On Stability of Linear Delay Differential Equations under Perron's Condition

Abstract

The stability of the zero solution of a system of first-order linear functional differential equations with nonconstant delay is considered. Sufficient conditions for stability, uniform stability, asymptotic stability, and uniform asymptotic stability are established.

1. Introduction

We begin with a classical result for the linear system 𝑥′=𝐴(𝑡)𝑥,(L1) where 𝐴 is an 𝑛×𝑛 matrix function defined and continuous on [0,∞). By 𝐶_𝐵[0,∞), we will denote the set of bounded functions defined and continuous on [0,∞) and by |⋅| the Euclidean norm.

In 1930, Perron first formulated the following definition being named after him.

Definition 1.1 (see [1]). System (L1) is said to satisfy Perron's condition (P1) if, for any given vector function 𝑓∈𝐶_𝐵[0,∞), the solution 𝑥(𝑡) of 𝑥^′=𝐴(𝑡)𝑥+𝑓(𝑡),𝑥(0)=0(N1) is bounded.

The following theorem by Bellman [2] is well known.

Theorem 1.2 (see [2]). If (P1) holds and |𝐴(𝑡)|≤𝑀₁ for some positive number 𝑀₁, then the zero solution of (L1) is uniformly asymptotically stable.

The proof is accomplished by making use of the basic properties of a fundamental matrix, the Banach-Steinhaus theorem, and the adjoint system 𝑥′=−𝐴^𝑇(𝑡)𝑥,(1.1) where 𝐴^𝑇 denotes the transpose of 𝐴.

It is shown by an example in [3] that Theorem 1.2 may not be valid if the function 𝑓 appearing in (N1) is replaced by a constant vector. However, such a theorem is later obtained in [4] under a Perron-like condition.

Theorem 1.2 is extended by Halanay [5] to linear delay systems of the form 𝑥^′(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥(𝑡−𝜏),(L2) where 𝐴, 𝐵 are 𝑛×𝑛 matrix functions defined and continuous on [0,∞) and 𝜏 is a positive real number.

Definition 1.3. System (L2) is said to satisfy Perron's condition (P2) if for any given vector function 𝑓∈𝐶_𝐵[0,∞), the solution 𝑥(𝑡) of 𝑥^′(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥(𝑡−𝜏)+𝑓(𝑡)(N2) satisfying 𝑥(𝑡)=0, 𝑡≤0, is bounded.

Theorem 1.4 (see [5]). If (P2) holds, |𝐴(𝑡)|≤𝑀₁, and |𝐵(𝑡)|≤𝑀₂ for some positive numbers 𝑀₁ and 𝑀₂, then the zero solution of (L2) is uniformly asymptotically stable.

The method used to prove Theorem 1.4 is similar to Bellman's except that the adjoint system 𝑦^′(𝑡)=−𝐴^𝑇(𝑡)𝑦(𝑡)−𝐵^𝑇(𝑡+𝜏)𝑦(𝑡+𝜏)(1.2) is not constructed with respect to an inner product but the functional𝐹𝑥,𝑦(𝑡)=_𝑡^𝑡+𝜏𝑦^𝑇(𝑠)𝐵(𝑠)𝑥(𝑠−𝜏)𝑑𝑠+𝑥^𝑇(𝑡)𝑦(𝑡).(1.3) For some extensions to impulsive differential equations, we refer the reader in particular to [6, 7].

In this paper, we consider the more general linear delay system 𝑥^′(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥𝑔(𝑡),(1.4) where 𝐴 and 𝐵 are 𝑛×𝑛 matrix functions defined and continuous on [0,∞) and 𝑔 is a continuously differentiable increasing function defined on [0,∞) satisfying 𝑔(𝑡)<𝑡 and 𝑔′(𝑡)≤1. We set ℎ∶=𝑔⁻¹. Obviously, ℎ∈𝐶¹[0,∞) and increases on [0,∞) and ℎ(𝑡)>𝑡.

Perron's condition takes the following form.

