# Asymptotic Stability of Differential Equations with Infinite Delay

Authors: D. Piriadarshani; T. Sengadir
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

A theorem on asymptotic stability is obtained for a differential equation with an infinite delay in a function space which is suitable for the numerical computation of the solution to the infinite delay equation.


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## Body

## 1. Introduction and Preliminaries

In this paper, we study the asymptotic stability of the solutions to the infinite delay differential equation given below:(1.1)x′(t)=ax(t)+∑i=1∞bix(t-τi),t≥0,x(θ)=ϕ(θ),θ∈(-∞,0],
under the following assumptions.  (i)
There existsp>0 with |bi|≤pγ-i for all i∈ℕ.(ii)
τi≤iτ1 for all i∈ℕ.The asymptotic stability of a linear infinite delay equation is studied in [1–5] in the context of abstract phase spaces which includes the space:
(1.2){ϕ∈C(-∞,0]:sup⁡θ∈(-∞,0]eγθ|ϕ(θ)|<∞,lim⁡θ→∞eγθϕ(θ)exists}.
The asymptotic constancy neutral equations are studied in [6]. Linear time-invariant systems with constant point delays are studied in [7] and in [8]; a Razumikhin approach is used to study exponential stability of delay equations. Asymptotic stability and stabilization of linear delay-differential equations are studied in [9].In this paper, the phase spaceCσ(-∞,0] for the initial function is chosen as follows. Let mi=iτ1>0 and βi=pγ-i. The space Cσ(-∞,0] is defined as
(1.3){ϕ∈C(-∞,0]:∑i=1∞βisup⁡θ∈[-mi,0]|ϕ(θ)|<∞}.
Here C(-∞,0] is the set of continuous complex valued functions defined on (-∞,0].The motivation to consider the above type of phase space is that for numerical computation of solutions it is enough to know the values of the initial data over a finite domain at every stage of computation. See [10, 11].The following definitions and results are well known, see for example [5] or [12].Definition 1.1.
The Kuratowski measure of noncompactnessα(V) of the subset V of a Banach space X is defined by
(1.4)α(V)=inf⁡⁡{∪j=1nd>0:thereexistsafinitenumberofsetsV1,V2,…,Vn,withdiam⁡Vj≤dsuchthatV=∪j=1nVj}.
For a bounded linear operator L:X→Y, |L|α is defined as
(1.5)|L|α=inf⁡⁡{k>0:α(L(V))≤kα(V)forallboundedsetsV}.Proposition 1.2.
LetX,Y,Z be Banach spaces and M:X→Y, L:Y→Z be bounded linear operators. Then, |M∘L|α≤|M|α|L|α. Further, if M:X→Y is compact, then |M|α=0.Theorem 1.3.
LetX be a Banach space and let A:D(A)→X be the infinitesimal generator of a semigroup of operators St:X→X. Then, the growth bound of the semigroup ω0 defined as
(1.6)ω0=lim⁡t→∞1tln⁡(‖St‖)=inf⁡{ω:∃M≥1suchthat‖St‖≤Meωt},
is given by
(1.7)ω0=max⁡{s(A),ωess},
where s(A)=sup{ℜ(λ):λ∈spec(A)} and
(1.8)ωess=lim⁡t→∞1tln⁡(|St|α).In Theorem1.3, spec(A) is the compliment of the resolvent set ρ(A) which is the set of all λ∈C such that the operator λI-A is one-one and onto and (λI-A)-1 is a bounded linear map.For a real numberr, ⌊r⌋=max{n∈Z:n≤r} and ⌈r⌉=min{n∈Z:n≥r}. We will make use of the observation ⌈r⌉≤⌊r⌋≤r+1 for r∈ℝ.


