Asymptotic Behavior of Linear Integrodifferential Systems
Viorel Barbu, Stanley I. Grossman · Transactions of the American Mathematical Society · 1972
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 $||{T_t}f|| \leqslant {M_{\epsilon }}{e^{\epsilon t}}||f||$. 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.