Asymptotically diagonal delay differential systems
The paper deals with the equation \[ x'(t)=(\tilde A+A(t))x(t)+(\tilde B+B(t))x(t-\tau), \] where \(\tau\) is a positive constant. Here \smallskip \((H_1)\) \(\tilde A,\tilde B\) are constant diagonal matrices \(\tilde A=\text{ diag }\big\{ \tilde a_1,\tilde a_2,\ldots,\tilde a_n\big\},\) \(\tilde B=\text{ diag }\big\{ \tilde b_1,\tilde b_2,\ldots,\tilde b_n\big\},\) such that \((|\tilde a_i|+|\tilde b_i|)\tau<1/e\) and the equation \(\lambda\tilde a_i+\tilde b_i e^{-\lambda\tau}\) has a unique real root \(\tilde\lambda_i\) in the interval \((\tilde a_i-1/\tau,\infty),\) \(i=1,2,\ldots,n.\) These roots do not coincide, i. e. \(\tilde\lambda_i\neq\tilde\lambda_j\) for \(i\neq j,\) \(i,j=1,2,\ldots,n.\) \smallskip \((H_2)\) \(A(\cdot)=(a_{ij}(\cdot))_{1\leq i,j\leq n},\) \(B(\cdot)=(b_{ij}(\cdot))_{1\leq i,j\leq n}\) are continuous matrix functions on \([0,\infty)\) such that \(\alpha(t):=\sup_{s\geq t}|A(s)|\in L^2,\) \(\beta(t):=\sup_{s\geq t}|B(s)|\in L^2,\) and \(a_{ii}(t)-{1\over\tau}\int_{t-\tau}^t a_{ii}(s) ds\in L^1,\) \(b_{ii}(t)-{1\over\tau}\int_{t-\tau}^t b_{ii}(s) ds\in L^1,\) \(i=1,2,\ldots,n.\) \medskip The main result is: Let hypotheses \((H_1)\) and \((H_2)\) be fulfilled. Then there exists a matrix function \(F\), \(F(t)\to0\) as \(t\to\infty,\) such that the following statements are valid. \smallskip \((A)\) For every constant vector \(c\), the function \(x\) defined by \[ x(t)=(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)c, \] where \[ \Lambda(t)=\tilde\Lambda+\big(I+(\tilde\Lambda-\tilde A)\tau\big)^{-1} \big(\text{diag}\{A(t)\}+\text{diag}\{B(t)\}e^{-\tilde\Lambda\tau}\big) \] and \(\tilde\Lambda=\text{diag} \{\tilde\lambda_1,\tilde\lambda_2,\ldots,\tilde\lambda_n\},\) is a solution of the considered equation. \smallskip \((B)\) For every solution \(x\) there exist a constant vector \(c\) and a vector function \(f,\) \(f(t)\to0\) as \(t\to\infty\), such that \[ x(t)=(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)(c+f(t)) \] and \[ \left|x(t)-(I+F(t))\exp\left(\int_0^t\Lambda(s) ds\right)c\right|e^{t/\tau}\quad\text{ is bounded for}\quad{t\to\infty}. \] \smallskip A theorem of Hartman--Winter establishes the asymptotic form of the fundamental matrix of a system of ordinary differential equations with asymptotically diagonal matrix. An extension of this theorem to a scalar delay differential equation is done by \textit{J. R Haddock} and \textit{R. J. Sacker} [J. Math. Anal. Appl. 76, 328-338 (1980; Zbl 0446.34071)]. The case of a ``quasi-triangular'' delay-system is considered by the first two authors in an earlier paper. The main result of this paper is an extension to the general case.
asymptotic form of the fundamental matrix
scientific article; zbMATH DE number 1005595