HYPERBOLICITY OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS WITH DELAY

ANDRÁS BÁTKAI

Abstract. Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed.

1. Introduction

Partial differential equations with delay have been studied for many years and by many different methods. In an abstract way and using the standard notation (see [27]), they can be written as

\begin{cases}u′(t) = Bu(t) + Φu_{t}, & t ≥ 0, \\ u(0) = x, \\ u_{0} = f, \end{cases} (DE)

in a Banach space X, where (B, D(B)) is a (unbounded) linear operator on X and the delay operator Φ is supposed to belong to, e.g., L(W^{1,p}([−1, 0], X), X) for some 1 ≤ p ≤ ∞. J. Hale [12] and G. Webb [26] were among the first to apply semigroup theory to delay equations, and we refer to [27] for more recent references on partial differential equations with delay.

As a first step one has to choose an appropriate state space. One of the possibilities is to work in the space of continuous X-valued functions. In this case, the relationship between solutions of (DE) and a corresponding semigroup has been studied intensively (see for example [13], [27] or [8, Section VI.6]) and is well understood. On the other hand, the state space E := X × L^{p}([−1, 0], X) turns out to be a better choice with regards to certain applications (e.g., to control theory, see [23], to numerical methods, see [17]) and will be used in this paper.

The aim of this work is to give robust hyperbolicity and stability results for linear partial differential equations with delay, especially for the cases where no spectral mapping theorems are available and we cannot use the powerful technics of characteristic equations. As an application, we analyze the effect of small delays to the asymptotic properties of feedback systems.

Date: April 18, 2001.

1991 Mathematics Subject Classification. 34K05, 34K20, 47D06.

Key words and phrases. C_{0}-semigroups, delay equations, exponential stability, hyperbolicity, effect of small delays.

The author thanks W. Desch (Graz), I. Győri (Veszprém) and R. Schnaubelt (Halle) for helpful discussions.

2 ANDRÁS BÁTKAI

In the next section we collect some results on the semigroup approach for delay equations in the L^{p} history space, mainly from [1]. This approach is especially useful in the Hilbert space case and allows stability results in the case where the semigroup generated by (B, D(B)) is not compact, see [2] for applications.

In Section 3, we present robust hyperbolicity results in the Hilbert space context of the following kind. Assume, that (B, D(B)) generates a hyperbolic semigroup and that the delay operator Φ is ”small” in some sense. Then the delay semigroup remains hyperbolic. As a special case we consider uniform exponential stability.

In the last section we investigate the important question of the effect of small delays. The problem is the following: We consider delays of the special form Φ := Cδ_{−τ}. The question is, knowing that the solutions of the system are exponentially stable for τ = 0, whether it follows that they remain stable for τ > 0. This question is motivated by feedback-systems and control theory and we give more references on this question in the text. First two examples are given to show how the stability can be destroyed and then a general approach to treat this question is provided. The question is considered for norm continuous semigroups and for commuting compact perturbations.

2. The semigroup approach to delay equations

Let us summarize here some results from [1] on the semigroup approach to linear partial differential equations with delay.

Consider the equation

(DE)  \begin{cases}u′(t) = Bu(t) + Φu_{t}, & t ≥ 0, \\ u(0) = x, \\ u_{0} = f, \end{cases}

where • x ∈ X, X is a Banach space,

• B : D(B) ⊆ X ⟶ X is a linear, closed, and densely defined operator,

• f ∈ L^{p}([−1, 0], X), p ≥ 1,

• Φ : W^{1,p}([−1, 0], X) ⟶ X is a linear, bounded operator,

• u : [−1, ∞) ⟶ X and u_{t} : [−1, 0] ⟶ X is defined by u_{t}(σ) := u(t + σ).

Definition 2.1. We say that a function u : [−1, ∞) ⟶ X is a (classical) solution of (DE) if

(i) u ∈ C([−1, ∞), X) ∩ C^{1}([0, ∞), X), (ii) u(t) ∈ D(B) and u_{t} ∈ W^{1,p}([−1, 0], X) for all t ≥ 0, and (iii) u satisfies (DE) for all t ≥ 0.