Definition 1.5. System (1.4) is said to satisfy Perron's condition (P) if, for any given vector function 𝑓∈𝐶_𝐵[0,∞), the solution 𝑥(𝑡) of 𝑥^′(𝑡)=𝐴(𝑡)𝑥(𝑡)+𝐵(𝑡)𝑥𝑔(𝑡)+𝑓(𝑡)(1.5) satisfying 𝑥(𝑡)=0, 𝑡≤0 is bounded.

A natural question is whether the zero solution of (1.4) is uniformly asymptotically stable under Perron's condition (P). It turns out that the answer depends on the delay function 𝑔.

The paper is organized as follows. In Section 2, we only state our results; the proofs are included in Section 5. We define an adjoint system and give a variation of parameters formula in Section 3 to be needed in proving the main results. Section 4 contains also some lemmas concerning Perron's condition and a relation useful for changing the order of integration.

2. Stability Theorems

The conclusion obtained by Bellman and Halanay for systems (L1) and (L2), respectively, is quite strong. We are only able to prove the stability of the zero solution for more general equation (1.4) under Perron's condition. To get uniform stability or asymptotic stability or uniform asymptotic stability, we impose restrictions on the delay function.

For our purpose, we denote ℎ_∗(𝑡)(𝑡)𝑔∶=ℎ−𝑡,𝑡≥0,_∗𝑡,𝑡₀∶=sup_{ℎ𝑡𝑟∈0,𝑡}𝑟−𝑔(𝑟),𝑡,𝑡₀≥0.(2.1)

Theorem 2.1. Let (𝑃) hold. If there are positive numbers 𝑀₁ and 𝑀₂ such that |𝐴(𝑡)|≤𝑀₁,|𝐵(𝑡)|≤𝑀₂∀𝑡≥0,(2.2) then the zero solution of (1.4) is stable.

Theorem 2.2. Let (P) hold. If (2.2) is satisfied and if there exists a positive real number 𝑀₃ such that ℎ_∗(𝑡)≤𝑀₃∀𝑡≥0,(2.3) then the zero solution of (1.4) is uniformly stable.

Theorem 2.3. Let (P) hold. If (2.2) and limsup_𝑡→∞𝑔_∗𝑡,𝑡₀𝑡−𝑡₀=0foreach𝑡₀≥0(2.4) are satisfied, then the zero solution of (1.4) is asymptotically stable.

Theorem 2.4. Let (P) hold. If (2.2), (2.3), and limsup_𝑡→∞𝑔_∗𝑡,𝑡₀𝑡−𝑡₀=0uniformlyfor𝑡₀≥0(2.5) are satisfied, then the zero solution of (1.4) is uniformly asymptotically stable.

Remark 2.5. Note that if 𝑔(𝑡)=𝑡−𝜏, then ℎ(𝑡)=𝑡+𝜏 and hence the conditions (2.3), (2.4), and (2.5) are automatically satisfied. In this case, all theorems become equivalent, that is, the zero solution is uniformly asymptotically stable. Thus, the results obtained by Bellman and Halanay are recovered.

3. Variation of Parameters Formula

To establish a variation of parameters formula to represent the solutions of (1.5), one needs an adjoint system. The following lemma helps to define the adjoint of (1.4).

Lemma 3.1. Let 𝑥(𝑡) be a solution of (1.4). If 𝑦(𝑡) is a solution of 𝑦^′(𝑡)=−𝐴^𝑇(𝑡)𝑦(𝑡)−𝐵^𝑇(ℎ(𝑡))𝑦(ℎ(𝑡))ℎ^′(𝑡),(3.1) then 𝑑𝐹𝑥(𝑡)(𝑡)𝑑𝑡,𝑦=0,(3.2) where 𝐹𝑥,𝑦(𝑡)=_𝑡^ℎ(𝑡)𝑦^𝑇(𝑠)𝐵(𝑠)𝑥𝑔(𝑠)𝑑𝑠+𝑥^𝑇(𝑡)𝑦(𝑡).(3.3)

Proof. Verify directly.

Definition 3.2. The system (3.1) is said to be adjoint to system (1.4).

