On a reduction of the order in a differential-algebraic system
This paper is concerned with the solvability of the Cauchy problem and the construction of the Green operator for linear differential-algebraic equations (DAEs) of the form \[ A(t)z'(t)=B(t)z(t) +f(t), \quad t\in [a,b],\tag{1} \] where $A(t)$ and $B(t)$ are continuous matrix functions of size $m$ by $n$, $f$ is a continuous vector function, and the unknown $z$ is a continuously differentiable function. In general, $m\neq n$ is assumed. In the case $m=n$, the leading matrix $A(t)$ is assumed to be singular and of constant rank. Under some certain assumptions, a procedure of reducing DAEs to a sequence of systems of differential and algebraic equations is proposed. The distance from the original system to a regular system of differential equations which is obtained at the end of the procedure is measured by the notion \textit{degeneration order}. The Green operator associated with DAEs (1) and an explicit formula for the solution of the Cauchy problem are constructed. Note that the theory of DAEs has been developed since early 80's and the solvability analysis of linear DAEs is well known. Unfortunately no comparison of the approach in this paper with the other existing approaches such as the projector-based analysis by R. März et.al. or the strangeness-index approach by P. Kunkel and V. Mehrmann has been mentioned.
scientific article; zbMATH DE number 7000120