0})$$ 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 $$\tau\in \mathbb{R}_{+}$$ there exists $$\delta(\tau) >0,$$ such that if $$\| x_{0}\| \leq \delta(\tau)$$ then for all $$\varphi\in S(\tau,x_{0}): \varphi(t)$$ is defined for $$t\geq\tau$$ and there exists $$0\leq T(\varphi)<\infty$$ such that $$\varphi(t) =0$$ for all $$t\geq \tau + T(\varphi).$$ $$T_{0}(\varphi)=\inf\{T(\varphi)\geq 0: \varphi(t)=0 \, \forall t\geq \tau +T(\varphi)\}$$ is called the setting time of the solution $$\varphi.$$ If $$T_{0}(\tau,x_{0})=\sup_{\varphi\in S(\tau,x_{0})}T_{0}(\varphi)< +\infty$$ then the origin is finite time stable for system (1). One of the theorems is based on a Lyapunov function $$V:\mathbb{R}_{+}\times\mathbb{R}^{n}\to \mathbb{R}$$ such that $\frac{\partial V}{\partial t}(t,x)+\sum_{i=1}^{n} \frac{\partial^{i}V}{\partial t_{i}}f_{i}(t,x)\leq r(t,x) \tag{2}$ for all $$(t,x)\in\mathbb{R}_{+}\times\mathbb{R}^{n},$$ where $$r:\mathbb{R}_{+}\to \mathbb{R}_{+}$$ is a continuous definite function, $$r(0)=0, r(t)>0 \, \forall t>0.$$
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 $$\varepsilon >0$$ $\int_{0}^{\varepsilon}\frac{dz}{r(z)} <+\infty,$ then the origin (1) is finite time stable for system (1).

##### MSC:
 34D20 Stability of solutions to ordinary differential equations 93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, $$L^p, l^p$$, etc.) in control theory
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