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Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

L. Berezansky·E. Braverman
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Abstract

Consider a linear impulsive equation in a Banach space $$\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0,$$ $$x(τ_i +0)= B_i x(τ_i -0) + α_i,$$ with $\lim_{i \rightarrow \infty} τ_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 $\{ α_i \}$. 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 $\{ α_i \}$ ; (c) $\lim_{t \rightarrow \infty}x(t)=0$ for any $f, α_i$ tending to zero; (d) exponential estimate of $f$ implies a similar estimate for $x$.

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BibTeXRIS

L. Berezansky, E. Braverman. 1993-12-06. Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space. https://doi.org/10.1007/bf00675173

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