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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Asymptotic behavior of linear integrodifferential systems


Authors: Viorel Barbu and Stanley I. Grossman
Journal: Trans. Amer. Math. Soc. 173 (1972), 277-288
MSC: Primary 45M05
MathSciNet review: 0308712
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Abstract: We consider the system $ ({\text{L)}}y'(t) = Ay(t) + \int_{ - \infty }^t {B(t - s)y(s)ds,y(t) = f(t),t \leqslant 0} $ where $ y(t)$ is an $ n$-vector and $ A$ and $ B(t)$ are $ n \times n$ matrices. System $ ({\text{L)}}$ generates a semigroup given by $ {T_t}f(s) = y(t + s;f)$ for $ f$ bounded, continuous and having a finite limit at $ - \infty $. Under hypotheses concerning the roots of $ \det (\lambda I - A - \hat B(\lambda ))$, where $ \hat B(\lambda )$ is the Laplace transform, various results about the asymptotic behavior of $ y(t)$ are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If $ B(t) \in {L^1}[0,\infty )$ and $ {(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $ \operatorname{Re} \lambda > 0$, then for every $ \epsilon > 0$, there is an $ {M_{\epsilon}}$ such that $ \vert\vert{T_t}f\vert\vert \leqslant {M_{\epsilon}}{e^{\epsilon t}}\vert\vert f\vert\vert$. Theorem 2. If $ {(\lambda I - A - \hat B(\lambda ))^{ - 1}}$ exists for $ \operatorname{Re} \lambda > - \alpha (\alpha > 0)$ and if $ B(t){e^{\alpha t}} \in {L^1}[0,\infty )$, then the solution to $ ({\text{L)}}$ is exponentially asymptotically stable.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0308712-2
Keywords: Volterra integrodifferential systems, infinite lag, exponential asymptotic stability, logarithmic growth, solutions in $ {L^2}[0,\infty ),({C_0})$ semigroup, resolvent of $ ({C_0})$ semigroup, Hardy class $ {H^2}$
Article copyright: © Copyright 1972 American Mathematical Society