Robustness of Exponential Dissipation with respect to Small Time Delay

Abstract

We investigate robustness of exponential dissipation for the following general nonlinear evolutionary equation with small time delay: 𝜕_𝑡𝑢+𝐴𝑢=𝑓(𝑢(𝑡),𝑢(𝑡−𝜏)). We firstly obtain a converse Lyapunov theorem. With the help of it, we establish an important result on robustness of exponential dissipation to small time delay assuming that the nonlinearity is globally Lipschitz.

1. Introduction

As is well known, time delays are usually encountered in practical control systems. The stability analysis has received attentions over the last several decades. Mathematically, it is also very important to understand the sensitivity of the dynamical behavior of the system to the introduction of small time delays. For linear systems, we well understand this problem, including both finite dimensional and infinite dimensional situations, see [1–5]. However, for nonlinear systems, the problem is much more difficult, but there are some very nice results in [6–10].

This paper is devoted to the following general nonlinear evolutionary equation with small delay: 𝜕_𝑡𝑢+𝐴𝑢=𝑓(𝑢(𝑡),𝑢(𝑡−𝜏)), (1) where the nonlinearity 𝑓∶𝑋^𝛼×𝑋^𝛼→𝑋 is assumed to be globally Lipschitz. Here, 𝐴 is a sectorial operator on a Banach space 𝑋. 𝑋^𝛼 is a fractional power space. Here, we investigate the effects of small time delay on the exponential dissipation of the corresponding evolutionary equation without delay: 𝜕_𝑡𝑢+𝐴𝑢=𝐹(𝑢), (2) where 𝐹(𝑢)=𝑓(𝑢,𝑢).

In [11], Lyapunov introduced his famous sufficient conditions for asymptotic stability of the following nonautonomous dynamical system: 𝑥^′(𝑡)=𝑓(𝑡,𝑥(𝑡)),𝑥(𝑡)∈𝑅^𝑛. (3) There, we can also find the first contribution to the converse question, known as converse Lyapunov theorems. In recent years, the answers have proved instrumental in establishing robustness of various stability notions and have served as the starting point for many nonlinear control systems design concepts.

In 2005, Li and Kloeden [8] presented a converse Lyapunov theorem for exponential dissipation of the following general nonlinear differential equations with multiple small time delays: 𝑥^′(𝑡)=𝑓𝑥𝑡−𝜏₁,𝑥𝑡−𝜏₂,…,𝑥𝑡−𝜏_𝑛, (4) where 𝑓 is assumed to be globally Lipschitz. They also prove that exponential dissipation remains under small time delays. This result can be seen as a generalization of some classical ones on global exponential asymptotic stability (e.g., [12]) and was used by the authors to study robustness of exponential dissipation with respect to small time delays.

Recently, Guo and Li [13] gave a nonautonomous analog of the result. They not only present a converse Lyapunov theorem but also prove robustness of the uniform exponential dissipation with respect to unbounded external perturbations.

In the dynamical theory, a basic problem concerns the robustness of global attractors under perturbations [14]. It is known that if a nonlinear system with a global attractor 𝒜 is perturbed, then the perturbed one also has an attractor 𝒜^′ near 𝒜, provided that the perturbation is sufficiently small [7, 15]. However, in general, we only know that 𝒜^′ is a local attractor. Whether the global feature can be preserved is still an open problem. To our great joy, a dissipative system usually implies the existence of the global attractor. So, if one wants to settle the above problem, he only needs to examine the robustness of dissipation under perturbations. In this present work, we will investigate the infinite dimensional situations which are more difficult than the finite ones. With the nonlinearity being globally Lipschitz, we obtain a converse Lyapunov theorem and prove that exponential dissipation has nice robustness properties under small time delay.

2. Preliminaries

In this paper, we study the following delayed initial value problem: 𝑢_𝑡+𝐴𝑢=𝑓(𝑢(𝑡),𝑢(𝑡−𝜏)),𝑡>0,𝑢|_[−𝜏,0]=𝑢₀(𝑡). (5) For simplicity, we use ‖⋅‖ and ‖⋅‖_𝛼 to denote the norm on 𝑋 and 𝑋^𝛼, respectively. We write 𝒞=𝐶([−𝜏,0],𝑋^𝛼) with the norm |‖⋅‖|_𝛼 defined by 𝑢_0𝛼=max_{[]𝑡∈−𝜏,0}𝑢₀(𝑡)_𝛼,∀𝑢₀∈𝒞. (6)

Next, we will recall some basic definitions and facts.

