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.

1. Introduction and Preliminaries

In this paper, we study the asymptotic stability of the solutions to the infinite delay differential equation given below:

()

under the following assumptions.

(i)
There exists p > 0 with |b_i | ≤ pγ⁻ⁱ for all i ∈ ℕ.
(ii)
τ_i ≤ 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:

()

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 space C_σ(−∞, 0] for the initial function is chosen as follows. Let m_i = iτ₁ > 0 and β_i = pγ⁻ⁱ. The space C_σ(−∞, 0] is defined as

()

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 noncompactness α(V) of the subset V of a Banach space X is defined by

()

For a bounded linear operator L : X → Y, |L|_α is defined as

()

Proposition 1.2. Let X, 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. Let X be a Banach space and let A : D(A) → X be the infinitesimal generator of a semigroup of operators S_t : X → X. Then, the growth bound of the semigroup ω₀ defined as

()

is given by

()

where s(A) = sup {ℜ(λ) : λ ∈ spec (A)} and

()

In Theorem 1.3, 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 number r, ⌊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:

()

where u₀ ∈ C_σ,0(−∞, 0] = {u ∈ C_σ(−∞, 0] : u(0) = 0}.

Its mild solution is given by the semigroup T_t : C_σ,0(−∞, 0] → C_σ,0(−∞, 0] defined as

()

Proposition 2.1. Let m_i = iτ₁ and β_i = pγ⁻ⁱ. The infinitesimal generator of the semigroup defined by (2.2) is given by B : D(B) → C_σ,0(−∞, 0], Bϕ = ϕ^′, where

()

Further, ρ(B) = {λ : ℜ(λ)>−ln γ/τ₁}.

Besides, if ℜ(λ)>−ln γ/τ₁, then e_λ ∈ C_σ(−∞, 0] and for every f ∈ C_σ(−∞, 0], h defined as and e_λ defined as e_λ(θ) = e^λθ are elements of C_σ(−∞, 0].

Finally, for the semigroup T_t defined in (2.2), ω₀ = −ln γ/τ₁.

Proof. Since θ ∈ [−iτ₁, 0]⇒t + θ ∈ [−iτ₁, t],

()

and hence ∥T_t∥_σ ≤ 1, T_t+s = T_tT_s is obvious, then

()

can be proved using Proposition 1.9 of [10]. The proof that B is the infinitesimal generator of T_t is also easy.

Note that λ = 0 trivially satisfies ℜ(λ)>−ln γ/τ₁. Let 0 ≠ λ ∈ ρ(B). Define ϕ, as ϕ(θ) = θ. Since , ϕ ∈ C_σ,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 C_σ,0(−∞, 0]. Let us note that ψ(0) = 0. Now, we find that ψ₁, defined as ψ₁(θ) = θ/λ + (1/λ²)(1 − e^λθ) is the unique continuously differentiable function such that and ψ₁(0) = 0. From this we infer that ψ₁ = (λI₀ − B) ⁻¹ϕ and hence ψ₁ ∈ C_σ,0(−∞, 0]. Now, since ϕ ∈ C_σ,0(−∞, 0], we obtain (1 − e_λ) ∈ C_σ,0(−∞, 0]⊆C_σ(−∞, 0]. Since the constant function 1 is an element of C_σ(−∞, 0], e_λ ∈ C_σ(−∞, 0]. Noting that −ln γ/τ₁ = inf {ℜ(λ) : e_λ ∈ C_σ(−∞, 0]}, we obtain ℜ(λ)>−ln γ/τ₁.

Let t ≥ τ₁. It is clear that for all i ≤ ⌊t/τ₁⌋, and θ ∈ [−iτ₁, 0], T_tϕ(θ) = 0. For i > ⌊t/τ₁⌋, and θ ∈ [−iτ₁, 0], we have t + θ ≥ t − iτ₁ ≥ −(i − ⌊t/τ₁⌋)τ₁. Thus,

()

Hence

()

Hence, the operator norm .

To prove the equality, we construct a function η ∈ C_σ,0(−∞, 0] such that and the result follows.

Let δ = (⌊t/τ₁ ⌋ + 1)τ₁ − t = τ₁(⌊t/τ₁ ⌋ + 1 − t/τ₁). We have, δ < τ₁. Define,

()

It is clear that

, Now,

()

Thus

Hence, .

Now, ω₀ = lim _t→∞(1/t)ln (∥T_t∥_σ) = −ln (γ)/τ₁.

Let ℜ(λ)>−ln γ/τ₁. Since

()

we have λ ∈ ρ(B).

Let f ∈ C_σ(−∞, 0]. Define g(θ) = f(θ) − f(0). Then g ∈ C_σ,0(−∞, 0].

Let ψ = (λI₀ − B) ⁻¹g. We have, ψ(0) = 0.

Define . Now ψ₁(0) = 0 and .

By the uniqueness of the solution to the initial value problem of the ODE:

()

it is now obvious that ψ₁ = ψ and hence ψ₁ ∈ C_σ,0(−∞, 0].

