Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

Details Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

1993-12-06

arxiv.org/abs/funct-an/9312001v1

Mathematics

Dynamical Systems

19 pages, LaTex-file

Scientific paper

10.1007/BF00675173

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 $x$.

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