The upper right Dini derivative of a function 𝑦∈𝐶((𝛼,𝛽),𝑋^𝛼) is defined as 𝑑⁺𝑦(𝑡)𝑑𝑡∶=limsup_ℎ→0+𝑦(𝑡+ℎ)−𝑦(𝑡)ℎ. (7)

Let 𝑥∈𝑋^𝛼 and 𝒩 be an open neighborhood of 𝑥. For 𝑉∈𝐶(𝒩,𝑅¹) and 𝑣∈𝑋^𝛼, we define 𝐷⁺_𝑣𝑉(𝑥)∶=limsup_ℎ→0+𝑉(𝑥+ℎ𝑣)−𝑉(𝑥)ℎ. (8)

We will denote by 𝑢(𝑡,𝑥) the solution of (2), where 𝑢(0,𝑥)=𝑥.

Definition 1. The system (2) is said to be exponentially dissipative, if there exist positive numbers 𝐵,𝜆, and 𝜌 such that ‖𝑢(𝑡,𝑥)‖_𝛼≤𝐵𝑒^−𝜆𝑡‖𝑥‖_𝛼+𝜌,∀𝑡≥0,𝑥∈𝑋^𝛼. (9)

Lemma 2. Let 𝒩 be an open subset of 𝑋^𝛼. Assume that the function 𝑉∶𝒩→𝑅⁺ is Lipschitz; that is, there exists a 𝐿_𝑉>0 such that 𝑉(𝑥)−𝑉𝑦≤𝐿_𝑉𝑥−𝑦_𝛼,∀𝑥,𝑦∈𝑋^𝛼. (10) Let 𝑢(𝑡) be a solution of (2). Then, 𝑑⁺𝑉(𝑢(𝑡𝑑𝑡))=𝐷⁺_𝑔(𝑢)𝑉(𝑢(𝑡)),where𝑔(𝑢)=𝐹(𝑢)−𝐴𝑢. (11)

Proof. The detailed proof is contained in [12, 16]. Here, we give a simple proof for the reader’s convenience. Making use of Taylor formula, we observe that 𝑉(𝑢((𝑢(𝑡𝑢(𝑡)(𝑢)(ℎ)(𝑢(𝑡𝑢(𝑡)(𝑢)(ℎ)𝑢(𝑡)(𝑢)ℎ𝑢(𝑡)(𝑢)ℎ(𝑢(𝑡.𝑡+ℎ))−𝑉))=𝑉+𝑔ℎ+𝑜−𝑉))=𝑉+𝑔ℎ+𝑜−𝑉+𝑔+𝑉+𝑔−𝑉)) (12) Since 𝑉(𝑥) is Lipschitz, one easily sees that 𝑉𝑢(𝑡)+𝑔(𝑢)ℎ+𝑜(ℎ)−𝑉𝑢(𝑡)+𝑔(𝑢)ℎ=𝑜(ℎ). (13) Therefore, by definition (7), we immediately deduce that 𝑑⁺𝑉(𝑢(𝑡𝑑𝑡))=limsup_ℎ→0+𝑉(𝑢(𝑡+ℎ))−𝑉(𝑢(𝑡))ℎ=limsup_ℎ→0+𝑉𝑢(𝑡)+𝑔(𝑢)ℎ−𝑉(𝑢(𝑡))ℎ=𝐷⁺_𝑔(𝑢)𝑉(𝑢(𝑡)). (14) The proof is finished.

At last, we come to the main theorem on analytic semigroup which is extremely important in the study of the dynamics of nonlinear evolutionary equations [17].

Theorem 3 (fundamental theorem on sectorial operators). Let 𝐴 be a positive, sectorial operator on a Banach space 𝑋 and let 𝑒^−𝐴𝑡 be the analytic semigroup generated by −𝐴. Then, the following statements hold.(1)For any 𝛼≥0, there is a constant 𝐶_𝛼>0 such that for all 𝑡>0𝐴^𝛼𝑒^−𝐴𝑡_𝐿(𝑋)≤𝐶_𝛼𝑡^−𝛼𝑒^−𝑎𝑡(𝑎>0). (15)(2)For 0<𝛼≤1, there is a constant 𝐶_𝛼>0 such that for 𝑡≥0 and 𝑥∈𝐷(𝐴^𝛼)𝑒^−𝐴𝑡𝑥−𝑥≤𝐶_𝛼𝑡^𝛼𝐴^𝛼𝑥. (16)(3)For every 𝛼≥0, there is a constant 𝐶_𝛼>0 such that for all 𝑡>0 and 𝑥∈𝑋𝑒^{−𝐴(𝑡+ℎ)}−𝑒^−𝐴𝑡𝑥_𝛼≤𝐶_𝛼|ℎ|𝑡^−(1+𝛼)‖𝑥‖. (17)

3. Main Results

In this section, we will prove our two main results: one is converse Lyapunov theorem, and the other is robustness of exponential dissipation to small time delay.

