An exponentional estimate on the fundamental matrix of a linear impulsive system (Q2759327)

The authors consider the system of differential equations with impulses NEWLINE\[NEWLINE\frac{dx}{dt}=(A(\tau)+\overline{A}(\varphi,\tau))x,\;\tau\not=\tau_j;\quad \Delta x|_{\tau=\tau_j}=\varepsilon(B_j+B(\varphi,\tau_j))x;NEWLINE\]NEWLINE with \(\tau\in(t,+\infty),\) \(x\in\mathbb{R}^n,\) \(\varphi\in\mathbb{R}^m,\) \(\tau_{j+1}-\tau_j=\theta\varepsilon,\) \(t_1>t,\) \(\varepsilon\) a small parameter and the matrices \(B_j\) constant. The averaging of the matrices \({\overline A}(\varphi,\tau)\) and \(B(\varphi,\tau_j)\) are supposed to be small enough on cubes of the periods \(0\leq\varphi_\nu\leq 2\pi,\) \(\nu=1,\dots,m.\) The matrizant \(Q(\tau,t,\varepsilon)\) of the linear impulsive system \(\frac{dx}{dt}= A(\tau)x,\) \(\tau\not=\tau_j;\) \(\Delta x|_{\tau=\tau_j}=\varepsilon B_j x,\) is supposed to satisfy the inequality \(\|Q(\tau,t,\varepsilon)\|\leq K\exp\{-\gamma(\tau - t)\}.\) By means of a uniform estimate on oscillating integrals and oscillating sums, the exponential estimate \(\|\Omega_t^{\tau}(\psi, \varepsilon) \|\leq K_1\|\exp\{-\gamma_1(\tau-t)\}\|\) is proved, where \(K_1\) does not depend on \(\varepsilon,\) \(\gamma_1\in(0,\gamma),\) and the function \(\varphi=\varphi_t^{\tau}(\psi,\varepsilon)\) is a solution to the impulsive Cauchy problem NEWLINE\[NEWLINE\frac{d\varphi}{dt} = \frac{\omega(\tau)}{\varepsilon} + b(\varphi, \tau), \;\tau\not=\tau_j; \quad \Delta\varphi|_{\tau= \tau_j} = \varepsilon \Phi_j (\varphi), \;\varphi|_{\tau=t}=\psi.NEWLINE\]