## 2. Asymptotic Stability of a PDE

Consider the following simple initial boundary value problem for a PDE:(2.1)∂u∂t=∂u∂θ,t≥0,θ≤0,u(t,0)=0,t≥0,u(0,θ)=u0(θ),θ≤0,
where u0∈Cσ,0(-∞,0]={u∈Cσ(-∞,0]:u(0)=0}.Its mild solution is given by the semigroupTt:Cσ,0(-∞,0]→Cσ,0(-∞,0] defined as(2.2)Ttu0(θ)=u0(t+θ),t+θ<0=0,t+θ≥0.Proposition 2.1.
Letmi=iτ1 and βi=pγ-i. The infinitesimal generator of the semigroup defined by (2.2) is given by B:D(B)→Cσ,0(-∞,0], Bϕ=ϕ′, where
(2.3)D(B)={ϕ∈Cσ,0(-∞,0]:ϕ′∈Cσ,0(-∞,0]}.
Further, ρ(B)={λ:ℜ(λ)>-lnγ/τ1}.Besides, ifℜ(λ)>-lnγ/τ1, then eλ∈Cσ(-∞,0] and for every f∈Cσ(-∞,0], h defined as h(θ)=∫0θeλ(θ-ξ)f(ξ)dξ and eλ defined as eλ(θ)=eλθ are elements of Cσ(-∞,0].Finally, for the semigroupTt defined in (2.2), ω0=-lnγ/τ1.Proof.
Sinceθ∈[-iτ1,0]⇒t+θ∈[-iτ1,t],
(2.4)sup⁡θ∈[-iτ1,0]|Ttϕ(θ)|≤sup⁡θ∈[-iτ1,0]|ϕ(θ)|,
and hence ∥Tt∥σ≤1,Tt+s=TtTs is obvious, then
(2.5)lim⁡t→0‖Ttϕ-ϕ‖σ=0
can be proved using Proposition 1.9 of [10]. The proof that B is the infinitesimal generator of Tt is also easy.
Note thatλ=0 trivially satisfies ℜ(λ)>-lnγ/τ1. Let 0≠λ∈ρ(B). Define ϕ, as ϕ(θ)=θ. Since ∑i=1∞pγ-i<∞, ϕ∈Cσ,0(-∞,0] and hence there is a unique ψ∈D(B), such that λψ-ψ′=ϕ. Indeed, ψ=(λI0-B)-1ϕ. Here, I0 is the identity on Cσ,0(-∞,0]. Let us note that ψ(0)=0. Now, we find that ψ1, defined as ψ1(θ)=θ/λ+(1/λ2)(1-eλθ) is the unique continuously differentiable function such that λψ1-ψ1′=ϕ and ψ1(0)=0. From this we infer that ψ1=(λI0-B)-1ϕ and hence ψ1∈Cσ,0(-∞,0]. Now, since ϕ∈Cσ,0(-∞,0], we obtain (1-eλ)∈Cσ,0(-∞,0]⊆Cσ(-∞,0]. Since the constant function 1 is an element of Cσ(-∞,0], eλ∈Cσ(-∞,0]. Noting that -lnγ/τ1=inf{ℜ(λ):eλ∈Cσ(-∞,0]}, we obtain ℜ(λ)>-lnγ/τ1.Lett≥τ1. It is clear that for all i≤⌊t/τ1⌋, and θ∈[-iτ1,0], Ttϕ(θ)=0. For i>⌊t/τ1⌋, and θ∈[-iτ1,0], we have t+θ≥t-iτ1≥-(i-⌊t/τ1⌋)τ1. Thus, (2.6)sup⁡θ∈[-iτ1,0]|Ttϕ(θ)|≤sup⁡θ∈[-iτ1,0]|ϕ(t+θ)|≤sup⁡θ∈[-i-⌊t/τ1⌋τ1,0]|ϕ(θ)|.Hence(2.7)‖Ttϕ‖σ≤∑i=1∞|βi|sup⁡θ∈[-iτ1,0]|Ttϕ(θ)|≤∑i=⌊t/τ1⌋+1∞|βi|sup⁡θ∈[-i-⌊t/τ1⌋τ1,0]|ϕ(θ)|≤∑i=⌊t/τ1⌋+1∞|βi-⌊t/τ1⌋||βi||βi-⌊t/τ1⌋|sup⁡θ∈[-i-⌊t/τ1⌋τ1,0]|ϕ(θ)|≤sup⁡i>⌊t/τ1⌋|βi||βi-⌊t/τ1⌋|∑i=⌊t/τ1⌋+1∞sup⁡θ∈[-i-⌊t/τ1⌋τ1,0]|βi-⌊t/τ1⌋||ϕ(θ)|≤sup⁡i>⌊t/τ1⌋|βi||βi-⌊t/τ1⌋|‖ϕ‖σ≤sup⁡i>⌊t/τ1⌋γ-iγ-i+⌊t/τ1⌋‖ϕ‖σ≤γ-⌊t/τ1⌋‖ϕ‖σ.Hence, the operator norm∥Tt∥σ≤γ-⌊t/τ1⌋.To prove the equality, we construct a functionη∈Cσ,0(-∞,0] such that ∥Ttη∥σ=γ-⌊t/τ1⌋∥η∥σ and the result follows.Letδ=(⌊t/τ1⌋+1)τ1-t=τ1(⌊t/τ1⌋+1-t/τ1). We have, δ<τ1. Define, (2.8)η(θ)=-θδ,-δ≤θ≤0=1,θ<-δ.It is clear that∥η∥σ=∑i=1∞pγ-i, Now,
(2.9)Ttη(θ)=-(θ+tδ),(-δ-t)≤θ≤-t=1,θ<-δ-t.
Thus ∥Ttη∥σ=p∑i=⌊t/τ1⌋+1∞γ-i.Hence,∥Ttη∥σ=γ-⌊t/τ1⌋∥η∥σ.Now,ω0=limt→∞(1/t)ln(∥Tt∥σ)=-ln(γ)/τ1.Letℜ(λ)>-lnγ/τ1. Since
(2.10)‖(λI0-B)-1g‖σ=‖∫0∞e-λtTtgdt‖σ≤∫0∞e-R(λ)t‖Ttg‖σdt≤∫0∞e-R(λ)teω0t‖g‖σdt=∫0∞e(ω0-R(λ))t‖g‖σdt≤∫0∞e(-ln⁡(γ)/τ1-R(λ))t‖g‖σdt,
we have λ∈ρ(B).Letf∈Cσ(-∞,0]. Define g(θ)=f(θ)-f(0). Then g∈Cσ,0(-∞,0].Letψ=(λI0-B)-1g. We have, ψ(0)=0.Defineψ1(θ)=-∫0θeλ(θ-ξ)g(ξ)dξ. Now ψ1(0)=0 and ψ1′(0)=0.By the uniqueness of the solution to the initial value problem of the ODE:(2.11)λψ-ψ′=g,ψ(0)=0,
it is now obvious that ψ1=ψ and hence ψ1∈Cσ,0(-∞,0].Now,(2.12)∫0θeλ(θ-ξ)g(ξ)dξ=∫0θeλ(θ-ξ)[f(ξ)-f(0)]dξ=∫0θeλ(θ-ξ)f(ξ)dξ+1λ(1-eλθ)f(0).
Since 1-eλ∈Cσ,0(-∞,0], h∈Cσ,0(-∞,0]⊂Cσ(-∞,0], where h is defined as h(θ)=∫0θeλ(θ-ξ)f(ξ)dξ.


## 3. Stability of the Infinite Delay Equation

The proof of the next theorem assuring the existence of a unique solution to (1.1) is similar to the proof of Theorem 2.2 of [10].Theorem 3.1.
Leta∈ℝ and the sequences bi and βi be as in Section 1. Assume that τi≤iτ1. Then there exists a unique solution x:ℝ→ℝ to (1.1) such that its restriction to [0,∞), denoted by y, is in C1[0,∞). Further, for any t∈[0,∞), there is a constant c(t)>0 such that
(3.1)sup⁡s∈[0,t]|y(s)|≤c(t)‖ϕ‖σ.In addition, the family of operators{St:t≥0} defined as(3.2)Stϕ(θ)=x(t+θ),t+θ≥0=ϕ(t+θ),t+θ<0
forms a semigroup. Also, the infinitesimal generator of St is given by A:D(A)→Cσ(-∞,0], where
(3.3)D(A)={ϕ∈Cσ(-∞,0]:ϕ′∈Cσ(-∞,0],ϕ′(0)=aϕ(0)+∑i=1∞biϕ(-τi)}Aϕ=ϕ′.
Further, D(A) is dense and A is a closed operator.Theorem 3.2.
For the semigroupSt defined by (3.2)
(3.4)|St|α≤γ-⌊t/τ1⌋.Further, assume thata+∑i=1∞bi≠0. Then for the generator of the semigroup St defined by (3.3) and
(3.5)spec(A)={λ:R(λ)≤-ln⁡(γ)τ1}∪{λ:R(λ)>-ln⁡⁡(γ)τ1:λ=a+∑i=1∞bie-λτi}.Besides, suppose that for anyλ∈C with λ=a+∑i=1∞bie-λτi, we have ℜ(λ)<-μ1 for some μ1>0. Then, the semigroup St is asymptotically stable.Proof.
LetTt be as in Proposition 2.1. Fix t>0. Define Vt:Cσ(-∞,0]→Cσ(-∞,0] as
(3.6)Vtϕ(θ)=0,t+θ≥0=ϕ(t+θ)-ϕ(0),t+θ<0.
Define Kt:C[0,t]→Cσ(-∞,0] as
(3.7)[Ktz](θ)=z(t+θ)-z(0),t+θ≥0=0,t+θ<0.It is easy to see that(3.8)‖Ktz‖σ≤2∑i=1∞|βi|(sup⁡s∈[0,t]|z(s)|).
Thus, Kt is a bounded linear map.DefineK1:Cσ(-∞,0]→Cσ(-∞,0] as [K1ϕ](θ)=ϕ(0) for all θ∈(-∞,0]. It is clear that K1 is compact. Define Bt:Cσ(-∞,0]→C[0,t] as Btϕ=z, where z is the restriction of y to [0,t]. From (3.1), Bt is a bounded linear map. Let St be as in (3.3). Then, (3.9)St=Vt+KtBt+K1.Now, ifI is the identity on Cσ(-∞,0] and J:Cσ,0(-∞,0]→Cσ(-∞,0] is the inclusion map, then Vt=JTt(I-K1), and, finally, (3.10)St=JTt(I-K1)+KtBt+K1.Next, we show thatBt is, in fact, a compact map. Let x be the solution to (1.1) as in Theorem 3.1:(3.11)z(s)=easϕ(0)+eas∫0se-aη∑i=1∞bix(η-τi)dη,s∈[0,t].Thus,(3.12)z′(s)=az(s)+∑i=1∞bix(s-τi).Considern∈ℕ such that t∈[nτ1,(n+1)τ1]. From (3.1) and (3.11), we obtain existence of c1(t)≥0 such that