It is easy to see that the adjoint of system (3.1) is system (1.4); thus the systems are mutually adjoint to each other.

Lemma 3.3. Let 𝑌(𝑡,𝑠) be a matrix solution of (3.1) for 𝑡<𝑠 satisfying 𝑌(𝑠,𝑠)=𝐼 and 𝑌(𝑡,𝑠)=0 for 𝑡>𝑠. Then 𝑥(𝑡) is a solution of (1.5) if and only if 𝑥(𝑡)=𝑌^𝑇(𝑠,𝑡)𝑥(𝑠)+^𝑠_𝑔(𝑠)𝑌^𝑇ℎ𝛽,𝑡𝐵ℎ𝛽𝑥𝛽ℎ^′𝛽𝑑𝛽+^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝑓𝛽𝑑𝛽.(3.4)

Proof. Replacing 𝑡 by 𝛽 in (1.5) and then integrating the resulting equation multiplied by 𝑌^𝑇(𝛽,𝑡) over 𝛽∈[𝑠,𝑡], we have ^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝐴𝛽𝑥𝛽𝑑𝛽+^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝐵𝛽𝑥𝑔𝛽𝑑𝛽+^𝑡_𝑠𝑌^𝑇𝑓𝛽=𝛽,𝑡𝑑𝛽_𝑡𝑠𝑌^𝑇𝛽,𝑡𝑥^′𝛽(𝑡)𝑑𝛽=𝑥−𝑌^𝑇(𝑠,𝑡)𝑥(𝑠)−^𝑡_𝑠𝜕𝑌𝜕𝛽^𝑇𝑥𝛽(𝑡)𝛽,𝑡𝑑𝛽=𝑥−𝑌^𝑇(𝑠,𝑡)𝑥(𝑠)+^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝐴𝛽+𝑌^𝑇ℎ𝛽,𝑡𝐵ℎ𝛽ℎ^′𝛽𝑥𝛽(𝑡)𝑑𝛽=𝑥−𝑌^𝑇(𝑠,𝑡)𝑥(𝑠)+^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝐴𝛽𝑥𝛽𝑑𝛽+_ℎ(𝑠)^ℎ(𝑡)𝑌^𝑇𝛽,𝑡𝐵𝛽𝑥𝑔𝛽𝑑𝛽.(3.5) Comparing both sides and using _𝑡^ℎ(𝑡)𝑌^𝑇𝛽,𝑡𝐵𝛽𝑥𝑔𝛽𝑑𝛽=0,(3.6) which is true in view of 𝑌(𝛽,𝑡)=0 for 𝛽>𝑡, we get 𝑥(𝑡)=𝑌^𝑇(𝑠,𝑡)𝑥(𝑠)−^𝑠_ℎ(𝑠)𝑌^𝑇𝛽,𝑡𝐵𝛽𝑥𝑔𝛽𝑑𝛽+^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝑓𝛽𝑑𝛽(3.7) and hence 𝑥(𝑡)=𝑌^𝑇(𝑠,𝑡)𝑥(𝑠)+^𝑠_𝑔(𝑠)𝑌^𝑇ℎ𝛽,𝑡𝐵ℎ𝛽𝑥𝛽ℎ^′𝛽𝑑𝛽+^𝑡_𝑠𝑌^𝑇𝛽,𝑡𝑓𝛽𝑑𝛽.(3.8)

It is not difficult to see from (3.4) that if 𝑋(𝑡,𝑠) is a matrix solution of (1.4) for 𝑡>𝑠 satisfying 𝑋(𝑠,𝑠)=𝐼 and 𝑋(𝑡,𝑠)=0 for 𝑡<𝑠, then 𝑋(𝑡,𝑠)=𝑌^𝑇(𝑠,𝑡).(3.9) Using this relation in Lemma 3.3 leads to the following variation of parameters formula.

