### On inconsistent initial conditions for linear time-invariant differential-algebraic equations

G. Reissig, H. Boche, P.I. Barton
2002 IEEE Transactions on Circuits and Systems I Fundamental Theory and Applications
Given an arbitrary initial value for the differential-algebraic equation _ ( ) + ( ) = ( ), an initial value can be selected from among all consistent initial values by means of the Laplace transform. We show that this choice is the only one that fulfills some simple, physically reasonable assumptions on linear systems' behavior. Our derivation is elementary compared to previous justifications of the above Laplace transform based method. We also characterize by means of a system of linear
more » ... tem of linear equations involving , , derivatives of , and , which gives a new method to numerically calculate . Index Terms-Differential-algebraic equations (DAEs). I. BACKGROUND For several reasons, differential-algebraic equations (DAEs) of the form are preferred models in many fields, even if an equivalent explicit equation _ x(t) = f (x(t); t) could be obtained [4]. (Here, F : n 2 n 2 ! n is sufficiently smooth and, typically, the derivative of F with respect to its second argument is singular.) On the other hand, DAEs may exhibit troublesome phenomena, including order reduction in numerical integration [4], impasse points [22], extreme sensitivity with respect to perturbations even if all eigenvalues are stable [23], as well as inconsistent initial values [4], [18]. To discuss the particular difficulty of the consistency of initial values, we start with some basic terminology: A C 1 -mapping x: I ! n is a solution of DAE (1), if I is an open interval and F (x(t); _ x(t); t) = 0 for all t 2 I. A point x 0 2 n is called a consistent initial value for DAE (1) at t0 if there is some solution x of (1) defined on a neighborhood of t 0 such that x(t 0 ) = x 0 . DAE (1) is called uniquely solvable if the following holds for all t0 2 . There is a consistent initial value for (1) at t 0 . Any two solutions x 1 and x 2 of (1) that are defined in a neighborhood of t 0 and fulfill x 1 (t 0 ) = x 2 (t 0 ) coincide on the intersection of their domains of definition. In contrast to the explicit case, the set of consistent initial values for (1) at some t 0 usually forms a submanifold M of n , M 6 = n , e.g., [19] . Hence, obtaining a consistent initial value x0 2 M is part of any simulation of DAE (1) [4] . Assume now that DAE (1) is an appropriate model for the physical system under investigation for t > t0 only, and that this system has some history that is not adequately described by (1). More precisely, let there be given some continuous mapping x 0 : ]t 0 0 "; t 0 ] ! n for some " > 0, with x 0 0 = x 0 (t0), not necessarily consistent for (1) at t 0 . We would like to find not only some solution x + of (on some neighborhood of t0, but one that extends the past x 0 to a physically reasonable mapping t 7 ! x 0 (t); if t t 0 x + (t); otherwise. If DAE (1) is uniquely solvable, and, in addition, the system is adequately described by some (other) uniquely solvable DAE for t < t0, this amounts to the selection of a consistent initial value x + 0 = x + (t 0 ) for (1) at t 0 from among all such consistent values on grounds of the past, x 0 0 . In other words, inconsistent initial conditions require to define a mapping that is physically reasonable. Although the above setting is quite common in several fields of application [3], [15], [16], the above problem has been solved for very special nonlinear DAEs only [5], [20] . For the linear time-invariant case of DAE (1), which we assume to be uniquely solvable, it has been proposed to define for the special case t 0 = 0, where x is the inverse of s 7 ! (sA + B) 01 (f (s) + Ax 0 0 ) under the Laplace transform, andf is the image under the Laplace transform of f restricted to the non-negative reals [8], [11], [13]. With only few exceptions [14], the same value (4) is obtained from all other methods proposed in the literature [2], [5], [19]-[21], [24]-[26] for those DAEs (3) to which they apply. In fact, the value (4) is physically reasonable as it is also obtained from distributional solutions [9], [10], and from singular perturbations if A is nilpotent and B is the identity matrix [7]. This note makes the following contributions. First, we show that the four conditions listed in Section II, which are physically reasonable to impose, ensure the existence of a unique mapping (2), thereby extending ideas from [2], [17], [25] . As we use simple linear algebra only, and the value of x + 0 obtained from the unique mapping (2) coincides with (4), our result can be seen as a further, elementary justification of (4). Further, we characterize (4) by means of a linear system of equations involving A, B, derivatives of f at 0, and x 0 0 , which gives a new method to numerically calculate x + 0 .