Theorem 4 (converse Lyapunov theorem). Suppose that 𝐹∶𝑋^𝛼→𝑋 in (2) is globally Lipschitz with Lipschitz constant 𝐿. Suppose that the system without delay (2) is exponentially dissipative. Then, there exists a function 𝑉∶𝑋^𝛼→𝑅⁺ satisfying ‖𝑥‖²_𝛼−𝑎≤𝑉(𝑥)≤𝑏‖𝑥‖²_𝛼𝐷+𝑐,_+𝑔(𝑥)𝑉(𝑥)≤−𝑑‖𝑥‖²_𝛼𝑉(𝑥)𝑦+𝜎,−𝑉≤𝐿_𝑉‖𝑥‖_𝛼+𝑦_𝛼+1𝑥−𝑦_𝛼 (18)(19)(20) for all 𝑥,𝑦∈𝑋^𝛼, where 𝑔(𝑥)=𝐹(𝑥)−𝐴𝑥, 𝑎,𝑏,𝑐,𝑑,𝜎, and 𝐿_𝑉 are appropriate positive constants.

Proof. Since the system (2) is exponentially dissipative, there exist positive constants 𝐵, 𝜆, and 𝜌 such that ‖𝑢(𝑡,𝑥)‖_𝛼≤𝐵𝑒^−𝜆𝑡‖𝑥‖_𝛼+𝜌,∀𝑡≥0,𝑥∈𝑋^𝛼. (21) Let 𝑇=ln(2𝐵)/𝜆, and define 𝑉₁ as follows: 𝑉₁(𝑥)∶=^𝑇₀‖𝑢(𝑠,𝑥)‖²_𝛼𝑑𝑠,𝑥∈𝑋^𝛼. (22) By (21) and the elementary inequality, it is easy to check that 0≤𝑉₁(𝑥)≤^𝑇₀𝐵𝑒^−𝜆𝑠‖𝑥‖_𝛼+𝜌²𝑑𝑠≤2^𝑇₀𝐵²𝑒^−2𝜆𝑠‖𝑥‖²_𝛼+𝜌²≤𝐵𝑑𝑠₂𝜆‖𝑥‖²_𝛼+2𝑇𝜌². (23) So, 𝑉₁(𝑥) satisfies the right inequality of (18).
Next, by the Lipschitz continuity of 𝐹, it is easy to verify that there exists a constant 𝐶(𝑇)>0 such that 𝑢(𝑡,𝑥)−𝑢𝑡,𝑦_𝛼≤𝐶(𝑇)𝑥−𝑦_𝛼,∀𝑥,𝑦∈𝑋^𝛼,𝑡∈[0,𝑇]. (24) Considering (21) and (23), for any 𝑥,𝑦∈𝑋^𝛼, we have 𝑉₁(𝑥)−𝑉₁𝑦=^𝑇₀‖𝑢(𝑠,𝑥)‖²_𝛼−𝑢𝑠,𝑦²_𝛼(𝑇𝑑𝑠≤𝐶)𝑥−𝑦_𝛼^𝑇₀‖𝑢(𝑠,𝑥)‖_𝛼+𝑢𝑠,𝑦_𝛼(𝑇𝑑𝑠≤𝐶)𝑥−𝑦_𝛼^𝑇₀𝐵𝑒^−𝜆𝑠‖𝑥‖_𝛼+𝑦_𝛼(𝑇+2𝜌𝑑𝑠≤𝐶)𝑥−𝑦_𝛼𝐵𝜆‖𝑥‖_𝛼+𝑦_𝛼+2𝜌𝑇𝑑𝑠≤𝐿₁‖𝑥‖_𝛼+𝑦_𝛼+1𝑥−𝑦_𝛼. (25) So, 𝑉₁(𝑥) satisfies (20).
Since 𝑉₁(𝑢(𝑡,𝑥))=^𝑇₀‖𝑢(𝑠,𝑢(𝑡,𝑥))‖²_𝛼=𝑑𝑠_𝑇0‖𝑢(𝑡+𝑠,𝑥)‖²_𝛼=𝑑𝑠_{𝑡𝑡+𝑇}‖𝑢(𝑠,𝑥)‖²_𝛼𝑑𝑠, (26) by the choice of 𝑇 and (21), we have that 𝑑𝑉𝑑𝑡¹(𝑢(=‖𝑢(𝑡,𝑥))𝑡+𝑇,𝑥)‖²_𝛼−‖𝑢(𝑡,𝑥)‖²_𝛼=‖𝑢(𝑇,𝑢(𝑡,𝑥))‖²_𝛼−‖𝑢(𝑡,𝑥)‖²_𝛼≤−‖𝑢(𝑡,𝑥)‖²_𝛼+𝐵𝑒^−𝜆𝑇‖𝑢(𝑡,𝑥)‖_𝛼+𝜌²≤−‖𝑢(𝑡,𝑥)‖²_𝛼+2𝐵²𝑒^−2𝜆𝑇‖𝑢(𝑡,𝑥)‖²_𝛼+2𝜌²1≤−2‖𝑢(𝑡,𝑥)‖²_𝛼+2𝜌². (27) Consequently, by Lemma 2, 𝐷⁺_{𝑔(𝑢(𝑡,𝑥))}𝑉₁(𝑢(=𝑑𝑡,𝑥))⁺𝑉𝑑𝑡¹(𝑢(=𝑑𝑡,𝑥))𝑉𝑑𝑡¹(𝑢(𝑡,𝑥)). (28) In particular, setting 𝑡=0, one obtains that 𝐷⁺_𝑔(𝑥)𝑉₁(𝑥)=𝑑𝑉𝑑𝑡¹(𝑢(𝑡,𝑥))_𝑡=01≤−2‖𝑥‖²_𝛼+2𝜌², (29) which indicates that 𝑉₁ satisfies (19).
