The method of Lyapunov functionals and exponential stability of impulsive systems with time delay (Q875258)

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. 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.