Linear algebraic-differential systems with varying delay (Q1398489)

Consider the linear algebraic-differential systems (ADS) with varying delay of the form \[ A(t)x'(t)=B(t)x(t)+\sum_{i=0}^pD_i(t)x^{(i)}(t-\sigma(t))+f(t), \quad t\in T=[t_0,t_k),\tag{1} \] \[ x(t)=\psi(t), \quad t\in [t_n,t_0), \tag{2} \] where \(t_n=\min_{t\in T}(t-\sigma(t))\). On the basis of a theory of linear ADS [see, \textit{Yu. E. Boyarintsev}, Regular and singular systems of linear ordinary differential equations. Novosibirsk: Izdatel'stvo ``Nauka'', Sibirskoe Otdelenie (1980; Zbl 0453.34004); and \textit{V. F. Chistyakov}, Algebraic-differential operators with finite-differential kernel, Novosibirsk; Nauka, Sibirskaya Izdatel'skaya Firma Rossijskoj Akademii Nauk (1996; Zbl 0999.34002)], it is shown that under some constraints upon the input data (2), a linear differential operators exists transforming equation (1) into the form \[ y'(t)=P(t)y(t)+ \sum_{j=1}^s G_j(t)y (t-\widetilde{\sigma}(t))+ \widetilde{f}(t),\quad t\in T, \] \[ y(t)=\psi(t),\quad t\in [t_n,t_0),\;t_n=\min_{t\in T,\,1\leq j\leq s} (t-\widetilde{\sigma}_j(t)). \] This allows one to determine the index of the initial system.