Now, let us define another Lyapunov function 𝑉₂. We firstly take a nonnegative function 𝛾(𝑠) as 𝛾(𝑠)=max𝑠²−𝜌²₀,0,𝑠≥0, (30) where 𝜌₀=(2𝐵+1)𝜌. It is easy to check that 𝛾(𝑠) satisfies 𝛾(𝑠)−𝛾(𝑟)≤(𝑠+𝑟)|𝑠−𝑟|,∀𝑠,𝑟≥0. (31) Now, we let 𝑉₂(𝑥)=sup_𝑠≥0𝛾‖𝑢(𝑠,𝑥)‖_𝛼,∀𝑥∈𝑋^𝛼. (32) We firstly verify the following fact: 𝑉₂(𝑥)=sup_{0≤𝑠≤𝑇}𝛾‖𝑢(𝑠,𝑥)‖_𝛼,∀𝑥∈𝑋^𝛼. (33) Indeed, if ‖𝑥‖_𝛼≤2𝜌, then by (21) ‖𝑢(𝑡,𝑥)‖_𝛼≤𝐵𝑒^−𝜆𝑡‖𝑥‖_𝛼+𝜌≤2𝜌𝐵+𝜌=𝜌₀. (34) According to the definition of 𝛾(𝑠), we know that 𝛾(‖𝑢(𝑠,𝑥)‖_𝛼)=0. Therefore, in case of ‖𝑥‖_𝛼≤2𝜌, one trivially has 𝑉₂(𝑥)=0=sup_{0≤𝑠≤𝑇}𝛾‖𝑢(𝑠,𝑥)‖_𝛼. (35) If ‖𝑥‖_𝛼≥2𝜌, then by the choice of 𝑇 we find that ‖𝑢(𝑠,𝑥)‖_𝛼≤𝐵𝑒^−𝜆𝑇‖𝑥‖_𝛼‖𝑥‖+𝜌=^𝛼2+𝜌<‖𝑥‖_𝛼,∀𝑠≥𝑇. (36) Since 𝑢(0,𝑥)=𝑥 and 𝛾(𝑠) is nondecreasing in 𝑠, one can deduce the correctness of (33).
Next, we will check that 𝑉₂ also satisfies (20). By (33), (31), (24), and (21) 𝑉₂(𝑥)=sup_{0≤𝑠≤𝑇}𝛾‖𝑢(𝑠,𝑥)‖_𝛼=sup_{0≤𝑠≤𝑇}𝛾‖𝑢(𝑠,𝑥)‖_𝛼−𝛾𝑢𝑠,𝑦_𝛼+𝛾𝑢𝑠,𝑦_𝛼≤sup_{0≤𝑠≤𝑇}𝛾‖𝑢(𝑠,𝑥)‖_𝛼−𝛾𝑢𝑠,𝑦_𝛼+𝑉₂𝑦≤sup_{0≤𝑠≤𝑇}‖𝑢(𝑠,𝑥)‖_𝛼+𝑢𝑠,𝑦_𝛼×‖𝑢(𝑠,𝑥)‖_𝛼−𝑢𝑠,𝑦_𝛼+𝑉₂𝑦≤𝐶(𝑇)𝑥−𝑦_𝛼×sup_{0≤𝑠≤𝑇}𝐵𝑒^−𝜆𝑠‖𝑥‖_𝛼+𝑦_𝛼+2𝜌+𝑉₂𝑦≤𝐶(𝑇)𝐵‖𝑥‖_𝛼+𝑦_𝛼+2𝜌𝑥−𝑦_𝛼+𝑉₂𝑦≤𝐿₂‖𝑥‖_𝛼+𝑦_𝛼+1𝑥−𝑦_𝛼+𝑉₂𝑦. (37)
Next, we will check that for arbitrary 𝑥∈𝑋^𝛼, 𝑉₂(𝑥) is bounded by ‖𝑥‖²_𝛼−𝜌²₀≤𝑉₂(𝑥)≤2𝐵²‖𝑥‖²_𝛼+2𝜌². (38) Firstly, according to the definition of 𝛾(𝑠), it is obvious to see that 𝑠²−𝜌²₀≤𝛾(𝑠)≤𝑠²,∀𝑠≥0. (39) So, it follows that 𝑉₂(𝑥)=sup_𝑠≥0𝛾‖𝑢(𝑡,𝑥)‖_𝛼≥𝛾‖𝑢(0,𝑥)‖_𝛼=𝛾‖𝑥‖_𝛼≥‖𝑥‖²_𝛼−𝜌²₀. (40) Recalling (21), we infer ‖𝑢(𝑠,𝑥)‖_𝛼≤𝐵𝑒^−𝜆𝑠‖𝑥‖_𝛼+𝜌≤𝐵‖𝑥‖_𝛼,∀𝑠≥0. (41) Frequently, by the definition of 𝑉₂ and the monotonicity property of 𝛾(𝑠), we get 𝑉₂(𝑥)≤𝛾𝐵‖𝑥‖_𝛼≤𝐵‖𝑥‖+𝜌_𝛼+𝜌²≤2𝐵²‖𝑥‖²_𝛼+2𝜌². (42) So, we verify the correctness of (38).
Lastly we need to check that 𝑉₂(𝑢(𝑡,𝑥)) is nonincreasing in 𝑡. Note that 𝑉₂(𝑢(𝑡,𝑥))=sup_𝑠≥0𝛾‖𝑢(𝑠,𝑢(𝑡,𝑥))‖_𝛼=sup_𝑠≥0𝛾‖𝑢(𝑠+𝑡,𝑥)‖_𝛼=sup_𝑠≥𝑡𝛾‖𝑢(𝑠,𝑥)‖_𝛼. (43) It is easy to see the validity of our checking.
Now, let 𝑉(𝑥)=𝑉₁(𝑥)+𝑉₂(𝑥). (44) Considering (23), (38), (25), (37), and (31), we can get the validity of (18), (19), and (20). The proof is complete.

