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.