Decomposition of time-varying implicit linear systems into dynamical and static parts (Q1202269)

The paper considers the time varying, linear descriptor system \(E(t)\dot x(t)=A(t)x(t)+B(t)u(t)\), where \(x(t)\in R^ n\) is the semistate vector and \(u(t)\in R^ m\) is the input vector. If the system is time invariant and regular, i.e. if \(E(t)\), \(A(t)\), \(B(t)\) do not depend on time and \(\text{det}(sE-A)\neq 0\) then it was shown by \textit{D. G. Luenberger} [Automatica No. 14, 473-485 (1978; Zbl 0398.93040)] that this system can be decomposed into a dynamical part and a static part. In this paper the author derives necessary and sufficient conditions under which a similar decomposition holds in the general time varying case.