Algebraic-differential systems with deviating argument (Q1397525)

The article deals with algebraic differential systems (ADS) of the form \[ A(t)x'(t)+B(t)x(t)+D(t)x(t-\sigma)=f(t),\quad t\in T=[t_0,+\infty), \tag{1} \] \[ x(t)=\psi (t),\quad t\in [t_0-\sigma,t_0), \tag{2} \] \(\det A(t)\equiv 0\) on \(T\), \(\sigma= \text{const}>0.\) In studying existence and uniqueness of solution of problem (1), (2), naturally there arises the question to transform the equations (1), (2) into the system of ordinary differential equations with delay of the form \[ x^r(t)+\sum_{j=0}^{r-1} A_j(t)x^{(j)}(t)+ \sum_{j=0}^sD_j(t)x^{(j)}(t-\sigma)= \widetilde{f}(t),\quad t\in T, \tag{3} \] \[ x(t)=\psi(t),\quad t\in [t_0-\sigma,t_0), \tag{4} \] for which already there are theorems of existence. This problem is rather easy if the matrices \(A(t)\) and \(B(t)\) are such that it exists a linear differential operator \(L\), such that \[ L[A(t)x'(t)+B(t)x(t)]= x'(t)+ \widetilde{B}(t)x(t),\quad t\in T. \tag{5} \] The purpose of the work is to find conditions for the transformation of the system (1), (2) into (3), (4), if the operator does not exist. The approach used in the article follows the basic ideas of the theory of left regularizing operators [see, for example, \textit{V. F. Chistyakov}, Algebraic-differential operators with a finite-dimension kernel (Russian), Novosibirsk: Nauka, Sibirskaya Izdatel'skaya Firma Rossijskoj Akademii Nauk (1996; Zbl 0999.34002)].