# Generalized Halanay Inequalities with Applications to Generalized Exponential Stability and Boundedness of Time-Delay Systems

Authors: Dehao Ruan; Yue Liu
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, a class of generalized Halanay inequalities is investigated. Under some assumptions, the generalized exponential stability and boundedness are discussed by means of integral inequalities. We do not require the boundedness of the time-varying delays and coefficients. In addition, our results improve some previous works. At last, three examples and simulations are given to illustrate the effectiveness of the main results.


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## Body

## 1. Introduction

In recent years, dynamical systems have come to play a more and more important role in many areas such as physics, population dynamics, electrical engineering, medicine biology, ecology, economics, and other areas of science and engineering. The main reason for this is that it allows us to model some kinds of natural scientific, social scientific, and engineering problems appropriately. Because of their great applications, dynamical systems have been developed very fast (see, e.g., [1–21]).As particular interest, the stability and boundedness are always one of the most important issues in the theory of dynamical systems. In many dynamical systems such as telecommunication systems and population dynamics systems, time delays are not avoidable; it is well known that the delay may cause oscillation and instability in systems. It is important to consider the influence of the time delays in the investigation of these systems. Therefore, stability and boundedness of dynamical systems with time delays have been extensively studied by many authors (see, for instance, [4–6, 8–15, 17, 18, 20, 21]).In the stability and boundedness analysis of dynamical systems, one powerful technique is the Lyapunov’s direct method. The key of Lyapunov’s direct method is to construct a suitable Lyapunov function or functional. However, finding a suitable Lyapunov function or functional is not an easy case. On the contrary, Halanay inequalities and their generalizations can be also used in such analysis [5, 8, 9, 14, 15, 20, 21]. In [5, 8, 20], the authors studied the following generalized Halanay inequality:(1)D+ut≤γt−αtut+βtsupt−τt≤s≤tus,t≥t0.In [20], the authors studied the exponential stability and boundedness of the above inequality and gave application to study dissipativity of the Volterra functional differential equations; in [8], the authors used the above inequality to consider the exponential stability and boundedness of neural networks with unbounded delays; in [5], the authors investigated the generalized exponential stability and boundedness of the above inequality and gave application to a class of nonlinear nonautonomous time-delay systems. However, the existing works are not general enough or require that γt/αt is bounded. Motivated by the aforementioned discussions, we also study the above inequality and derive new generalized exponential stability and boundedness conditions; the obtained results also improve and generalize some existing works.The contents of this work are organized as follows. In Section2, we introduce some new generalized Halanay inequalities. In Section 3, there are examples and simulations given to illustrate the effectiveness of the theoretical results. At last, in Section 4, our conclusion is presented.


