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

Consider the system

x ˙=f(t,x),x(τ)=x 0 ,(1)

where f: + × n n is continuous, τ0,x 0 n ; S(τ,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 τ + there exists δ(τ)>0, such that if x 0 δ(τ) then for all φS(τ,x 0 ):φ(t) is defined for tτ and there exists 0T(φ)< such that φ(t)=0 for all tτ+T(φ)· T 0 (φ)=inf{T(φ)0:φ(t)=0tτ+T(φ)} is called the setting time of the solution φ· If T 0 (τ,x 0 )=sup φS(τ,x 0 ) T 0 (φ)<+ then the origin is finite time stable for system (1). One of the theorems is based on a Lyapunov function V: + × n such that

V t(t,x)+ i=1 n i V t i f i (t,x)r(t,x)(2)

for all (t,x) + × n , where r: + + is a continuous definite function, r(0)=0,r(t)>0t>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 ε>0

0 ε dz r(z)<+,

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

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