To be able to solve (DE) by semigroup methods, we introduce the Banach space

E := X × L^{p}([−1, 0], X)

with an arbitrary product norm, usually the p-norm, and the operator

A := (\begin{matrix}B & Φ \\ 0 & \frac{d}{dσ} \end{matrix}) (2.1)

HYPERBOLICITY OF DELAY EQUATIONS 3

with domain

D(A) := \{( \begin{matrix}x \\ f \end{matrix} ) ∈ D(B) × W^{1,p}([−1, 0], X) : f(0) = x\} . (2.2)

Consider now the abstract Cauchy problem

(ACP)  \begin{cases}v′(t) = A v(t), & t ≥ 0, \\ v(0) = v_{0} \end{cases}

associated to the operator matrix (A, D(A)) on the Banach space E with initial value v_{0} := ( \begin{matrix}x \\ f \end{matrix} ). There is a natural correspondence between the solutions of the two problems (see [1, Proposition 2.3 and 2.4]).

Lemma 2.2. (i) If u is a solution of (DE), then t ↦ ( \begin{matrix}u(t) \\ u_{t} \end{matrix} ) is a solution of the equation

(ACP). (ii) If t ↦ ( \begin{matrix}u(t) \\ v(t) \end{matrix} ) is a solution of (ACP), then v(t) = u_{t} for all t ≥ 0 and u is a solution

of (DE).

We can then give the following definition for well-posedness.

Definition 2.3. We say that (DE) is well-posed if

(i) for every ( \begin{matrix}x \\ f \end{matrix} ) ∈ D(A) there is a unique solution u(x, f, ·), and (ii) the solutions depend continuously on the initial values, i.e., if a sequence ( \begin{matrix}x_{n} \\ f_{n} \end{matrix} ) in

D(A) converges to ( \begin{matrix}x \\ f \end{matrix} ) ∈ D(A), then u(x_{n}, f_{n}, t) converges to u(x, f, t) uniformly for t in compact intervals.

There is also a correspondence between the well-posedness of equation (DE) and of the abstract Cauchy problem (ACP), see also [1, Theorem 2.8].

Proposition 2.4. Let (A, D(A)) be the operator matrix defined by (2.1) and (2.2). Then the following assertions are equivalent.

(i) Equation (DE) is well-posed. (ii) (A, D(A)) is the generator of a strongly continuous semigroup on E.

As a consequence of Lemma 2.2 and Proposition 2.4, we have that if (A, D(A)) generates a strongly continuous semigroup (T (t))_{t≥0}, then the solutions of equation (DE) are given by the first component of the function t ↦ T (t)( \begin{matrix}x \\ f \end{matrix} ) for ( \begin{matrix}x \\ f \end{matrix} ) ∈ D(A).

By means of the perturbation theorem of Miyadera-Voigt (see [8, Corollary III.3.16]) one can formulate the following sufficient condition for the well-posedness of (DE), see [1, Theorem 3.3, Examples 3.4], [20].

Corollary 2.5. Assume that (B, D(B)) generates a strongly continuous semigroup (S(t))_{t≥0} on X, ∞ > p ≥ 1, and that there exists a function η : [−1, 0] → L(X) of bounded variation such that Φ : C([−1, 0], X) → X is given by the Riemann-Stieltjes integral

Φ(f) := ∫_{−1}^{0} dηf. (2.3)

Then (A, D(A)) is a generator on E.

An important special case is the operator Φ defined by

Φ(f) := ∑_{k=0}^{n} B_{k}f(h_{k}), f ∈ W^{1,p}([−1, 0], X),

where B_{k} ∈ L(X) and h_{k} ∈ [−1, 0] for k = 0, …, n.

It was also shown in [1] that the class of delay operators considered in Corollary 2.5 satisfies the following.

Definition 2.6. We call the delay operator Φ ∈ L(W^{1,p}([−1, 0], X), X) admissible if (a) the operator (A, D(A)) is a generator for each generator (B, D(B)) and

(b) the function λ ↦ ΦR(λ, A_{0}) is a bounded analytic function on the halfplane {λ ∈ ℂ : ℜλ > ω} for all ω ∈ ℝ.

