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

Moulay, E.; Perruquetti, W.

Int. J. Control 81, No. 5, 797-803 (2008).

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).

Reviewer: Oleg Filatov (Samara)

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

Keywords:

Finite time stability; non-autonomous systems; Lyapunov function

References:

[1]	DOI: 10.1109/9.668834 · Zbl 0925.93821
[2]	DOI: 10.1137/S0363012997321358 · Zbl 0945.34039
[3]	Hahn W, Theory and Application of Liapunov’s Direct Method (1963) · Zbl 0119.07403
[4]	DOI: 10.1137/0324047 · Zbl 0603.93005
[5]	Hale JK, Ordinary Differential Equations, 2. ed. (1980)
[6]	DOI: 10.1016/S0167-6911(02)00119-6 · Zbl 0994.93049
[7]	DOI: 10.1109/9.905699 · Zbl 0992.93075
[8]	DOI: 10.1016/S0167-6911(02)00130-5 · Zbl 0994.93041
[9]	Khalil HK, Nonlinear Systems (1996)
[10]	Lyapunov AM, Internat. J. Contr 55 pp 520– (1992)
[11]	Moulay, E and Perruquetti, W. 2003. Finite time stability of non-linear systems. IEEE Conference on Decision and Control. 2003, Hawaii, USA. pp.3641–3646.
[12]	DOI: 10.1016/j.jmaa.2005.11.046 · Zbl 1131.93043
[13]	Moulay E, Advances in Variable Structure and Sliding Mode Control, 334, Lecture Notes in Control and Information Sciences (2006)
[14]	Perruquetti, W and Drakunov, S. 2000. Finite time stability and stabilisation. IEEE Conference on Decision and Control. 2000, Sydney, Australia. pp.1894–1899.
[15]	DOI: 10.1080/00207177908922792 · Zbl 0422.49006
[16]	Utkin VI, Sliding Modes in Control Optimization (1992)