## 2. Main Results

In this section, we consider the following generalized Halanay inequality:(2)D+ut≤γt−αtut+βtsupt−τt≤s≤tus,t≥t0,ut=ϕt∈BC−∞,t0,ℝ0,t≤t0,where D+ut is the upper-right Dini derivative, ut≥0, αt≥0, βt≥0 and γt≥0 for t∈t0,+∞, τt≥0 is a general (bounded or unbounded) time-delay function, and BC−∞,t0,ℝ0 is the Banach space of all non-negative bounded continuous functions defined on −∞,t0 equipped with the norm ϕ∞=sups≤t0ϕs. In order to set the problem, we complementally define(3)α∗t=αt,t≥t0,0,t<t0.Theorem 1.
Assume the following assumptions hold(A.1)
limt⟶+∞t−τt=+∞(A.2)
limt⟶+∞∫t0tα∗sds=+∞(A.3)
supt≥t0∫t−τttα∗sds≔N<+∞(A.4)
supt≥t0βt/α∗t≔δ<1(A.5)
There exists a non-negative numberM such that ∫t0te−∫stα∗uduγsds≤M, t≥t0; then, there exists a constant λ∗∈0,1such that(4)ut≤ϕ∞e−λ∗∫t0tα∗sds+M1−δ,t∈ℝ.Proof.
Due to assumption (A.3), consider the following function:(5)Fλ=λ+δeλN−1,λ∈0,+∞.
Note thatFλ is continuous on 0,+∞, F0=δ−1<0, limλ⟶+∞Fλ=+∞, and F′λ>0 for λ∈0,+∞, so the equation Fλ=0 has a unique solution λ∗∈0,+∞. Thus,(6)λ∗α∗t+δα∗teλ∗N−α∗t=α∗tFλ∗=0,t≥t0.
From (A.4) and (6), we get(7)λ∗α∗t+βteλ∗∫t−τttα∗sds−α∗t≤α∗tFλ∗=0,t≥t0.
Then, we can obtain(8)λ∗α∗t+βteλ∗∫t−τttα∗sds≤α∗t,t≥t0.
By the formula for the variation of parameters, we have(9)ut≤ut0e−∫t0tα∗sds+∫t0te−∫stα∗uduγsds+∫t0te−∫stα∗uduβssups−τs≤v≤suvds.
Now, we prove (4). Obviously, (4) holds for t≤t0. Assume there exist t1>t0 and ϵ1>0 such that(10)ut<ϕ∞e−λ∗∫t0tα∗sds+M1−δ+ϵ1,t<t1,(11)ut1=ϕ∞e−λ∗∫t0t1α∗sds+M1−δ+ϵ1.
Thus, from (8) and (9), we have(12)ut1≤ut0e−∫t0t1α∗sds+M+∫t0t1e−∫st1α∗uduβsϕ∞e−λ∗∫t0s−τsα∗sds+M1−δ+ϵ1ds≤ut0e−∫t0t1α∗sds+ϕ∞e−∫t0t1α∗sds∫t0t1e1−λ∗∫t0sα∗uduβseλ∗∫s−τssα∗ududs+M+δM1−δ+δϵ1≤ut0e−∫t0t1α∗sds+ϕ∞e−∫t0t1α∗sds∫t0t1e1−λ∗∫t0sα∗udu1−λ∗α∗sds+M1−δ+δϵ1≤ϕ∞e−λ∗∫t0t1α∗sds+M1−δ+δϵ1,which is a contradiction. This shows that(13)ut≤ϕ∞e−λ∗∫t0tα∗sds+M1−δ,t∈ℝ.
The proof is completed.Remark 1.
Especially whenγt≡0 and αs is replaced by 1, 1/s−t0+1 and 1/e+s−t0lne+s−t0 lead to the usual stability with exponential, polynomial, and logarithmic decays, respectively.Theorem 2.
Assume assumptions (A.1)–(A.4) hold and the following assumption holds.(A.6)
supt≥t0γt/α∗t≔γ<+∞. Then,(14)ut≤ϕ−γ1−δ+e−λ∗∫t0tα∗sds+γ1−δ,t∈ℝ,where λ∗ was introduced previously and ϕ−γ/1−δ+=maxϕ−γ/1−δ,0.Proof.
Whenϕ≤γ/1−δ, then us≤γ/1−δ for t≥t0. Assume there exist t2>t0 and ϵ2>0 such that(15)ut<γ1−δ+ϵ2,t<t2,(16)ut2=γ1−δ+ϵ2.
