Lyapunov regularity of impulsive differential equations
The authors consider a linear impulsive differential equation in \(\mathbb{R}^n\) of the form \[ x'= A(t)x,\quad t\neq\tau_i,\quad\Delta x|_{t=\tau_i}= B_i x.\tag{1} \] \(A(t)\) are \(n\times n\) matrices varying continuously for \(t\geq 0\), \(B_1,B_2,\dots\) are given matrices and \(0<\tau_1< \tau_2<\cdots\)\,. A solution \(x\) of (1) is a function of class \(C^1\) except the times \(\tau_i\) and having discontinuities of the first kind at these times with \[ x'(t)= A(t) x\qquad (t\neq \tau_i), \] \[ \Delta x|_{t=\tau_i}= x(\tau^+_i)- x(\tau^-_i)\quad\text{for }i\in\mathbb{N}. \] The notion of Lyapunov regularity for an impulsive differential equation is introduced. Sufficiently conditions for the existence of stable and unstable invariant manifolds for the solution of the nonlinear differential equation \[ x'= A(t) x+ f(t,x) \] are derived. By means of so-called exponential contraction sufficient conditions for the existence of a solution \(x(t)\) of the equation \[ x'= A(t)x+ f(t,x(t)),\quad t\neq \tau_i,\quad\Delta x|_{t=\tau_i}= B_i x+ q_i(x) \] are found. The following estimate is valid: \[ \| x(t)\|\leq Ke^{\widetilde a(t- s)+ as}\| x(s)\|,\quad t\geq s, \] \(K\), \(\widetilde a\), \(a\)-const.
linear impulse differential equations in \(\mathbb{R}^n\)
scientific article; zbMATH DE number 5793315