Exponential Stability of Linear Delay Impulsive Differential Equations

TL;DR

This paper extends a known stability result to linear delay impulsive differential equations, proving exponential stability under certain conditions using a new solution representation formula.

Contribution

It introduces a novel proof for exponential stability of linear delay impulsive differential equations with impulses, generalizing existing results for non-impulsive cases.

Findings

Established exponential stability conditions for delay impulsive differential equations.

Derived a new solution representation formula for these equations.

Confirmed stability results through theoretical analysis.

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