Lemma 3.4. Let 𝑋(𝑡,𝑠) be a matrix solution of (1.4) for 𝑡>𝑠 satisfying 𝑋(𝑠,𝑠)=𝐼 and 𝑋(𝑡,𝑠)=0 for 𝑡<𝑠. Then 𝑥(𝑡) is a solution of (1.5) if and only if 𝑥(𝑡)=𝑋(𝑡,𝑠)𝑥(𝑠)+^𝑠_𝑔(𝑠)𝑋𝑡,ℎ𝛽𝐵ℎ𝛽𝑥𝛽ℎ^′𝛽𝑑𝛽+^𝑡_𝑠𝑋𝑡,𝛽𝑓𝛽𝑑𝛽.(3.10)

4. Auxiliary Results

Lemma 4.1. If (P) holds, then there is a positive number 𝐾₁ such that ^𝑡₀|𝑋(𝑡,𝑠)|𝑑𝑠≤𝐾₁∀𝑡>0.(4.1)

Proof. The proof follows as in [5]. We provide only the steps for the reader's convenience.
Define 𝑆𝑓(𝑡)=^𝑡₀𝑋𝑡,𝛽𝑓𝛽𝑑𝛽,𝑓∈𝐶_𝐵[),𝑆0,∞_𝑘𝑓=^𝑡𝑘₀𝑋𝑡_𝑘,𝛽𝑓𝛽𝑑𝛽,𝑓∈𝐶_𝐵[0,∞),(4.2) for each rational number 𝑡_𝑘, 𝑘∈ℕ.
In view of (P), the family of continuous linear operators {𝑆_𝑘} from 𝐶_𝐵[0,∞) to 𝐶_𝐵[0,∞) is pointwise-bounded. For the space of bounded continuous functions 𝐶_𝐵[0,∞), the usual sup norm ‖⋅‖ is used.
By the Banach-Steinhaus theorem, the family is uniformly bounded. Thus, there is a positive number 𝑀 such that ‖𝑆_𝑘(𝑓)‖≤𝑀‖𝑓‖ for every 𝑓∈𝐶_𝐵[0,∞).
As the rational numbers are dense in the real numbers, for each 𝑡 there is 𝑡_𝑘 such that 𝑡_𝑘→𝑡 as 𝑘→∞ and so ^𝑡₀𝑋𝑡,𝛽𝑓𝛽𝑑𝛽≤𝑀𝑓∀𝑓∈𝐶_𝐵[0,∞).(4.3)
The final step involves choosing a sequence of functions and using a limiting process.

Lemma 4.2. If (2.2) and (4.1) are true, then there is a positive number 𝐾₂ such that |𝑌(𝑠,𝑡)|≤𝐾₂∀0≤s<𝑡.(4.4)

Proof. From (3.1), we have 𝑌(𝑠,𝑡)=𝐼+^𝑡_𝑠𝐴^𝑇𝛽𝑌𝛽,𝑡𝑑𝛽+^𝑡_𝑠𝐵^𝑇ℎ𝛽𝑌ℎ𝛽,𝑡ℎ^′𝛽𝑑𝛽.(4.5) Hence, by using (4.1), we see that for all 0≤𝑠<𝑡, |𝑌(𝑠,𝑡)|≤1+𝑀₁𝐾₁+𝑀₂𝐾₁=∶𝐾₂.(4.6)

Lemma 4.3. Let 𝐺(𝑟,𝑡) be a continuous function satisfying 𝐺(𝑟,𝑡)=0 for 𝑟>𝑡. Then ^𝑡_𝑡0𝑠^ℎ(𝑠)𝐺(𝑟,𝑡)𝑑𝑟𝑑𝑠=^𝑡_ℎ𝑡0𝑟−𝑔(𝑟)𝐺(𝑟,𝑡)𝑑𝑟+^ℎ(𝑡0)_𝑡0𝑟−𝑡₀𝐺(𝑟,𝑡)𝑑𝑟.(4.7)

5. Proofs of Theorems

Let 𝑡₀≥0 be given. For a given continuous vector function 𝜙 defined on [𝑔(𝑡₀),𝑡₀], let 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝜙) denote the solution of (1.4) satisfying 𝑥(𝑡)=𝜙(𝑡),𝑡≤𝑡₀.(5.1) As usual, 𝜙_𝑔=sup_{𝑔𝑡𝑡∈0,𝑡0}𝜙(𝑡).(5.2)