(3.13)sup⁡s∈[0,t]|z′(s)|≤c1(t)‖ϕ‖σ.Hence by Arzela-Ascoli theorem,Bt is a compact operator.It is easy to show that|J|α≤∥J∥σ=1. By the compactness of K1 and Bt, |I-K1|α=1 and |KtBt|α=|K1|α=0. Thus, from the relation
(3.14)St=JTt(I-K1)+KtBt+K1,
and Propositions 1.2 and 2.1 of this paper, we obtain (3.15)|St|α≤|Tt|α≤‖Tt‖σ≤γ-⌊t/τ1⌋.
So,
(3.16)ωess=lim⁡t→∞1tln⁡(|St|α)≤-ln⁡⁡γτ1.Let0≠λ∈ρ(A).There is a uniqueψ∈D(A) such that
(3.17)λψ-ψ′=-1,ψ′(0)=aψ(0)+∑i=1∞biψ(-τi).It is clear that there isc∈C such that ψ(θ)=(c-1/λ)eλθ-1/λ. Now, we claim that c≠1/λ. If c=1/λ, then ψ(θ)=-1/λ for all θ∈(-∞,0]. Since ψ∈D(A), we must have ψ′(0)=aψ(0)+∑i=1∞biψ(-τi). But this would imply that a+∑i=1∞bi=0 which is a contradiction, to the hypothesis that a+∑i=1∞bi≠0. Now, since c-1/λ≠0, it is obvious that eλ∈Cσ(-∞,0]. But this implies that ℜ(λ)>-ln(γ)/τ1. If 0∈ρ(A), the condition ℜ(λ)>-ln(γ)/τ1 is obvious. Thus,
(3.18)ρ(A)⊆{λ:R(λ)>-ln⁡(γ)τ1}.We now infer that{λ:ℜ(λ)≤-ln(γ)/τ1}⊆spec(A). Next, if λ=a+∑i=1∞bie-λτi, and ℜ(λ)>-ln(γ)/τ1, then eλ∈Cσ(-∞,0] and hence eλ∈D(A) with λeλ=Aeλ. Thus, λ∈spec(A). So,
(3.19){λ:R(λ)>-ln⁡(γ)τ1,λ=a+∑i=1∞bie-λτi}⊆spec(A).Let us assume thatℜ(λ)>-ln(γ)/τ1 and λ≠a+∑i=1∞bie-λτi.Then, by Proposition2.1, we have eλ∈Cσ(-∞,0] and the function h defined as h(θ)=∫0θeλ(θ-ξ)f(ξ)dξ is in Cσ(-∞,0].DefiningΛ:Cσ(-∞,0]→C as Λ(ϕ)=aϕ(0)+∑i=1∞biϕ(-τi) and taking c=(Λ(h)-f(0))/(Λ(eλ)-λ), we find that ϕ defined as ϕ(θ)=∫0θeλ(θ-ξ)f(ξ)dξ+ceλθ is (λI-A)-1(f). Thus,
(3.20){λ:R(λ)>-ln⁡⁡(γ)τ1,λ≠a+∑i=1∞bie-λτi}⊆ρ(A).
From (3.18), (3.19), and (3.20), we finally conclude that (3.21)spec(A)={λ:R(λ)≤-ln⁡(γ)τ1}∪{λ:R(λ)>-ln⁡⁡(γ)τ1,λ=a+∑i=1∞bie-λτi},
or (3.22)ρ(A)={λ:R(λ)>-ln⁡⁡(γ)τ1,λ≠a+∑i=1∞bie-λτi}.
Since ω0=max{s(A),ωess}≤max{-μ1,-ln(γ)/τ1}, the result follows.Remark 3.3.
Consider the PDE:(3.23)∂u∂t=∂u∂θ,u(0,θ)=ϕ(θ).
Let B be as in Proposition 2.1 and A be as in Theorem 3.1. For ϕ∈D(B), u(t,θ)=Ttϕ∈Cσ,0(-∞,0] is the solution to the above PDE. For ϕ∈D(A), u(t,θ)=Stϕ∈Cσ(-∞,0] is the solution to the above PDE. For the first solution u(t+θ)=0, t+θ≥0 and for the second solution u(t+θ)=x(t+θ), t+θ≥0. Here x is the solution to the delay equation.



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