In order to prove the second result, we need to verify the following lemma.

Lemma 5. Suppose that 𝑓 is globally Lipschitz with Lipschitz constant 𝐿>0, that is, 𝑓𝑥₁,𝑥₂−𝑓𝑦₁,𝑦₂≤𝐿𝑥₁−𝑦_1𝛼+𝑥₂−𝑦_2𝛼, (45) for any 𝑥_𝑖,𝑦_𝑖∈𝑋^𝛼, and that the system (2) is exponentially dissipative. Then, there exist 𝐵₀>1 and 𝜏₀>0 such that when 𝜏≤𝜏₀, any solution of (1) with initial value 𝑢₀∈𝒞=𝐶([−𝜏,0],𝑋^𝛼) satisfies 𝑢𝑡,𝑢_0𝛼<𝐵₀𝑢_0𝛼+1,∀𝑡≥0,𝑢₀∈𝒞. (46)

Proof. According to (45), it is easy to see that there is an 𝑀>0 such that 𝑓𝑥,𝑦≤𝑀‖𝑥‖_𝛼+𝑦_𝛼+1,∀𝑥,𝑦∈𝑋^𝛼. (47) Firstly, we prove that for arbitrary 𝑡∈[0,𝜏], there exists 𝐵₁ such that any solution of (1) with initial value 𝑢₀∈𝒞=𝐶([−𝜏,0],𝑋^𝛼) satisfies 𝑢𝑡,𝑢_0𝛼<𝐵₁𝑢_0𝛼+1, (48) because 𝑢(𝑡,𝑢₀) can be expressed as follows: 𝑢(𝑡)=𝑒^−𝑡𝐴𝑢₀(0)+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑓(𝑢(𝑠),𝑢(𝑠−𝜏))𝑑𝑠=𝑒^−𝑡𝐴𝑢₀(0)+^𝑡₀𝑒^{−(𝑡−𝑠)𝐴}𝑓𝑢(𝑠),𝑢₀(𝑠−𝜏)𝑑𝑠. (49) By (47) and (15), we can obtain ‖𝑢(𝑡)‖_𝛼=𝐴^𝛼𝑢(𝑡)≤𝑒^−𝑡𝐴𝐴^𝛼𝑢₀(0)+^𝑡₀𝐴^𝛼𝑒^{−(𝑡−𝑠)𝐴}⋅𝑀‖𝑢(𝑠)‖_𝛼+𝑢₀(𝑠−𝜏)_𝛼+1𝑑𝑠≤𝐶₁𝑢_0𝛼+𝑀𝐶_𝛼^𝑡₀(𝑡−𝑠)^−𝛼‖𝑢(𝑠)‖_𝛼𝑑𝑠+𝑀𝐶_𝛼𝑢_0𝛼^𝑡₀(𝑡−𝑠)^−𝛼𝑢𝑑𝑠+𝑀_0𝛼^𝑡₀(𝑡−𝑠)^−𝛼𝑑𝑠≤𝐶₂𝑢_0𝛼+𝑀𝐶_𝛼^𝑡₀(𝑡−𝑠)^−𝛼𝑢₀(𝑠)_𝛼𝑑𝑠. (50) According to the Gronwall inequality, one easily sees that 𝑢𝑡,𝑢_0𝛼≤𝐶₃𝑢_0𝛼<𝐵₁𝑢_0𝛼+1. (51)
Now, we choose and fix 𝐵₂ and 𝜏₀ with 𝐵₂>3(𝑏𝑑+𝑎𝑑+𝑐𝑑+𝑏𝜎+𝑐)𝑑,𝜏^𝛿₀<𝑑96𝑏𝑀₀𝐿_𝑉𝐿. (52) Let 𝑣(𝑡)=𝑢(𝑡,𝑢₀). We will show that ‖𝑣(𝑡)‖_𝛼<𝐵₂𝑣_𝜏𝛼+1,∀𝑡≥𝜏, (53) where ‖𝑣_𝜏‖_𝛼=max_[0,𝜏]‖𝑣(𝑠)‖_𝛼.
We argue by contradiction and suppose that for some solution 𝑣(𝑡)=𝑢(𝑡,𝑢₀) of (1), it holds that 𝑣(𝑡₁)_𝛼≥𝐵₂𝑣_𝜏𝛼+1 (54) for some 𝑡₁>𝜏. Observing that 𝐵₂>1, we deduce that there exists a 𝑡₀>𝜏 such that ‖𝑣(𝑡)‖_𝛼<𝐵₂𝑣_𝜏𝛼+1,for𝑡∈𝜏,𝑡₀,𝑣𝑡_0𝛼=𝐵₂𝑣_𝜏𝛼+1. (55)(56)
