New exponential stability conditions for linear delayed systems of differential equations

Abstract

New explicit results on exponential stability, improving recently published results by the authors, are derived for linear delayed systems $$ \dot{x}_i(t)=-\sum_{j=1}^m \sum_{k=1}^{r_{ij}}a_{ij}^{k}(t)x_j(h_{ij}^{k}(t)),\qquad i=1,\dots,m $$ where $t\ge 0$, $m$ and $r_{ij}$, $i,j=1,\dots,m$ are natural numbers, $a_{ij}^{k}\colon [0,\infty)\to\mathbb{R}$ are measurable coefficients, and $h_{ij}^{k}\colon [0,\infty)\to\mathbb{R}$ are measurable delays. The progress was achieved by using a new technique making it possible to replace the constant $1$ by the constant $1+{1}/{\mathrm{e}}$ on the right-hand sides of crucial inequalities ensuring exponential stability.

Keywords

• exponential stability
• linear delayed differential system
• estimate of fundamental function
• bohl-perron theorem