Proof of Theorem 2.1. From Lemma 3.3, we may write 𝑥(𝑡)=𝑌^𝑇𝑡₀,𝑡𝜙𝑡₀+^𝑡0_𝑔𝑡0𝑌^𝑇ℎ𝛽,𝑡𝐵ℎ𝛽𝜙𝛽ℎ^′𝛽𝑑𝛽.(5.3) In view of Lemma 4.2, it follows that |𝑥(𝑡)|≤𝐾₂+ℎ𝑡₀−𝑡₀𝐾₂𝑀₂𝜙_𝑔.(5.4) Hence, the zero solution is stable.

Proof of Theorem 2.2. Using (2.3) in (5.4), we get |𝑥(𝑡)|≤𝐾₃𝜙_𝑔,𝐾₃=𝐾₂+𝐾₂𝑀₂𝑀₃,(5.5) from which the uniform stability follows.

Proof of Theorem 2.3. By Theorem 2.1, the zero solution is stable. We need to show the attractivity property.
From Lemma 3.3, for 𝑠≥𝑡₀, we can write 𝑥𝑡,𝑡₀,𝜙=𝑌^𝑇(𝑠,𝑡)𝑥𝑠,𝑡₀,𝜙+^𝑠_𝑔(𝑠)𝐺ℎ𝛽,𝑡𝑥𝛽,𝑡₀,𝜙ℎ^′𝛽𝑑𝛽,(5.6) where 𝐺(𝑠,𝑡)=𝑌^𝑇(𝑠,𝑡)𝐵(𝑠).(5.7) Integrating with respect to 𝑠 from 𝑡₀ to 𝑡, we have 𝑡−𝑡₀𝑥𝑡,𝑡₀,𝜙=^𝑡_𝑡0𝑌^𝑇(𝑠,𝑡)𝑥𝑠,𝑡₀,𝜙+_𝑠^ℎ(𝑠)𝐺(𝑟,𝑡)𝑥𝑔(𝑟),𝑡₀,𝜙𝑑𝑟𝑑𝑠.(5.8) We change the order of integration by employing Lemma 4.3. After some rearrangements, we obtain 𝑡−𝑡₀𝑥𝑡,𝑡₀,𝜙=^𝑡_𝑡0𝑌^𝑇(𝑠,𝑡)𝑥𝑠,𝑡₀,𝜙𝑑𝑠+^𝑡_ℎ𝑡0𝑠−𝑔(𝑠)𝐺(𝑠,𝑡)𝑥𝑔(𝑠),𝑡₀,𝜙𝑑𝑠+^ℎ(𝑡0)_𝑡0𝑠−𝑡₀𝐺(𝑠,𝑡)𝑥𝑔(𝑠),𝑡₀,𝜙𝑑𝑠.(5.9) It follows that 𝑡−𝑡₀𝑥𝑡,𝑡₀,𝜙≤𝐾₁𝐾₃𝜙_𝑔+𝑔_∗𝑡,𝑡₀𝑀₂𝐾₁𝜙_𝑔+ℎ_∗𝑡₀𝑀₂𝐾₁𝜙_𝑔.(5.10) In view of condition (2.4), we see from (5.10) that lim_𝑡→∞𝑥𝑡,𝑡₀,𝜙=0.(5.11)

Proof of Theorem 2.4. By Theorem 2.2, the zero solution is uniformly stable. From (5.10) and (2.3), we have 𝑡−𝑡₀𝑥𝑡,𝑡₀,𝜙≤𝐾₁𝐾₃𝜙_𝑔+𝑔_∗𝑡,𝑡₀𝑀₂𝐾₁𝜙_𝑔+𝑀₃𝑀₂𝐾₁𝜙_𝑔.(5.12) Using condition (2.4) in the above inequality, we see that the zero solution is uniformly asymptotically stable as 𝑡→∞.

Acknowledgments

This research was supported by Grant P201/11/0768 of the Czech Grant Agency (Prague), by the Council of Czech Government MSM 0021630503 and MSM 00216 30519, and by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology.