On the standard canonical form of time-varying linear DAEs

Abstract

We introduce a solution theory for time-varying linear differential-algebraic equations (DAEs) E ( t ) x ˙ = A ( t ) x E(t)\dot x=A(t)x which can be transformed into standard canonical form (SCF); i.e., the DAE is decoupled into an ODE z ˙ 1 = J ( t ) z 1 \dot z_1 = J(t)z_1 and a pure DAE N ( t ) z ˙ 2 = z 2 N(t) \dot z_2 = z_2 , where  N N is pointwise strictly lower triangular. This class is a time-varying generalization of time-invariant DAEs where the corresponding matrix pencil is regular. It will be shown in which sense the SCF is a canonical form, that it allows for a transition matrix similar to the one for ODEs, and how this can be exploited to derive a variation of constants formula. Furthermore, we show in which sense the class of systems transferable into SCF is equivalent to DAEs which are analytically solvable, and relate SCF to the derivative array approach, differentiation index and strangeness index. Finally, an algorithm is presented which determines the transformation matrices which put a DAE into SCF.