We now characterize the resolvent set and the resolvent operator of A (see [1, Lemma 4.1]). Here ϵ_{λ}(t) := e^{λt} and Φ_{λ} ∈ L(X) is defined by Φ_{λ}x := Φ(ϵ_{λ} × Id)x = Φ(e^{λ·}x) for x ∈ X. The operator (A_{0}, D(A_{0})) is the generator of the nilpotent left shift semigroup (T_{0}(t))_{t≥0} in L^{p}([−1, 0], X).

Lemma 2.7. Let X be a Banach space, (B, D(B)) be linear, closed and densely defined, and Φ : W^{1,p}([−1, 0], X) ⟶ X be linear and bounded. Let (A, D(A)) be the operator matrix defined in (2.1) and (2.2). Then λ ∈ ρ(A) if and only if λ ∈ ρ(B +Φ_{λ}). Moreover, for λ ∈ ρ(A) the resolvent R(λ, A) is given by

(\begin{matrix}R(λ, B + Φ_{λ}) & R(λ, B + Φ_{λ})ΦR(λ, A_{0}) \\ ϵ_{λ} ⊗ R(λ, B + Φ_{λ}) & [ϵ_{λ} ⊗ R(λ, B + Φ_{λ})Φ + Id]R(λ, A_{0}) \end{matrix}) . (2.4)

3. Hyperbolicity and stability

In the following, assume that X is a Hilbert space and p = 2. It follows that E will be also a Hilbert space and we may use the powerful Gearhart spectral mapping theorem to characterize hyperbolicity and exponential stability of the delay semigroup. The theorem can be found, e.g., in [8, Theorem V.1.11] and [8, Theorem V.1.18]. For the hyperbolicity, this means that a semigroup (T (t))_{t≥0} with generator (G, D(G)) in a Hilbert space X is hyperbolic if and only if iℝ ⊂ ρ(G) and sup{∥R(iω, G)∥ : ω ∈ ℝ} < ∞. There are recent generalizations of this result to the Banach space case, see [16, 19], which may in the near future allow to generalize the results presented here to the Banach space case.

Theorem 3.1. Let X be a Hilbert space and consider the equation (DE). Assume that Φ is admissible, the semigroup (B, D(B)) generates a hyperbolic semigroup and consider

a_{n} := sup_{ω∈ℝ}∥(Φ_{iω}R(iω, B))^{n}∥ < ∞. (3.1)

If

a := ∑_{n=0}^{∞} a_{n} < ∞, (3.2)

then (A, D(A)) generates a hyperbolic semigroup.

HYPERBOLICITY OF DELAY EQUATIONS 5

Proof. As a consequence of the above mentioned Theorem of Gearhart, the numbers a_{n} are defined for all n ∈ ℕ and we have to show the boundedness of the resolvent operator given in (2.4) on the line iℝ. Under our assumptions, this is equivalent to the existence and boundedness of R(λ, B + Φ_{λ}) on the line {λ ∈ iℝ}.

Defining M := sup_{λ∈iℝ} ∥R(λ, B)∥, we obtain for all λ ∈ iℝ that

R(λ, B) ∑_{n=0}^{∞} (Φ_{λ}R(λ, B))^{n} ∈ L(X)

and

∥R(λ, B) ∑_{n=0}^{∞} (Φ_{λ}R(λ, B))^{n}∥ ≤ M ∑_{n=0}^{∞} ∥ (Φ_{λ}R(λ, B))^{n} ∥ ≤ M ∑_{n=0}^{∞} a_{n} = M · a.

Easy calculations show that this operator defines an inverse for (λ−B−Φ_{λ}) being bounded on the line iℝ.

Corollary 3.2. Assume that X is a Hilbert space, Φ is admissible, and (B, D(B)) generates a hyperbolic semigroup. If

sup_{ω∈ℝ}∥Φ_{iω}R(iω, B)∥ < 1, (3.3)

or in particular if

sup_{ω∈ℝ}∥Φ_{iω}∥ < \frac{1}{sup_{ω∈ℝ} ∥R(iω, B)∥}, (3.4)

then (A, D(A)) generates a hyperbolic semigroup.