Thanks to Theorem 4, there is a Lyapunov function 𝑉 satisfying (18)–(20). By Lemma 2, we find that 𝑑⁺𝑉(𝑣(𝑡𝑑𝑡))=limsup_ℎ→0+1ℎ𝑉𝑣(𝑡)(𝑣(𝑡)((𝑡)(𝑣(𝑡+ℎ𝑓,𝑣𝑡−𝜏))−𝐴𝑣−𝑉))≤limsup_ℎ→0+1ℎ[𝑉(𝑣(𝑡)+ℎ𝐹(𝑣(𝑡))−𝐴𝑣(𝑡))−𝑉(𝑣(𝑡))]+limsup_ℎ→0+1ℎ𝑉𝑣(𝑡)(𝑣(𝑡)((𝑣(𝑡)(𝑣(𝑡+ℎ𝑓,𝑣𝑡−𝜏))−𝑉+ℎ𝐹))=𝐷⁺_{𝑔(𝑣(𝑡))}𝑉(𝑣(𝑡))+limsup_ℎ→0+1ℎ𝑉𝑣(𝑡)(𝑣(𝑡)((𝑣(𝑡)(𝑣(𝑡.+ℎ𝑓,𝑣𝑡−𝜏))−𝑉+ℎ𝐹)) (57) By (47) and (55), we see that for 𝑡∈[𝜏,𝑡₀]‖𝐹(𝑣(𝑡))‖,𝑓(𝑣(𝑡),𝑣(𝑡−𝜏))≤𝑀2𝐵₂𝑣_𝜏𝛼+1+1=𝑅. (58) Denote by 𝐿_ℎ the Lipschitz constant of 𝑉 on ℬ(𝑣(𝑡),ℎ𝑅). Then, we infer from (20) that limsup_ℎ→0+𝐿_ℎ≤𝐿_𝑉2‖𝑣(𝑡)‖_𝛼+1. (59) At the same time, from Lemma 3.3.2 of [18], we can show that 𝑣(𝑡) is locally Hölder. That is to say, ‖𝑣(𝑡−𝜏)−𝑣(𝑡)‖_𝛼≤𝑀₀𝜏^𝛿,𝛿∈(0,1−𝛼). (60) Therefore, on [𝜏,𝑡₀] we have that 𝑑⁺𝑉(𝑣(𝑡𝑑𝑡))≤𝐷⁺_{𝑔(𝑣(𝑡))}𝑉(𝑣(𝑡))+limsup_ℎ→0+𝐿_ℎ𝑓(𝑣(𝑡),𝑣(𝑡−𝜏))−𝑓(𝑣(𝑡),𝑣(𝑡))≤𝐷⁺_{𝑔(𝑣(𝑡))}𝑉(𝑣(𝑡))+limsup_ℎ→0+𝐿_ℎ𝐿‖𝑣(𝑡−𝜏)−𝑣(𝑡)‖_𝛼≤−𝑑‖𝑣(𝑡)‖²_𝛼+𝜎+𝑀₀𝜏^𝛿𝐿𝐿_𝑉2‖𝑣(𝑡)‖_𝛼‖𝑣(𝑡+1≤−𝑑)‖²_𝛼+𝜎+𝑀₀𝜏^𝛿𝐿𝐿_𝑉2𝐵₂𝑣_𝜏𝛼+1+1, (61) because 2𝐵₂𝑣_𝜏𝛼≤+1+12𝐵₂(𝑣_𝜏𝛼+1)+1²≤2𝐵₂𝑣_𝜏𝛼+1+2𝐵₂²=4𝐵²₂𝑣_𝜏𝛼+2²≤16𝐵²₂𝑣_𝜏𝛼+1². (62) According to (18) and (61), we find that 𝑑⁺𝑉(𝑣(𝑡𝑑𝑑𝑡))≤−𝑏𝑉(𝑣(𝑡))+𝑑𝑐𝑏+𝜎+32𝑀₀𝜏^𝛿𝐿𝐿_𝑉𝐵²₂𝑣_𝜏𝛼+1²,∀𝑡∈𝜏,𝑡₀. (63) If we denote that 𝜆₁=𝑑𝑏,𝜎₁=𝑐𝑑𝑏,𝐶𝑣_𝜏𝛼=32𝑀₀𝜏^𝛿𝐿𝐿_𝑉𝐵²₂𝑣_𝜏𝛼+1², (64) then by the Gronwall inequality 𝑉(𝑣(𝑡))≤𝑉(𝑣(𝜏))𝑒^{−𝜆1(𝑡−𝜏)}+1𝜆₁𝜎₁+𝐶𝑣_𝜏𝛼1−𝑒^{−𝜆1(𝑡−𝜏)}(𝑣(𝜏+1≤𝑉))𝜆₁𝜎₁+𝐶𝑣_𝜏𝛼. (65) Utilizing (18) again, we conclude that for 𝑡∈[𝜏,𝑡₀], ‖𝑣(𝑡)‖²_𝛼≤𝑏𝑣_𝜏²_𝛼+𝜎𝑎+𝑐+¹𝜆₁+32𝜆₁𝑀₀𝜏^𝛿𝐿𝐿_𝑉𝐵²₂𝑣_𝜏𝛼+1². (66) By the choice of 𝐵₁ and 𝜏₀, one easily checks that 1𝑏<3𝐵²₂𝜎,𝑎+𝑐+¹𝜆₁<13𝐵²₂,32𝜆₁𝑀₀𝜏^𝛿𝐿𝐿_𝑉<13. (67) Hence, in particular, for 𝑡=𝑡₀, we find that 𝑣𝑡₀²_𝛼<𝐵²₂𝑣_𝜏𝛼+1². (68) This contradicts (56).
Now, the conclusion of the theorem follows immediately from (48) and (53). And the proof is complete.

