On generalized linear regular delay systems
The authors consider generalized linear regular delay systems of the form \[ Ex'(t)=Ax(t-r)+Bu(t), \quad t>t_0,\;r>0, \qquad x(t)=\phi(t), \quad t_0-r\leq t \leq t_0, \tag{1} \] with \(E,A\in \mathbb{R} ^{n\times n}\), \(B\in \mathbb{R} ^{n\times l}\), \(u\in C([t_0, +\infty))\), \(\phi\in C([t_0-r,t_0))\). The matrices \(E\) and \(A\) may be degenerated but the matrix pencil \(sE-A\) is supposed to be regular. First, system (1) is reduced to a suitable homogeneous one. Then the homogeneous system is decomposed in two subsystems, whose solutions are obtained. The conditions on the function \(\phi\) are presented for initial value problem (1) to have a unique solution. The main tools are the matrix pencil theory and the theory of systems of linear delay differential equations. Some necessary preliminaries from these theories are also included.
generalized regular linear delay systems
scientific article; zbMATH DE number 1346374
scientific article