Thus, from (8) and (9), we have(17)ut2≤ut0e−∫t0t2α∗udu+γ∫t0t2e−∫st2α∗uduα∗sds+∫t0t2e−∫st2α∗uduβsγ1−δ+ϵ2ds≤γ1−δe−∫t0t2α∗udu+γ−γe−∫t0t2α∗udu+δγ1−δ+δϵ2−δγ1−δ+δϵ2e−∫t0t2α∗udu≤γ1−δ+δϵ2,which is a contradiction. Now, assume ϕ>γ/1−δ. Then,(18)ut≤ϕ−γ1−δe−λ∗∫t0tα∗sds+γ1−δ,t∈ℝ.
Assume there existt2∗>t0 and ϵ2∗>0 such that(19)ut<ϕ−γ1−δe−λ∗∫t0tα∗sds+γ1−δ+ϵ2∗,t<t2∗,(20)ut2∗=ϕ−γ1−δe−λ∗∫t0t2∗α∗sds+γ1−δ+ϵ2∗.
Thus, from (8) and (9), we get(21)ut2∗≤ut0e−∫t0t2∗α∗udu+γ∫t0t2∗e−∫st2∗α∗uduα∗sds+∫t0t2∗e−∫st2∗α∗uduβsϕ∞−γ1−δe−λ∗∫t0s−τsα∗sds+γ1−δ+ϵ2∗ds≤ut0e−∫t0t2∗α∗udu+ϕ∞−γ1−δe−∫t0t2∗α∗udu∫t0t2∗e1−λ∗∫​t0sα∗uduβse∫s−τssα∗ududs+γ−γe−∫t0t2∗α∗udu+δγ1−δ+δϵ2∗−δγ1−δ+δϵ2∗e−∫t0t2α∗udu≤ϕ−γ1−δe−∫t0t2∗α∗udu+ϕ∞−γ1−δe−λ∗∫t0t2∗α∗udu−e−∫t0t2∗α∗udu+γ1−δ+δϵ2∗≤ϕ∞−γ1−δe−λ∗∫t0t2∗α∗udu+γ1−δ+δϵ2∗,which is a contradiction, and then,(22)ut≤ϕ−γ1−δ+e−λ∗∫t0tα∗sds+γ1−δ,t∈ℝ.
The proof is completed.Remark 2.
Theorem 2 requires pointwise condition. Theorem 1 asks for averaging nature (see conditions (A.5) and (A.6)). Suppose (A.6) holds, one can deduce thatγ≤M; under this circumstance, we can say Theorem 2 is sharper than Theorem 1. However, when αt=t−t0 and γt=1 for t≥t0, then (A.6) is invalid, but (A.5) still holds. This means that Theorem 1 can solve the case which Theorem 2 cannot. This is a balance, we introduce some results about generalized exponential stability and boundedness of time-delay systems, which makes the conditions be more feasible.Theorem 3.
Assume assumptions (A.1)–(A.4) hold and the following assumption holds.(A.7)
There exists a pair of constantsa≥0 and 0≤b<λ∗ such that(23)∫t0te−∫stα∗uduγsds≤ae−b∫t0tα∗sds,t≥t0.
Then,(24)ut≤ϕ∞e−λ∗∫t0tα∗sds+a1−δ/1−b1−λ∗/δb/λ∗e−b∫t0tα∗sds,t∈ℝ,where λ∗ was introduced previously.Proof.
Now we prove (24). Assume there exist t3>t0 and ϵ3>0 such that(25)ut<ϕ∞e−λ∗∫t0tα∗sds+a1−δ/1−b1−λ∗/δb/λ∗e−b∫t0tα∗sds+ϵ3,t<t3,(26)ut3=ϕ∞e−λ∗∫t0t3α∗sds+a1−δ/1−b1−λ∗/δb/λ∗e−b∫t0t3α∗sds+ϵ3.
Thus, from (8) and (9),(27)ut3≤ut0e−∫t0t3α∗sds+ae−b∫t0t3α∗sds+∫t0t3e−∫st3α∗uduβsϕ∞e−λ∗∫t0s−τsα∗udu+a1−δ/1−b1−λ∗/δb/λ∗e−b∫t0s−τsα∗udu+ϵ3ds≤ϕ∞e−λ∗∫t0t3α∗sds+ae−b∫t0t3α∗sds+a1−δ/1−b1−λ∗/δb/λ∗e−∫t0t3α∗udu∫t0t3e∫t0s1−bα∗uduβseb∫s−τssα∗ududs+δϵ3≤ϕ∞e−λ∗∫t0t3α∗sds+ae−b∫t0t3α∗sds+aδ/1−b1−λ∗/δb/λ∗1−δ/1−b1−λ∗/δb/λ∗e−b∫t0t3α∗udu−e−∫t0t3α∗udu+δϵ3≤ϕ∞e−λ∗∫t0t3α∗sds+a1−δ/1−b1−λ∗/δb/λ∗e−b∫t0t3α∗sds+δϵ3,we get which is a contradiction. This shows that(28)ut≤ϕ∞e−λ∗∫t0tα∗sds+a1−δ/1−b1−λ∗/δb/λ∗e−b∫t0tα∗sds.
The proof is completed.Remark 3.
It is obvious that Theorem 1 is a special case of Theorem 3 forb=0.Remark 4.
Theorems 1–3 remain true if we replace the first part of (2) by (9).