Theorem 6. Assume that 𝑓 is globally Lipschitz and the system (2) without delay is exponentially dissipative. Then, the system (1) with time delay is also exponentially dissipative.

Proof. Let 𝑢₀∈𝒞 and 𝑣(𝑡)=𝑢(𝑡,𝑢₀) be the solution of (1). According to Lemma 5, ‖𝑣(𝑡)‖_𝛼≤𝐵₁(‖𝑣_𝜏‖_𝛼+1) for all 𝑡≥𝜏, repeating the same argument as in (65), one easily sees that the first inequality in (65) remains valid for all 𝑡≥𝜏. Furthermore, making use of (18), we deduce that ‖𝑣(𝑡)‖²_𝛼(𝑣(𝑡(𝑣(𝜏𝑒−𝑎≤𝑉))≤𝑉))^{−𝜆1(𝑡−𝜏)}+1𝜆₁𝜎₁+𝐶𝑣_𝜏𝛼≤𝑏𝑣_𝜏²_𝛼+𝑐𝑒^{−𝜆1(𝑡−𝜏)}+1𝜆₁𝜎₁+𝐶𝑣_𝜏𝛼. (69) Frequently, ‖𝑣(𝑡)‖²_𝛼≤𝑏𝑣_𝜏²_𝛼+𝑐𝑒^{−𝜆1(𝑡−𝜏)}1+𝑎+𝜆₁𝜎₁+𝐶𝑣_𝜏𝛼,∀𝑡≥𝜏. (70) By (46), it can be easily seen that ‖𝑣(𝑡)‖²_𝛼≤𝐶₁𝑢_0𝛼+1²𝑒^−𝜆1(𝑡−𝜏)+𝜏^𝛿𝐶₂𝑢_0𝛼+1²+𝐶₃≤𝐶₁𝑒^𝜆1𝜏0𝑢_0𝛼+1²𝑒^−𝜆1𝑡+𝜏^𝛿𝐶₂𝑢_0𝛼+1²+𝐶₃,∀𝑡≥𝜏, (71) where 𝐶_𝑖 (𝑖=1,2,3) are appropriate positive constants independent of 𝜏 and 𝑢₀. For 𝑡∈[0,𝜏], we have by (46) that ‖𝑣(𝑡)‖²_𝛼≤𝐵²₀𝑒^𝜆1𝜏0𝑢_0𝛼+1²𝑒^−𝜆1𝑡. (72) Therefore, taking 𝐶^′₁=(𝐶₁+𝐵²₀)𝑒^𝜆1𝜏0, one concludes that ‖𝑣(𝑡)‖²_𝛼≤𝐶^′₁𝑢_0𝛼+1²𝑒^−𝜆1𝑡+𝜏^𝛿𝐶₂𝑢_0𝛼+1²+𝐶₃,∀𝑡≥0. (73)
Now, we fix a 𝑇>0 and 𝜏≤𝜏₀ independent of 𝑢₀ such that 𝐶^′₁𝑒^−𝜆1𝑇<18,𝜏^𝛿𝐶₂<18. (74) So, ‖𝑣(𝑡)‖²_𝛼≤14𝑢_0𝛼+1²+𝐶₃≤𝑢_0𝛼2+12+𝐶₃²,∀𝑡≥𝑇. (75) Setting 𝐶₀=1/2+𝐶₃, we find that ‖𝑣(𝑡)‖_𝛼≤12𝑢_0𝛼+𝐶₀,∀𝑡≥𝑇,𝑢₀∈𝒞. (76)