The proof is an easy consequence of the previous theorem using Weierstrass’ criterion on the convergence of infinite series.

As an important special case of hyperbolicity, we may now consider uniform exponential stability.

Theorem 3.3. Let X be a Hilbert space and consider the equation (DE). Assume that Φ is admissible, that ω_{0}(B) < 0, and consider

a_{n} := sup_{ω∈ℝ}∥(Φ_{iω}R(iω, B))^{n}∥ < ∞. (3.5)

If

a := ∑_{n=0}^{∞} a_{n} < ∞, (3.6)

then ω_{0}(A) < 0.

Proof. It follows from the inequality

sup_{ω∈ℝ}∥(Φ_{α+iω}R(α + iω, B))^{n}∥ ≤ sup_{ω∈ℝ}∥(Φ_{iω}R(iω, B))^{n}∥

for all α ≥ 0, which is a consequence of the generalized maximum principle, and from Theorem 3.1 that the semigroup generated by (A − α) is hyperbolic for all α ≥ 0. Thus, ω_{0}(A) < 0.

6 ANDRÁS BÁTKAI

Corollary 3.4. Assume that X is a Hilbert space, Φ is admissible, ω_{0}(B) < 0 and let α ∈ (ω_{0}(B), 0]. If

sup_{ω∈ℝ}∥Φ_{α+iω}R(α + iω, B)∥ < 1, (3.7)

or in particular if

sup_{ω∈ℝ}∥Φ_{α+iω}∥ < \frac{1}{sup_{ω∈ℝ} ∥R(α + iω, B)∥}, (3.8)

then ω_{0}(A) < α ≤ 0.

We demonstrate in the following example that the results obtained in Theorem 3.3 are more general then the ones in [1].

Corollary 3.5. Assume that X is a Hilbert space, Φ = Cδ_{−1} for C ∈ L(X) commuting with (B, D(B)), that ω_{0}(B) < 0 and that

r(C) < \frac{1}{sup_{ω∈ℝ} ∥R(iω, B)∥}.

Then ω_{0}(A) < 0.

Proof. We use ideas analogous to [18, Theorem IV.3.6]. Let us denote by M := sup_{ω∈ℝ} ∥R(iω, B)∥. Our assumption means that there exists 0 < q < 1 such that r(C) · M < q < 1. We obtain that there exists n_{0} ∈ ℕ such that

∥C^{n}∥^{\frac{1}{n}} · M < q < 1 for all n ≥ n_{0}.

This means that

a_{0,n} = sup_{ω∈ℝ}∥(Φ_{iω}R(iω, B))^{n}∥ ≤ sup_{ω∈ℝ}∥R(iω, B)^{n}∥ sup_{ω∈ℝ}∥Φ_{iω}^{n}∥ ≤ ∥C^{n}∥ · M^{n} < q^{n}

for n ≥ n_{0}, and the assertion follows by Weierstrass’ criterion.

4. An application: the effect of small delays

The problem considered in this section is the following: Assume that hyperbolicity or uniform exponential stability is known for the solutions of the equation

(DE)_{0}  \begin{cases}u′(t) = (B + C)u(t), & t ≥ 0, \\ u(0) = x, \\ u_{0} = f, \end{cases}

where C ∈ L(X).

The question is, whether the same type of asymptotics holds for the solutions of the equation

(DE)_{τ}  \begin{cases}u′(t) = Bu(t) + Cu(t − τ), & t ≥ 0, \\ u(0) = x, \\ u_{0} = f, \end{cases}

where τ > 0 is “small”.

It is known, see e.g. [22, Example B-IV.3.10], that if X is a Banach lattice, C is a positive operator and (B, D(B)) generates a positive semigroup, then the solutions of

HYPERBOLICITY OF DELAY EQUATIONS 7

(DE)_{τ} are uniformly exponentially stable if and only if the solutions of (DE)_{0} are uniformly exponentially stable.

It is an open question however, what happens to the hyperbolicity in the positive case. The example of Montgomery-Smith [21] suggests that this result may not remain true.