## 3. Examples and Simulations

In this section, there are examples and simulations given to illustrate our main results.Example 1.
(29)
d
x
t
=
t
+
1
−
2
t
x
t
+
0.4
e
−
1.6
t
x
t
−
1
1
+
t
d
t
,
t
≥
0
,
x
t
=
10
,
t
∈
−
1,0
.
Note that(30)αt=2t,βt=0.4e−1.6t,γt=t+1,δ=0.2e−1.6,supt≥0∫t−1/1+ttα∗sds=2.
Then, we haveλ∗=0.8 and(31)∫0te−t2+s2s+1ds≤0.9,t≥0.
Then, all the assumptions of theorem 1 are satisfied, so we have(32)xt≤10e−0.8t2+910−2e−1.6,t≥0.
In Figure1, yt=10e−0.8t2+9/10−2e−1.6.Figure 1
Example 1.Remark 5.
Theorem 2.4 in [20] and Theorem 1 in [8] require that −αt+βt≤−σ<0 for t≥0, and γt is a positive bounded function. Theorem 3.2 in [5] asks for supt≥0γt/αt<+∞. It is obvious that in Example 1, α0+β0=0, limt⟶+∞γt=+∞, and supt≥0γt/αt=+∞; thus, Theorem 2.4 in [20], Theorem 1 in [8], and Theorem 3.2 in [5] are invalid for this example.Example 2.
(33)
d
x
t
=
1
1
+
t
−
1
1
+
t
x
t
+
0.5
e
−
0.5
6
−
2
1
4
+
5
t
x
t
−
t
d
t
,
t
≥
1
,
x
t
=
10
,
t
∈
0,1
.
Note that(34)αt=11+t,βt=e−0.56−214+5t,γt=11+t,supt≥1∫t−ttα∗sds=6−2.
Then, we haveλ∗=0.5 and γ=1. Then, all the assumptions of Theorem 2 are satisfied, so we have(35)xt≤10−11−0.5e−0.56−2e−1+t−2+11−0.5e−0.56−2,t≥1.
In Figure2, yt=10−1/1−0.5e−0.56−2e−1+t−2+1/1−0.5e−0.56−2.Figure 2
Example 2.Remark 6.
It should point out that the delay functionτt=t is unbounded.Example 3.
(36)
d
x
t
=
e
−
0.3
t
−
x
t
+
0.4
e
−
0.6
x
t
−
1
d
t
,
t
≥
0
,
x
t
=
1
,
t
∈
−
1,0
.
Note that(37)αt=1,βt=0.4e−0.6,γt=e−0.3t,δ=0.4e−0.6,supt≥0∫t−1tα∗sds=1.
Then, we haveλ∗=0.6 and(38)supt≥0∫0te−t−se−0.3sds≤107e−0.3t.
Then, all the assumptions of Theorem 3 are satisfied, so we have(39)xt≤10e−0.6t+107−4e−0.3e−0.3t,t≥0.
In Figure3, yt=10e−0.6t+10/7−4e−0.3e−0.3t.Figure 3
Example 3.Remark 7.
Obviously, system (36) is exponential stability; however, the existing works [5, 8, 20] just can obtain the boundedness of system (36). In this sense, this paper is sharper than those of [5, 8, 20].


## 4. Conclusion

In this paper, by using integral inequalities, we obtain some sufficient conditions ensuring the generalized exponential stability and boundedness of a class of differential inequalities. Then, the obtained results can be used to investigate the generalized exponential stability and boundedness of some dynamical systems. In our next paper, we will investigate some inequalities which can be applied to investigate some neutral functional differential equations.



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