Stability of impulsive delay differential equations
In this paper, the authors introduce the impulsive delayed Cauchy matrix and seek the formula of solutions for the following linear impulsive delay differential equation \[ \begin{aligned} \dot x(t) & = Ax(t)+Bx(t-\tau),\quad t\geq0, \tau>0, t\neq t_i,\\ \Delta x(t_i) & = C_ix(t_i),\quad i=1, 2, \dots,\\ x(t) & = \varphi(t), \quad -\tau\leq t\leq0,\end{aligned} \] where \(A, B, C_i\) are constant matrices, \(AB=BA\), \(AC_i=C_iA\) and \(BC_i=C_iB\) for each \(i=1, 2,\dots\), \(\varphi\in C^1([-\tau, 0], \mathbb{R}^n)\), \(x(t)\in \mathbb{R}^n\), and time sequences \(\{t_k\}_{k=1}^{\infty}\) satisfy \(0=t_0<t_1<\dots<t_k<\dots\), \(\Delta x(t_k)=x(t_k^+)-x(t_k^-)\).
scientific article; zbMATH DE number 6859484
scientific article