The first who examined this effect was R. Datko [3, 4, 5, 6, 7]. It is known for finite dimensional equations that the stability cannot be destroyed and there exists an extensive literature on delay dependent stability conditions, see e.g. [10, 11]. For similar questions in the parabolic case we refer to [9, 25]. There is a recent exposition of this problem by J. Hale and S. Verduyn Lunel [14, 15], where many examples of functional differential and difference equations are considered. A control theoretical investigation using transfer functions was made for compact feedback in [24].

Before considering the abstract problem, we demonstrate on some simple examples how the stability can be destroyed. Though the following example seems to be known, we include it here because it is the simplest example and we could not find it written in the literature.

Example 4.1. Let (B, D(B)) be the (unbounded) generator of a unitary group in an infinite dimensional Hilbert space H and let C := d · Id for d < 0. Then (B + C, D(B)) generates an exponentially stable semigroup. We show that there exists a sequence (τ_{k}), τ_{k} → 0, such that the solution semigroup of the equation (DE)_{τ_{k}} does not decay exponentially for each k ∈ ℕ.

To construct this sequence, take (μ_{k}) ⊂ ℝ, iμ_{k} ∈ σ(B) such that |μ_{k}| → ∞ and μ_{k} ≠ −d. Defining the numbers

τ_{k} := \begin{cases}\frac{3π}{2(μ_{k}+d)}, & μ_{k} + d > 0, \\ \frac{−π}{2(μ_{k}+d)}, & μ_{k} + d < 0, \end{cases}

and the operators Φ^{(k)} := d · Idδ_{−τ_{k}}, Φ_{λ}^{(k)} := e^{−λτ_{k}}d · Id, we obtain for the numbers λ_{k} := (μ_{k} + d)i ∈ iℝ that

λ_{k} ∈ σ(B + Φ^{(k)}i_{λ_{k}}) = σ(B) + d · e^{−λ_{k}τ_{k}} = σ(B) + d · i.

By the spectral characterization in Lemma 2.7 it follows that the associated operator (A, D(A)) can not generate a uniformly exponentially stable semigroup. If we assume further that iμ_{k} ∈ P σ(B), which e.g., is satisfied if (B, D(B)) has compact resolvent, then we also find classical solutions of (DE)_{τ_{k}} which are not decaying exponentially.

The essence of this example can be formulated as follows.

Theorem 4.2. Let X be a Hilbert space and assume that (B, D(B)) generates a strongly continuous semigroup such that there exists ρ ∈ ℝ and μ_{k} ∈ ℝ, |μ_{k}| → ∞ such that ρ + iμ_{k} ∈ σ(B), i.e., the spectrum is unbounded along an imaginary line. Then there exists C ∈ L(X) and (τ_{k}) ⊂ ℝ^{+}, τ_{k} → 0 such that ω_{0}(B + C) < 0 but the solutions of (DE)_{τ} are not uniformly exponentially stable.

Proof. As in the previous example, take C := −μ · Id, μ > 0, μ > −ρ and define

τ_{k} := \frac{π}{|μ_{k}|}

8 ANDRÁS BÁTKAI

for some k ∈ ℕ. Then the corresponding characteristic equations are again

z − λ + μe^{−zτ_{k}} = 0, λ ∈ σ(B).

Now take λ = ρ + iμ_{k} and put z = ε + iμ_{k} for some ε > 0. Then we obtain

iμ_{k} + ε − ρ − iμ_{k} + μe^{−iμ_{k}τ_{k}} · e^{−ετ_{k}} = 0,

and hence

ε = μe^{−ετ_{k}} + ρ.

Since μ > −ρ, there exists a positive real solution ε.

Unfortunately, if the stabilizing operator is not the identity, the preceding technique can be applied only with enormous difficulty even in cases where the spectral mapping theorem holds. This is because we have in general no easy characterization of σ(B + Φ_{λ}), see [18, Example IV. 3.8].

Turning our attention now to the general problem of small delays, we use an idea similar to [13, Section 5.4 (4.9)] and transform the equation (DE)_{τ} into

u′(t) = (B + C)u(t) + C (u(t − τ) − u(t)) .

