Robust global exponential stability of uncertain impulsive systems (Q1774251)

Some sufficient conditions are derived for global exponential stability of the system \[ \dot x(t) =f(x(t)),\quad t\in (t_k,t_{k+1}],\quad \Delta x(t_k) =I_k(x(t_k)), \] where \(t\geq t_0\), \(x(t)\in \mathbb{R}^n\), \(\Delta x(t_k)=x(t^+_k)-x(t_k)\), \(f: \mathbb R^n\to \mathbb R^n\) is Lipschitz continuous and \(f(0)=0\), \(I_k:\mathbb R^n\to \mathbb R^n\) is continuous, \(I_k(0)=0\) and \(0<t_0<t_1<t_2<\dots\); \(\lim t_k=\infty\), for \(k\to\infty\). Further, a nonlinear system of the form \[ \begin{aligned} \dot x & =f(x)+\varphi_c(t,x),\quad t\in(t_k,t_{k+1}],\\ \Delta x & =I_k(x)+\varphi_d(t,x),\quad t =t_k,\\ x(t_0)& =x_0,\end{aligned}\tag{2} \] is considered, where \(\varphi_c(t,x)\) and \(\varphi_d(t,x)\) represent structural uncertainties. Sufficient conditions for robust global exponential stability of (2) are given.