Next, we will use mathematical induction to prove that ‖𝑣(𝑡)‖_𝛼≤12^𝑘𝑢_0𝛼+𝐶₀11+21+⋯+2^𝑘−1,∀𝑡≥𝑇_𝑘,𝑢₀∈𝒞, (77) where 𝑇_𝑘=𝑘(𝑇+1).
Indeed, let 𝑢₀∈𝒞. If 𝑘=1, then (77) clearly holds true. Suppose that (77) holds for 𝑘=𝑚; that is, ‖𝑣(𝑡)‖_𝛼≤12^𝑚𝑢_0𝛼+𝐶₀11+21+⋯+2^𝑚−1,∀𝑡≥𝑇_𝑚,𝑢₀∈𝒞. (78) Then, in particular, ‖𝑣(𝑡)‖_𝛼≤12^𝑚𝑢_0𝛼+𝐶₀11+21+⋯+2^𝑚−1,∀𝑡∈𝑇_𝑚,𝑇_𝑚+𝜏. (79) From (76), we know that 𝑣𝑡+𝑇_𝑚+𝜏_𝛼≤12max_{[]𝑡∈−𝜏,0}𝑣𝑡+𝑇_𝑚+𝜏_𝛼+𝐶₀,∀𝑡≥𝑇. (80) If we consider 𝑡+𝑇_𝑚+𝜏 as 𝑡, then the above can be rewritten as ‖𝑣(𝑡)‖_𝛼≤12max_{𝑇𝑡∈𝑚,𝑇𝑚+𝜏}‖𝑣(𝑡)‖_𝛼+𝐶₀,∀𝑡≥𝑇_𝑚+𝜏+𝑇. (81) From (79), ‖𝑣(𝑡)‖_𝛼≤1212^𝑚𝑢_0𝛼+𝐶₀11+21+⋯+2^𝑚−1+𝜌₀=12^𝑚+1𝑢_0𝛼+𝐶₀11+21+⋯+2^𝑚,∀𝑡≥𝑇_𝑚+𝜏+𝑇. (82) Choosing 𝜏≤1, we conclude that ‖𝑣(𝑡)‖_𝛼≤12^𝑚+1𝑢_0𝛼+𝐶₀11+21+⋯+2^𝑚,∀𝑡≥𝑇_𝑚+1+𝑇=𝑇_𝑚+1. (83) Thus, we see that (77) holds for 𝑚+1.
By (77), we know that ‖𝑣(𝑡)‖_𝛼≤12^𝑘𝑢_0𝛼+2𝐶₀,∀𝑡∈𝑇_𝑘,𝑇_𝑘+1,𝑢₀∈𝒞,𝑘=1,2,…. (84) Furthermore, by Lemma 5, we see that ‖𝑣(𝑡)‖_𝛼≤𝐵₀2^𝑘𝑢_0𝛼+𝜌₀,∀𝑡∈𝑇_𝑘,𝑇_𝑘+1,𝑢₀∈𝒞,𝑘=0,1,2,…, (85) where 𝜌₀=2𝐶₀+1.
We observe that 𝑡/(𝑇+1)∈[𝑘,𝑘+1] when 𝑡∈[𝑇_𝑘,𝑇_𝑘+1], so we infer from (85) that ‖𝑣(𝑡)‖_𝛼≤𝐵₀2^𝑘𝑢_0𝛼+𝜌₀=2𝐵₀2^𝑘+1𝑢_0𝛼+𝜌₀≤2𝐵₀2^−𝛼𝑡𝑢_0𝛼+𝜌₀,∀𝑡∈𝑇_𝑘,𝑇_𝑘+1,𝑢₀∈𝒞,𝑘=0,1,2,…, (86) where 𝛼=1/(𝑇+1). So, we easily see that ‖𝑣(𝑡)‖_𝛼≤2𝐵₀2^−𝛼𝑡𝑢_0𝛼+𝜌₀,∀𝑡≥0,𝑢₀∈𝒞. (87) This completes the proof of the theorem.