Asymptotic behavior of linear integrodifferential systems

Author:

Barbu Viorel,Grossman Stanley I.

Abstract

We consider the system ( L) y ( t ) = A y ( t ) + t B ( t s ) y ( s ) d s , y ( t ) = f ( t ) , t 0 ({\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 ) y(t) is an n n -vector and A A and B ( t ) B(t) are n × n n \times n matrices. System ( L) ({\text {L)}} generates a semigroup given by T t f ( s ) = y ( t + s ; f ) {T_t}f(s) = y(t + s;f) for f f bounded, continuous and having a finite limit at - \infty . Under hypotheses concerning the roots of det ( λ I A B ^ ( λ ) ) \det (\lambda I - A - \hat B(\lambda )) , where B ^ ( λ ) \hat B(\lambda ) is the Laplace transform, various results about the asymptotic behavior of y ( t ) y(t) are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If B ( t ) L 1 [ 0 , ) B(t) \in {L^1}[0,\infty ) and ( λ I A B ^ ( λ ) ) 1 {(\lambda I - A - \hat B(\lambda ))^{ - 1}} exists for Re λ > 0 \operatorname {Re} \lambda > 0 , then for every ϵ > 0 \epsilon > 0 , there is an M ϵ {M_{\epsilon }} such that | | T t f | | M ϵ e ϵ t | | f | | ||{T_t}f|| \leqslant {M_{\epsilon }}{e^{\epsilon t}}||f|| . Theorem 2. If ( λ I A B ^ ( λ ) ) 1 {(\lambda I - A - \hat B(\lambda ))^{ - 1}} exists for Re λ > α ( α > 0 ) \operatorname {Re} \lambda > - \alpha (\alpha > 0) and if B ( t ) e α t L 1 [ 0 , ) B(t){e^{\alpha t}} \in {L^1}[0,\infty ) , then the solution to ( L) ({\text {L)}} is exponentially asymptotically stable.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference6 articles.

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