Exponential Stability of Linear Delay Impulsive Differential Equations

Details Exponential Stability of Linear Delay Impulsive Differential Equations Exponential Stability of Linear Delay Impulsive Differential Equations

1993-11-26

arxiv.org/abs/funct-an/9311004v1

Mathematics

Dynamical Systems

29 pp., LaTex-file

Scientific paper

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.

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