Exponential Stability of Linear Delay Impulsive Differential Equations

For ordinary differential equations and functional differential equations the following result is well known. Suppose any solution is bounded on the half-line for each bounded on the half-line right-hand side. Then under certain conditions the equations is exponentially stable. We prove the same result for a delay differential equation $$\dot{x}(t)+\sum _{i=1}^k{A}_i(t)x[h_i(t)]=f(t),$$ with impulses $$x({\tau }_i+0)=B_{i}x({\tau }_i-0)$$ at fixed moments ${\tau }_i$. The proof is based on a solution representation formula obtained here.