Analysis of linear variable coefficient delay differential-algebraic equations
This article studies delay-differential algebraic equations (DDAEs) described by \[ A \dot{x}(t) + B x(t) + C x(t-\tau) = f, \] where \(A\), \(B\), \(C\) and \(f\) are time-varying matrices and a vector. Explicit DDAEs, given by \[ \begin{align*}{ \dot{x}_1(t) &= D x (t) + E x(t-\tau) + f_1 \cr x_2(t) &= F x_1(t) + G x(t-\tau) + f_2 \cr}\end{align*}, \; x = (x_1, x_2), \] are a well-known subset of this class which encompasses -- the explicit versions of -- delay-differential equations, neutral equations and difference equations. For explicit DDAEs, the initial value problem is well-posed given appropriate consistency conditions for many functional spaces of solutions (see e.g. [\textit{M. C. Delfour} and \textit{J. Karrakchou}, J. Math. Anal. Appl. 125, 361--399, 400--450 (1987; Zbl 0643.93034)] for the time-invariant case). On the contrary, the original system -- or implicit -- may be over or underdetermined, hence the analysis of the initial value problem is more complex. Sometimes these patterns are obvious (the matrices \(A\), \(B\) and \(C\) are not necessarily square or nonsingular), but they may also result from an implicit \textit{redundancy} between the equations. This article describes a step-by-step methodology to explicit the system redundancies, using only linear transformations, differentiations and time shifts. This general process is first exposed for differential-algebraic equations and difference equations; in the simplest cases, the transformation leads to a system with the structure \[ \left[ \begin{matrix} A \\ 0 \\ 0 \end{matrix}\right] \dot{x}(t) + \left[ \begin{matrix} ? \\ {B} \\ 0 \end{matrix}\right] x(t) + \left[ \begin{matrix} ? \\ ? \\ 0 \end{matrix} \right] x(t-\tau) = \left[ \begin{matrix} f_1 \\ f_2 \\ f_3 \end{matrix} \right], \] where the absence of implicit redundancy is characterized by a matrix \([A^t, B^t]^t\) of full row rank. Note that when additionally this matrix is square and \(f_3\) is an empty vector, the system -- that is neither over nor underdetermined -- is then equivalent to a system of explicit DDAEs. In the general case, the structure of the system obtained after transformation may be more complex than the example above; in particular, derivatives of the delayed variables may appear in the right-hand side.
scientific article; zbMATH DE number 6389979
scientific article