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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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