We use the equality

u(t_{2}) − u(t_{1}) = [S(t_{2} − t_{1}) − Id] u(t_{1}) + ∫_{t_{1}}^{t_{2}} S(t_{2} − s)Cu(s − τ)ds (4.1)

for t_{2} > t_{1} ≥ 0 following from (DE)_{τ}. Substituting t_{1} = t − τ and t_{2} = t, we obtain that

u(t) − u(t − τ) = [S(τ) − Id] u(t − τ) + ∫_{−τ}^{0} S(−s)Cu(t + s − τ)ds.

Thus, (DE)_{τ} can be written in the form

(4.2) u′(t) = (B + C)u(t)

− C ([S(τ) − Id] u(t − τ) + ∫_{−τ}^{0} S(−s)Cu(t + s − τ)ds) .

Defining

Φf := −C [S(τ) − Id] δ_{−τ}f − ∫_{−τ}^{0} CS(−s)Cδ_{s−τ}fds, (4.3)

where δ_{r} ∈ L (W^{1,p}([−1, 0], X), X) is given by δ_{r}(f) := f(r) for r ∈ [−1, 0], we see that the previous stability results in Corollary 3.4 are applicable to our original problem and that (DE)_{τ} has the form

u′(t) = (B + C) u(t) + Φu_{t}. (4.4)

In order to be able to apply the stability results of Corollary 3.4, or the hyperbolicity results of Corollary 3.2, we have to calculate

(4.5) Φ_{λ}R(λ, B + C)x = −C [S(τ) − Id] e^{−λτ}R(λ, B + C)x

− ∫_{−τ}^{0} CS(−s)Ce^{−λ(s−τ)}R(λ, B + C)xds.

HYPERBOLICITY OF DELAY EQUATIONS 9

Defining

I_{1}^{ω}(τ) := C [S(τ) − Id] e^{−iωτ}R(iω, B + C) (4.6)

and

I_{2}^{ω}(τ)x := ∫_{−τ}^{0} CS(−s)Ce^{−iω(s−τ)}R(iω, B + C)xds, (4.7)

it would be sufficient to show that there exists κ > 0 such that sup_{ω∈ℝ} ∥I_{i}^{ω}(τ)∥ < \frac{1}{2} for i = 1, 2 and all τ ∈ (0, κ). Then, using Corollary 3.4 or Corollary 3.2 and (4.5), the assertion follows since sup_{ω∈ℝ} ∥Φ_{iω}R(iω, B + C)∥ ≤ sup_{ω∈ℝ} ∥I_{1}^{ω}(τ)∥ + sup_{ω∈ℝ} ∥I_{2}^{ω}(τ)∥ < 1.

The estimate on I_{2}^{ω} is

∥I_{2}^{ω}(τ)∥ ≤ τ∥C∥^{2}K∥R(iω, B + C)∥, (4.8)

where K := sup_{0≤t≤1} ∥S(t)∥. Since ∥R(iω, B + C)∥ is uniformly bounded for all ω ∈ ℝ, there exists κ_{2} > 0 such that for all τ ∈ (0, κ_{2}) the estimate sup_{ω∈ℝ} ∥I_{2}^{ω}(τ)∥ < \frac{1}{2} holds.

The estimate on I_{1}^{ω} is

(4.9) ∥I_{1}^{ω}(τ)∥ ≤ ∥C∥ · ∥(S(τ) − Id)R(iω, B + C)∥ ≤ ∥C∥ · ∥(S(τ) − Id)R(λ, B)∥ · ∥(λ − B)R(iω, B + C)∥,

where λ > max {ω_{0}(B), 0} is fixed.

Since ∥(λ − B)R(iω, B + C)∥ is independent of τ, we only have to consider the term (S(τ) − Id)R(λ, B).

But then it follows from

∥(S(τ) − Id)R(λ, B)∥ ≤ ∥S(τ)∥(1 − e^{−λτ})∥R(λ, B)∥ + ∥ ∫_{0}^{τ} e^{−λs}S(s)ds∥

≤ τK (∥R(λ, B)∥|λ| + 1)

that for every ω ∈ ℝ

lim_{τ→0} ∥C [S(τ) − Id] e^{iωτ}R(iω, B + C)∥ = 0. (4.10)

Example 4.1 shows that in general this convergence cannot be uniform. This is the point where we need some extra assumptions.

