Exponential Stability of Linear Delay Impulsive Differential Equations

Abstract

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 $$ x(τ_i + 0) = B_i x(τ_i - 0) $$ at fixed moments $τ_i$. The proof is based on a solution representation formula obtained here.