Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

TL;DR

This paper investigates the boundedness and stability of solutions to impulsively perturbed linear systems in Banach spaces, establishing conditions for exponential stability, boundedness, and asymptotic behavior of solutions.

Contribution

It provides new criteria for stability and boundedness of impulsively perturbed systems in Banach spaces, including exponential estimates and asymptotic convergence conditions.

Findings

Conditions for exponential stability of homogeneous solutions.

Criteria ensuring solutions remain bounded with bounded inputs.

Results on asymptotic convergence to zero when inputs tend to zero.

Abstract

Consider a linear impulsive equation in a Banach space $$\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0,$$ $$x(\tau_i +0)= B_i x(\tau_i -0) + \alpha_i,$$ with $\lim_{i \rightarrow \infty} \tau_i = \infty $. Suppose each solution of the corresponding semi-homogeneous equation $$\dot{x}(t)+A(t)x(t) = 0,$$ (2) is bounded for any bounded sequence $\{ \alpha_i \}$. The conditions are determined ensuring (a) the solution of the corresponding homogeneous equation has an exponential estimate; (b) each solution of (1),(2) is bounded on the half-line for any bounded $f$ and bounded sequence $\{ \alpha_i \}$ ; (c) $\lim_{t \rightarrow \infty}x(t)=0$ for any $f, \alpha_i$ tending to zero; (d) exponential estimate of $f$ implies a similar estimate for $x$.