On the existence of almost periodic Lyapunov functions for impulsive differential equations (Q1578923)

The author investigates the existence of almost-periodic piecewise continuous functions of Lyapunov's type for the system of impulsive differential equations \[ \dot x= f(t, x),\;(t\neq \tau_k),\;\Delta x(\tau_k)= I_k(x(\tau_k)),\;k= 1,2,\dots,\;x(t_0+ 0)= x_0,\;t_0\in\mathbb{R}. \] The numbers \(\tau_k\) are fixed points of the time, and \(I_k\), \(k= 1,2,\dots\), are Lipschitz continuous functions. Many processes in physics and biology can be described by these impulsive systems of ordinary differential equations. The author proves a converse theorem of the type of Massera's theorem. It is shown that for impulsive differential equations there exists a piecewise continuous almost-periodic Lyapunov function with certain properties.