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

26 Chapter 2. Computing the spectrum
provided a motivation for the explicit expression of the solution operator which
is a combination of a shift and an ordinary differential equation (ODE),
�
ψ(θ) = ϕ(θ + h) θ ≤ −h
(T (h)ϕ)(θ) =
(2.15)
Solution of ψ(θ) ˙ = A0ψ(θ) + A1ϕ(θ + h − τ) θ ≥ −h,
for h < τ. Note that A1ϕ(θ + h − τ), i.e., the second term in the ODE-case, is a
previous time-point, which is known, and can be interpreted as an inhomogeneous
part of the ODE. The initial condition for the ODE in the second case is such that
T (t)ϕ is continuous, i.e., (T (h)ϕ)(−h) = ϕ(0), cf. Figure 2.2. This construction
is sometimes referred to the method of steps, and gives a natural way to integrate
a DDE by (numerically) solving the ODE-part in each step (but in this work we
focus on eigenvalues of DDEs and not the integration of DDEs).
We formalize this construction in a theorem.
Theorem 2.11 Consider the DDE (2.1) with the solution operator T (h) defined
by Definition 2.10. Suppose h ≤ τ, then for any ϕ ∈ C([−τ, 0]),
�
ψ(θ) = ϕ(θ + h) θ ∈ [−τ, −h]
(T (h)ϕ)(θ) =
Solution of ˙ ψ(θ) = A0ψ(θ) + A1ϕ(θ + h − τ) θ ∈ [−h, 0].
(2.16)
Proof: First, suppose θ ∈ [−τ, −h]. Then, the evaluation of ψ(θ) := x(h + θ) is
always the initial condition in the definition of the DDE (2.1) since h + θ ≤ 0.
Hence, x(h + θ) = ϕ(h + θ). This proves the first case in (2.16).
Now, suppose θ ∈ [−h, 0]. Since ψ(θ) := x(h + θ) and h + θ ≥ 0 we use the
first case in the definition of the DDE (2.1) to evaluate x(h + θ), i.e.,
˙ψ(θ) = A0ψ(θ) + A1x(h + θ − τ). (2.17)
Now note that since h ≤ τ, h+θ−τ is non-negative and the evaluation x(h+θ−τ)
is the second case (the initial condition) in the definition of the DDE (2.1), i.e.,
x(h + θ − τ) = ϕ(h + θ − τ). Also note that this case, i.e., the second case in
(2.16) is an inhomogeneous initial value problem where the initial value ψ(−h)
is taken from the first case. Existence and uniqueness of a solution ψ of (2.17)
for the interval θ ∈ [−h, 0] follows from the theorem of Picard-Lindelöf [Lin94].
The proof is completed by noting the fact that the left and the right hand side
in (2.16) are both uniquely defined. �