Now,

()

Since 1 − e_λ ∈ C_σ,0(−∞, 0], h ∈ C_σ,0(−∞, 0] ⊂ C_σ(−∞, 0], where h is defined as

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. Let a ∈ ℝ and the sequences b_i and β_i be as in Section 1. Assume that τ_i ≤ 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

()

In addition, the family of operators {S_t : t ≥ 0} defined as

()

forms a semigroup. Also, the infinitesimal generator of S_t is given by A : D(A) → C_σ(−∞, 0], where

()

Further, D(A) is dense and A is a closed operator.

Theorem 3.2. For the semigroup S_t defined by (3.2)

()

Further, assume that

. Then for the generator of the semigroup S_t defined by (3.3) and

()

Besides, suppose that for any λ ∈ C with , we have ℜ(λ)<−μ₁ for some μ₁ > 0. Then, the semigroup S_t is asymptotically stable.

Proof. Let T_t be as in Proposition 2.1. Fix t > 0. Define V_t : C_σ(−∞, 0] → C_σ(−∞, 0] as

()

Define K_t : C[0, t] → C_σ(−∞, 0] as

()

It is easy to see that

()

Thus, K_t is a bounded linear map.

Define K₁ : C_σ(−∞, 0] → C_σ(−∞, 0] as [K₁ϕ](θ) = ϕ(0) for all θ ∈ (−∞, 0]. It is clear that K₁ is compact. Define B_t : C_σ(−∞, 0] → C[0, t] as B_tϕ = z, where z is the restriction of y to [0, t]. From (3.1), B_t is a bounded linear map. Let S_t be as in (3.3). Then,

()

Now, if I is the identity on C_σ(−∞, 0] and J : C_σ,0(−∞, 0] → C_σ(−∞, 0] is the inclusion map, then V_t = JT_t(I − K₁), and, finally,

()

Next, we show that B_t is, in fact, a compact map. Let x be the solution to (1.1) as in Theorem 3.1:

()

Thus,

()

Consider n ∈ ℕ such that t ∈ [nτ₁, (n + 1)τ₁]. From (3.1) and (3.11), we obtain existence of c₁(t) ≥ 0 such that

()

Hence by Arzela-Ascoli theorem, B_t is a compact operator.

It is easy to show that |J|_α ≤ ∥J∥_σ = 1. By the compactness of K₁ and B_t, |I − K₁|_α = 1 and |K_tB_t|_α = |K₁|_α = 0. Thus, from the relation

()

and Propositions 1.2 and 2.1 of this paper, we obtain

()

So,

()

Let 0 ≠ λ ∈ ρ(A).

There is a unique ψ ∈ D(A) such that

()

It is clear that there is c ∈ C such that ψ(θ) = (c − 1/λ)e^λθ − 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

. But this would imply that

which is a contradiction, to the hypothesis that

. Now, since c − 1/λ ≠ 0, it is obvious that e_λ ∈ C_σ(−∞, 0]. But this implies that ℜ(λ)>−ln (γ)/τ₁. If 0 ∈ ρ(A), the condition ℜ(λ)>−ln (γ)/τ₁ is obvious. Thus,

()

We now infer that {λ : ℜ(λ)≤−ln (γ)/τ₁}⊆spec(A). Next, if

, and ℜ(λ)>−ln (γ)/τ₁, then e_λ ∈ C_σ(−∞, 0] and hence e_λ ∈ D(A) with λe_λ = Ae_λ. Thus, λ ∈ spec(A). So,

()

Let us assume that ℜ(λ)>−ln (γ)/τ₁ and .

Then, by Proposition 2.1, we have e_λ ∈ C_σ(−∞, 0] and the function h defined as is in C_σ(−∞, 0].

Defining Λ : C_σ(−∞, 0] → C as

and taking c = (Λ(h) − f(0))/(Λ(e_λ) − λ), we find that ϕ defined as

is (λI − A) ⁻¹(f). Thus,

()

From (3.18), (3.19), and (3.20), we finally conclude that

()

Since ω₀ = max {s(A), ω_ess} ≤ max {−μ₁, −ln (γ)/τ₁}, the result follows.

Remark 3.3. Consider the PDE:

()

Let B be as in Proposition 2.1 and A be as in Theorem 3.1. For ϕ ∈ D(B), u(t, θ) = T_tϕ ∈ C_σ,0(−∞, 0] is the solution to the above PDE. For ϕ ∈ D(A), u(t, θ) = S_tϕ ∈ C_σ(−∞, 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.

Acknowledgment

D. Piriadarshani would like to thank Professor B. Praba, Department of Mathematics, SSN College of Engineering, Kalavakkam, Tamil Nadu, India, for the support and advice received from her as a Cosupervisor.

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Asymptotic Stability of Differential Equations with Infinite Delay

Abstract

1. Introduction and Preliminaries

2. Asymptotic Stability of a PDE

3. Stability of the Infinite Delay Equation

Acknowledgment

References

References

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Asymptotic Stability of Differential Equations with Infinite Delay

Abstract

1. Introduction and Preliminaries

2. Asymptotic Stability of a PDE

3. Stability of the Infinite Delay Equation

Acknowledgment

References

References

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