As we could see in Example 4.1, the unboundedness of the spectrum of (B, D(B)) along imaginary axes may cause trouble if we allow any stabilizing operator C. In the following result on the independence of stability of small delays the spectrum of the generator also plays an important role.

Theorem 4.3. Assume that (B, D(B)) generates an immediately norm continuous semigroup and that the semigroup generated by (B + C, D(B)) is exponentially stable or hyperbolic in the Banach space X. Then there exists κ > 0 such that the solution semigroup of (DE)_{τ} is exponentially stable or hyperbolic, respectively, for all τ ∈ (0, κ). Thus, the stability and the hyperbolicity is not sensitive to small delays.

Proof. We have to show that the convergence in (4.10) is uniform in ω.

To this end we use the immediate norm continuity of the semigroup generated by (B + C, D(B)), see [8, Theorem III.1.16(i)]. An important consequence is that

10 ANDRÁS BÁTKAI

lim_{|ω|→∞} ∥R(iω, B + C)∥ = 0, see [8, Corollary II.4.19]. Thus, there exists L > 0 such that

∥R(iω, B + C)∥ < \frac{1}{2∥C∥(K + 1)} for |ω| > L,

where K := sup_{0≤t≤1} ∥S(t)∥.

For ω ∈ [−L, L], we recall that the function

(ω, τ) ↦ ∥C [S(τ) − Id] e^{iωτ}R(iω, B + C)∥

is uniformly continuous on [−L, L] × [0, 1]. Thus, there exists κ_{1} > 0 such that for all τ ∈ (0, κ_{1}) and for all ω ∈ [−L, L]

∥C [S(τ) − Id] e^{iωτ}R(iω, B + C)∥ < \frac{1}{2}.

Combining these estimates we obtain the desired statement.

The proof can be finished by choosing κ := min {κ_{1}, κ_{2}}.

We make the remark that the results of the previous theorem remain true also if X is a Bahach space. This is because under the conditions of the theorem, the delay semigroup will be eventually norm continuous, see [1, Proposition 5.3], and for eventually norm continuous semigroups the spectral mapping theorem holds, see [8, Theorem IV.3.10].

Analogous results were obtained by R. Schnaubelt [25] for non-autonomous equations in the parabolic case.

In the previous theorem we gave a condition on the generator (B, D(B)) without any restriction on the stabilizing operator C. In the following we also provide a condition involving C.

Proposition 4.4. Let (B, D(B)) be a generator of a strongly continuous semigroup (S(t))_{t≥0} in the Hilbert space X, C ∈ L(X) be a compact operator commuting with B and that the semigroup generated by (B + C, D(B)) is exponentially stable or hyperbolic in the Hilbert space X. Then there exists κ > 0 such that the solution semigroup of (DE)_{τ} is exponentially stable or hyperbolic for all τ ∈ (0, κ). Thus, the stability and the hyperbolicity is not sensitive to small delays.

Proof. Again we only have to show that the convergence in (4.10) is uniform in ω. Using that C commutes with B and hence with the semigroup, we obtain

C [S(τ) − Id] e^{iωτ}R(iω, B + C) = [S(τ) − Id] e^{iωτ}R(iω, B + C)C.

By our assumptions, the set CB(0, 1) ⊂ X is precompact in X. The proof can be finished by using the fact that on compact sets the strong and the uniform topology coincide.

Remark 4.5. To show that estimating I_{1}^{ω} and I_{2}^{ω} is not sharp, consider the well-known example B = 0 and C = d · Id for d < 0 in the Banach space ℂ. The direct calculation of the formula (4.5) shows that the solutions of (DE)_{τ} are exponentially stable if |d|τ < 1.

However, applying directly the spectral characterization of Lemma 2.7 and using that σ(B + Φ_{λ}) = {d · e^{−λτ}}, we obtain the well-known and best possible estimate (see [13, p. 135]) that the solutions decay exponentially if

|d|τ < \frac{π}{2} . (4.11)

References

[1] Bátkai, A., Piazzera, S., Semigroups and linear partial differential equations with delay, to appear in J. Math. Anal. Appl.

