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Zbl 1123.34065
Liu, Xinzhi; Wang, Qing
The method of Lyapunov functionals and exponential stability of impulsive systems with time delay. (English)
[J] Nonlinear Anal., Theory Methods Appl. 66, No. 7, A, 1465-1484 (2007). ISSN 0362-546X

Consider the system of functional differential equations with impulses $$ x'(t)=f(t,x_t),\, t\neq t_k, $$ $$ x(t_k^+)-x(t_k^-)=I_k(t_k,x_{t_k^-}),\, k\in\mathbb{N}, $$ $$ x_{t_0}=\phi, $$ where, as usual, $x_t(s)=x(t+s),\, s\in[-\tau,0]$. It is assumed that zero is a solution to this system. \par Using Lyapunov-like functionals, sufficient conditions are found to prove that the trivial solution is exponentially stable. The paper also contains several examples; in particular, it is shown that an unstable linear delay-differential equation may become exponentially stable by adding a suitable impulsive (nondelayed) perturbation.
[Eduardo Liz (Vigo)]

MSC 2000:: *34K45 Equations with impulses
34K20 Stability theory of functional-differential equations

Keywords: functional differential equation with impulses; Lyapunov functional; exponential stability

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