A simple success check for delay differential‐algebraic equations

Solutions of delay differential‐algebraic equations (DDAEs) may depend on derivatives and future evaluations of some of its equations. Structural analysis can be used to determine how often each equation needs to be differentiated and shifted. Unfortunately, these numbers are not always correct, such that a post‐processing step is required to validate the result. In this contribution, we present the first step towards a success check for DDAEs.


Introduction
We consider linear initial trajectory problems (ITPs) for DDAEs of the form on some time interval [0, T ] with τ > 0 consisting of n unknowns and n equations. Hereby, (∆ σ f ) (t) := f (t + σ) denotes the shift (forward) operator. Typically, the DDAE (1) is equipped with an initial trajectory It is well-known that a solution of a DDAE at time t may depend on derivatives and evaluations of (1) at t + kτ for some k ∈ N. For a numerical method, it is thus essential to understand how often equations need to be differentiated and shifted in time. The required number of differentiations and shifts can be determined using structural analysis [1]. Unfortunately, structural analysis might not detect the correct number of differentiations and shifts and consequently, a post-processing step is required to validate the result. For differential-algebraic equations (DAEs), i.e., equations of the form (1) that do not depend on x(t − τ ) andẋ(t − τ ), such a success check is presented in [4], which we generalize in the next section to DDAEs. Let us mention that the results presented below also apply to nonlinear DDAEs.

A simple success check for DDAEs
Suppose that numbers σ i and δ i are available that dictate how often we have to shift and differentiate, respectively, the ith equation in (1). These numbers might come from the structural analysis presented in [1] or may be obtained otherwise. The goal of the success check is to determine if σ i and δ i are sufficiently large. Let s i denote the largest non-negative number such that ∆ siτ x ( ) i appears in the set of shifted and differentiated variables for some ∈ N 0 and letd i denote the largest number such that ∆ siτ x (di) i is present. We call s i the highest shift for x i andd i the highest derivative for ∆ siτ x i . Moreover, we define the highest derivative of x i to be the largest number d i such that ∆ τ x (di) i appears for some ∈ N 0 . Note that by construction the highest derivatives satisfyd i ≤ d i for i = 1, . . . , n. With these preparations, we are able to present our simple success check in Algorithm 1.
Note that the relation ( ) constitutes a delay equation that might include positive and negative time delays, and thus it is not immediately clear if the associated ITP is solvable.
The assumption thatX does not depend on ∆ kτ x ( ) i for any k > 0 is a necessary condition to solve ( ) with the method of steps. Nevertheless,X in ( ) may contains variables x The implicit function theorem yields the explicit relation Shift the ith equation in ( ) s i times backwards in time to obtaiñ ifX does not depend on positive delay andd i = d i for i = 1, . . . , n then return success end if 6: else return failure 7: end if is equivalent to a regular DAE with index at most 1. We thus assume that Algorithm 1 applied to the DDAE (1) returns success. The assumptiond i = d i implies that if x (d) i appears inX , then d ≤d i . To obtain a strict inequality, we can proceed as follows. Suppose that the shifts are ordered increasingly, i.e., 0 ≤ s 1 ≤ s 2 ≤ · · · ≤ s n . Then it is easy to see that the Jacobian ∂ψ ∂X0 is strictly upper triangular and we conclude that ∂(X0−ψ) ∂X0 = I n − ∂ψ ∂X0 is nonsingular. The implicit function theorem applied to ( ) thus yields where the right-hand side is independent ofX 0 . To obtain a first-order formulation, we apply a trimmed linearization [3] to (4) and reorder the equations to obtain I 0 0 0 ż = J(t) 0 Q(t) I z + η(t, ∆ −τ z, . . . , ∆ −(sn−1)τ z, ∆ −τż , . . . , ∆ −(sn−1)τż ).
Clearly, the corresponding DAE has differentiation index at most 1. Using the shift index concept from [2] we thus conclude that the DDAE (5) has shift index 0. Summarizing our discussion yields our main result.
Theorem 2.2 If Algorithm 1 returns success, then the numbers δ j and σ j are sufficiently large, in the sense that (5) has differentiation index less or equal 1 and shift index 0.
If in addition the coefficient matrices in (1) are independent of time, then [6,Theorem 4] immediately implies that the initial trajectory problem associated with (5) has a unique solution within the space of piecewise smooth distributions [5]. If a continuous solution is required, then the initial trajectory has to satisfy some additional splicing conditions, see for instance [7].