[2] Bátkai, A., Piazzera, S., Damped wave equations with delay, Fields Institute Communications (2001).

[3] Datko, R., Is boundary control a realistic approach to the stabilization of vibrating elastic systems?, in: Ferreyra, Guillermo (ed.), “Evolution Equations”, Marcel Dekker, 133–140 (1994).

[4] Datko, R. Two questions concerning the boundary control of elastic systems, J. Diff. Eq. 92 (1991), 27–44.

[5] Datko, R. Not all feedback stabilized systems are robust with respect to small time delays in their feedback, SIAM J. Control and Optimization 26 (1988), 697–713.

[6] Datko, R., Lagnese, J., Polis, M. P., An example on the effect of time delays in boundary feedback of wave equations, SIAM J. Control and Optimization 24 (1986), 152–156.

[7] Datko, R., You, Y.C., Some second order vibrating systems cannot tolerate small time delays in their damping, J. Optim. Th. Appl. 70 (1991), 521–537.

[8] Engel, K.-J., Nagel R., “One-parameter Semigroups for Linear Evolution Equations”, Graduate Texts in Mathematics 194, Springer-Verlag, 1999.

[9] Gühring, G., Räbiger, F., Schnaubelt, R., A characteristic equation for non-autonomous partial functional differential equations, preprint, 2000.

[10] Győri, I., Pituk, M., Stability criteria for linear delay differential equations, Diff. Int. Eq. 10 (1997), 841–852.

[11] Győri, I., Hartung, F., Turi, J., Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl. 220 (1998), 290–313.

[12] Hale, J. K., “Functional Differential Equations”, Appl. Math. Sci., vol. 3, Springer-Verlag, 1971.

[13] Hale, J. K., Verduyn Lunel, S. M., “Introduction to Functional Differential Equations”, Appl. Math. Sci. 99, Springer-Verlag, 1993.

[14] Hale, J. K., Verduyn Lunel, S. M., Effects of small delays on stability and control, Preprint, 1999.

[15] Hale, J. K., Verduyn Lunel, S. M., Effects of time delays on the dynamics of feedback systems, Proceedings of EQUADIFF’99 Berlin

[16] Kaashoek, M. A., Verduyn Lunel, S. M., An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Diff. Eq. 112 (1994), 374–406.

[17] Kappel, F., Semigroups and delay equations, in: Brezis, H., Crandall, M. G., Kappel, F. (ed.), “Semigroups, Theory and Applications”, Vol. II., Pitman Research Notes in Mathematics 152, Longman, 136–176 (1986).

[18] Kato, T., “Perturbation Theory for Linear Operators”, Grundlehren Math. Wiss. 132, Springer-Verlag, 1980.

[19] Latushkin, Y., Shvydkoy, R., Hyperbolicity of semigroups and fourier multipliers, preprint, 2000.

[20] Maniar, L., Voigt, J., Linear delay equations in the L^{p} context, Preprint, 2000.

[21] Montgomery-Smith, S., Stability and dichotomy of positive semigroups in L^{p}, Proc. Am. Math. Soc. 124 (1996), 2433–2437.

[22] Nagel, R. (ed.), “One-parameter Semigroups of Positive Operators, Springer-Verlag, 1986.

[23] Nakagiri, S., Optimal control of linear retarded systems in Banach spaces, J. Math. Anal. Appl. 120 (1986), 169–210.

[24] Rebarber, R., Townly, S., Robustness with respect to delays for exponential stability of distributed parameter systems, SIAM J. Control and Optimization 37 (1998), 230–244.

[25] Schnaubelt, R., Parabolic evolution equations with asymptotically autonomous delay, preprint, 2001.

[26] Webb, G., Functional differential equations and nonlinear semigroups in L^{p}-spaces, J. Diff. Eq. 29 (1976), 71–89.

[27] Wu, J., “Theory and Applications of Partial Functional Differential Equations”, Springer-Verlag, Appl. Math. Sci. 119, 1996.

ELTE TTK, Department of Applied Analysis, 1053 Budapest, Kecskeméti u. 10-12, Hungary.

E-mail address: batka@cs.elte.hu