Large Time Behaviour of Neutral Delay Systems
Frasson, Miguel
Citation
Frasson, M. (2005, February 22). Large Time Behaviour of Neutral Delay Systems.
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belonging to the thesis
Large time behaviour of neutral delay systems
by Miguel V. S. Frasson.
1. Consider the functional differential equation (FDE)
d
dtM xt= Lxt, t ≥ 0, (1)
where M and L are continuous linear operators from C def= C([−r, 0], Cn) to Cn, and M is “atomic at zero”, i.e., M ϕ = ϕ(0) + M0ϕ where M0ϕ is “essentially”
independent of the value of ϕ at θ = 0. Let xt∈ C be defined by xt(θ) = x(t + θ).
Define the solution semigroup T (t) : C → C by T (t)ϕ = xt, t ≥ 0, where x is
the (unique) solution of (1) with initial condition x0 = ϕ. Let Λ be the (possibly
infinite) set of those λ ∈ C such that x(t) = ceλt is a solution of (1) for some
0 6= c ∈ Cn. If λ
0∈ Λ is a dominant eigenvalue, that is, if there exists > 0 such
that λ06= z ∈ Λ implies Re z < Re λ0− , then there exists a (unique) non-empty
T (t)-invariant manifold Mλ0 and K > 0 such that
e− Re λ0tkT (t)(I − P
λ0)ϕk ≤ Ke
−tkϕk, t ≥ 0, (2)
where Pλ0 is the spectral projection from C onto the finite dimensional space Mλ0.
Formula (2) yields the large time behaviour of solutions of (1). (See Lemma 2.1 and the figure on the cover of this thesis. See also [2].)
2. Explicit formulas for the spectral projections onto the eigenspaces of functional dif-ferential equations can easily be obtained using the associated infinitesimal genera-tor and Dunford calculus. This is an algorithmic approach that can be implemented using a computer algebra system such as Maple. (See Chapter 3 and Chapter 4 of this thesis.)
3. Sufficient conditions for the existence of a dominant root of the (characteristic) equation
∆(z) = z 1 +Pm
l=1cle−zσl − a − P k
j=1bje−zhj, (3)
can be obtained using the scalar-valued positive nondecreasing function V (λ) defined by
V (λ)def= Pm
l=1|cl|(1 + |λ|σl)e−λσl+P k
j=1|bj|hje−λhj, λ ∈ R,
which does not depend on the coefficient a of ∆(z). (See Chapter 5 of this thesis.) 4. Fix n, m ∈ N, bi, cj ∈ R and hi, σj ∈ (0, r] with i = 1, . . . , n and j = 1, . . . , m. If
λ0 = V−1(1), then for any λ > λ0, there exists a ∈ R such that λ is a real simple
dominant root of the characteristic equation (3). (See the comments on Chapter 5 of this thesis.)
5. Consider a non-linear FDE with only discrete time delays such that 0 ∈ C is an equilibrium and suppose that λ = 0 is the dominant root of the characteristic equation of the linearization around 0 ∈ C. One can explicitly compute the ODE that describes the flow on the center manifold, restricted to a neighbourhood of 0 ∈ C, using as elements the space M0 and the spectral projection P0. (Typically
Pλϕ depends on ϕ(0) and integrals involving ϕ.) Actually, the ODE only depends
6. Let η ∈ NBV ([0, r], Cn×n) and define the convolution product dη ∗ f to be dη ∗ f (t)def=
Z ∞
0
dη(θ)f (t − θ).
The map f 7→ dη ∗ f maps CR def
= {f ∈ C([0, r], Cn) : f (0) = 0} into itself. However, f 7→ dη ∗ f does not map C([0, r], Cn) into itself. (See item 4 of Remark 1.1 of this thesis.)
7. Let F and G be continuous functions from [t0, ∞) × Rn to Rn, and let D denote
the differentiation operator. Equations of the type
Dx = F (t, x) + G(t, x)Du, x(t0) = x0, (4)
where u and the solution x belong to NBV ([t0, ∞), Rn) and equality in (4) is
consid-ered in the space of distributions D(Rn), are called “measure differential equations”.
The solutions of (4) are piecewise differentiable with possible “jumps” in the dis-continuities of u. (See [1, 5].)
8. In a topological space T , consider the set functions A 7→ ¯A (closure of A) and A 7→ Ac (complement of A). For any set A ∈ T , at most 14 different sets are
obtained by repeated applications of closure and complementation to A. The subset A = (0, 1) ∪ (1, 2) ∪(2, 3) ∩ Q ∪ {4} of the real line is an example that the 14 sets can be obtained. This result is due to Kuratowski [3]. See also [4].
9. The Netherlands is a great country but not in size. The province of S˜ao Paulo in Brazil covers less than 3% of the area of Brazil, but is 6 times the size of The Netherlands.
10. One could conclude the existence of God from the testimony of the children of Israel who were freed from slavery in Egypt in a sequence of amazing events, like the crossing of the Red Sea on foot, among others. Confer the web page of the Israeli government
http://www.mfa.gov.il/MFA/History/History of Israel/ and the book Exodus of the Bible.
References
[1] Frasson M. V. S. (2000): Sistemas impulsivos do ponto de vista das equa¸c˜oes difer-enciais em medida. Master’s thesis, ICMC–USP (S˜ao Carlos, Brazil). In portuguese. [2] Frasson M. V. S., Verduyn Lunel S. M. (2003): Large time behaviour of linear functional differential equations. Integral Equations Operator Theory, 47 No. 1, 91– 121.
[3] Kuratowski C. (1922): Sur l’operation ¯A de l’analysis situs. Fund. Math., 3 182–199. [4] Langford E. (1971): Characterization of Kuratowski 14-sets. Amer. Math. Monthly,
78 362–367.
