On generalized solutions of linear differential-algebraic systems (Q1022199)

The system of differential-algebraic equations (DAS) \[ A(t) x'(t)= B(t)x(t)+ f(t),\quad t\in T= [0,+\infty),\tag{1} \] where \(A(t),\, B(t)\in\mathbb C^\infty(T)\) are \(n\times n\)-matrices, \(\text{det\,}A(t)\equiv 0\), \(t\in T\), is considered in the case when the posed Cauchy problem \[ x(0)= a\tag{2} \] is unsolvable in the space \(\mathbb C^1(T)\). In [Classical and generalized solutions of differential-algebraic systems with deviating argument, Funct. Differ. Equ. 11, No. 3--4, 485--510 (2004; Zbl 1060.34033)], the author proves the possibility of constructing a generalized solution in the sense of Sobolev-Schwartz of the initial problem for a DAS of the form (1) and for a linear system of functional-differential-algebraic equations (the initial data are inconsistent, the righthand side is either a smooth or a regular generalized function) under the assumption that the matrix coefficients are real analytic, because this assumption was required for existence of a central canonic form (CCF) of system (1), which is the basis for the analysis. In this case, the rank of the matrix \(A(t)\) is constant on the interval \(T\), with exception possibly of a countable set of points. In this paper, the substitution of the requirement of the existence of a CCF by the assumption of the existence for DAS (1) of a left regularizing operator (LRO) [\textit{V. F. Chistyakov}, Algebro-differential operators with a finite-dimensional kernel. Novosibirsk: Nauka, Sibirskaya Izdatel'skaya Firma Rossijskoj Akademii Nauk. 279 p. (1996; Zbl 0999.34002)] allows to obtain, results concerning generalized solutions of problem (1), (2) in the case when \(A(t),\,B(t)\in\mathbb{C}^\infty(T)\). The rank of the matrix \(A(t)\) may vary within the interval \(T\) in an arbitrary way because no additional restrictions on it are imposed. By an LRO it is understood a linear differential operator the action of which transforms system (1) to a form solved with respect to the derivative. It is shown that the solution of problem (1), (2) coincides in the space of distributions with the solution of the system obtained from (1) by the action of an LRO, which indicates that the results are of constructive nature because the determination of the coefficients of an LRO is a much more simple problem than the transformation of DAS (1) into a CCF. The author proves the existence of a generalized solution of problem (1), (2) in the case when the initial data are inconsistent and the right-hand side is given as the sum of a regular generalized function and a linear combination of the delta-function and its derivatives.