Impulsive control for stability of Volterra functional differential equations (Q2492098)

The main result in this paper establishes sufficient conditions for the uniform (aymptotic) stability of the zero solution to the functional-differential system with impulses \[ x'(t)=F(t,x),\, t\neq t_k,\, t>t_0,\, x\in\mathbb R^n\quad ;\quad x(t_k)-x(t_k^-)=I_k(t_k,x(t_k^-)),\, k\in\mathbb N, \] where \(F\) is a Volterra-type functional, and \(I_k\) are continuous functions. For proving this result, the authors use Lyapunov-Razumikhin techniques. An application is given for an impulsive FDE with infinite delay. The authors emphasize that the consideration of impulses helps to stabilize some unstable functional-differential systems.