The spectrum of delay-differential equations: numerical methods - KTH

14 Chapter 2. Computing the spectrum
2.2 Methods for DDEs
If the DDE is scalar or the matrices A0 and A1 are simultaneously triangularizable,
then σ(Σ) can be expressed in an explicit way. This is shown in Section
2.2.1. Apart from this special case, the eigenvalues and eigenvectors, i.e.,
s ∈ C, v ∈ C n \{0} such that
(−sI + A0 + A1e −τs )v = 0 (2.4)
can in general not be computed or expressed exactly with elementary or simple
operations. In order to numerically solve (2.4), it is natural to consider some
form of approximation. The type of approaches we will address next, are based
on approximations of (2.4) with an equation which is easier to solve.
Note that the presentation we use here is somewhat different from the typical
presentation of similar results in the literature. A lot of literature discuss different
types of approximations of the DDE (2.1), whereas the approach we take directly
involves aproximations of the characteristic equation (2.2). This allows us to
interpet the approximations in a more unified manner.
The problem of determining s ∈ C and v ∈ Cn \{0} such that,
m�
Bks k v = 0, (2.5)
k=0
for B0, . . . , Bm ∈ C n×n is called a polynomial eigenvalue problem. The polynomial
eigenvalue problem can be solved for problems of moderate dimension
and order using standard techniques, e.g., companion linearization. Companion
linearization is extensively used in this thesis and is further discussed in
Appendix A.1. In this section we will show that several methods for the delay
eigenvalue problem (2.4) are polynomial or rational approximations, such
that the resulting approximation is a polynomial eigenvalue problem (2.5). Obviously,
if we approximate e −τs with a rational function, we can rewrite the
resulting approximation (which is a rational eigenvalue problem) into a polynomial
eigenvalue problem (2.5) by multiplying both sides with the denominator.
We illustrate this idea with a very crude approximation. We know from the definition
of the exponential function that e −τs = limN→∞(1 + τs/N) −N . Clearly,
e −τs ≈ SN(s) := (1 + τs/N) −N should be an accurate rational approximation
for sufficiently large N. Replace e −τs in (2.4) with this approximation and multiply
by (1+τs/N) N . The resulting equation is a polynomial eigenvalue problem