Advances in Applied Mathematics
Vol. 08  No. 11 ( 2019 ), Article ID: 33111 , 7 pages
10.12677/AAM.2019.811214

The Stabilities for a Class of Nonlinear Differential Systems with Time-Delay

Ran Huo1, Xiaoli Wang2*

1College of Science, Inner Mongolia Agriculture University, Huhhot Inner Mongolia

2College of Statistics and Mathematics, Inner Mongolia University Financial and Economics, Huhhot Inner Mongolia

Received: Oct. 31st, 2019; accepted: Nov. 15th, 2019; published: Nov. 22nd, 2019

ABSTRACT

In this paper, we discuss the stability for a class of nonlinear differential systems by using integral inequalities.

Keywords:Integral Inequalities, Monotonous, Time-Delay Nonlinear Differential System, Stability

一类非线性时滞微分系统的稳定性

霍冉1,王晓丽2*

1内蒙古农业大学理学院数学与统计学系,内蒙古 呼和浩特

2内蒙古财经大学统计与数学学院数学系,内蒙古 呼和浩特

收稿日期:2019年10月31日;录用日期:2019年11月15日;发布日期:2019年11月22日

摘 要

由一类积分不等式推导给出一类非线性时滞微分系统Lipschitz稳定性判断准则。

关键词 :积分不等式,单调,非线性时滞系统,稳定

Copyright © 2019 by author(s) and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 准备工作

定义1 [1] :若时滞微分方程中未知函数的最高阶导数含有两个不同的变元值,称之为中立型时滞微分方程;

考虑时滞微分系统: d x d t = f ( t , x ( t ) , x ( t τ ) , x ˙ ( t τ ) ) (1.1)

其中 x R n y = c o l ( x 1 , x 2 , , x m ) z = c o l ( x m + 1 , x m + 2 , , x n ) x = c o l ( y , z ) f ( t , 0 , 0 , 0 ) 0 τ 0 为常数,给定初始函数:

x ( t ) = φ ( t ) x ˙ ( t ) = φ ˙ ( t ) t E t 0 = [ t 0 τ , t 0 ] 连续可微 (1.2)

我们总假定(1.1)满足初始条件(1.2)的解存在唯一,并用 x ( t , t 0 , φ ) 表示(1.1)满足条件(1.2)的解。

定义2 [2] :称(1.1)的平凡解关于部分变元y是Lipschitz稳定的,如果存在常数 M ( t 0 ) > 0 δ ( t 0 ) > 0 ,使当 φ + φ ˙ < δ (对 t E t 0 )时,有: y ( t ; t 0 , φ ) + y ˙ ( t ; t 0 , φ ) M ( t 0 ) ( φ + φ ˙ ) t t 0 0 成立,简记为LS;

定义3 [3] [4] :称(1.1)的平凡解关于部分变元y是一致Lipschitz稳定的,若定义2中的M和 δ 均与 t 0 无关,简记为ULS;

引理1:设如下条件于 t 0 成立

(I):常数 c > 0

(II): u ( t ) c + 0 t a ( s ) u ( s ) d s + i = 1 I 0 t b i ( s ) u α i + 1 ( s ) d s + j = 1 J 0 t c j ( s ) 0 s d j ( σ ) u ( σ ) d σ d s + k = 1 K 0 t e k ( s ) 0 s f k ( σ ) u β k + 1 ( σ ) d σ d s + 0 t i = 1 I b i ( s ) u α i ( s ) [ l = 1 L 0 s g l ( σ ) 0 σ h l ( τ ) u ( τ ) d τ d σ ] d s + 0 t i = 1 I b i ( s ) u α i ( s ) [ m = 1 M 0 s p m ( σ ) 0 σ q m ( τ ) u γ m + 1 ( τ ) d τ d σ ] d s

其中: u ( t ) a ( t ) b i ( t ) ( i = 1 , 2 , , I ) c j ( t ) , d j ( t ) ( j = 1 , 2 , , J )

e k ( t ) , f k ( t ) ( k = 1 , 2 , , K ) g l ( t ) , h l ( t ) ( l = 1 , 2 , , L )

p m ( t ) , q m ( t ) ( m = 1 , 2 , , M ) 均为 R + 上非负连续函数,

且: α i ( i = 1 , 2 , , I ) β k ( k = 1 , 2 , , K ) γ m ( m = 1 , 2 , , M ) 均为大于1的常数, 1 α 1 α 2 α I 1 β 1 β 2 β K 1 γ 1 γ 2 γ M

令: α ¯ = max ( α I , β K , γ M ) α _ = min ( α 1 , β 1 , γ 1 )

(III) 设 A ( t ) = a ( t ) + i = 1 I b i ( t ) + j = 1 J c j ( t ) 0 t d j ( s ) d s + 2 k = 1 K e k ( t ) 0 t f k ( s ) d s + i = 1 I b i ( t ) 0 t l = 1 L g l ( s ) 0 s h l ( σ ) d σ d s + i = 1 I b i ( t ) 0 t m = 1 M p m ( s ) 0 s q m ( σ ) d σ d s

B ( t ) = i = 1 I b i ( t ) + k = 1 K e k ( t ) 0 t f k ( s ) d s + i = 1 I b i ( t ) 0 t l = 1 L g l ( s ) 0 s h l ( σ ) d σ d s + 2 i = 1 I b i ( t ) 0 t m = 1 M p m ( s ) 0 s q m ( σ ) d σ d s

C ( t ) = 0 t m = 1 M p m ( s ) 0 s q m ( σ ) d σ d s

E 1 ( t ) = 1 α ¯ c α ¯ 0 t [ B ( τ ) + C ( τ ) ] exp ( α ¯ 0 τ A ( σ ) d σ ) d τ

F ( t ) = a ( t ) + j = 1 J c j ( t ) 0 t d j ( s ) d s

G ( t ) = i = 1 I b i ( t ) Щ α i α _ + k = 1 K e k ( t ) 0 t f k ( s ) Щ β k α _ d s + i = 1 I b i ( t ) N α i α _ 0 t l = 1 L g l ( s ) 0 s h l ( σ ) d σ

H ( t ) = m = 1 M p m ( t ) 0 t q m ( s ) Щ γ m α _ d s

E 2 ( t ) = 1 α _ c α _ 0 t [ G ( τ ) + H ( τ ) ] exp ( α _ 0 τ F ( σ ) d σ ) d τ

0 + A ( s ) d s < +

E 1 ( t ) > 0 [ 1 ( α ¯ 1 ) c α ¯ 0 t B ( s ) E 1 1 α ¯ ( s ) exp ( α ¯ 0 s A ( τ ) d τ ) d s ] > 0

E 2 ( t ) > 0 [ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 2 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] > 0

则有: u ( t ) c exp ( 0 t F ( s ) d s ) [ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 2 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] 1 α _ 1

注:该引理已在文献 [5] 中证明。

2. 一类非线性时滞微分系统的稳定性

考虑如下微分系统:

d x d t = A ( t ) x ( t ) + f ( t , x ( t δ ( t ) ) , 0 t h { s , x ( s δ ( s ) ) , 0 s g [ σ , x ( σ δ ( σ ) ) ] d σ } d s ) (2.1)

其中: A ( t ) = ( a i j ( t ) ) n × n f C [ I × R n × R n × R n , R n ]

h C [ I × R n × R n × R n , R n ] g C [ I × R n × R n , R n ]

0 < δ 0 δ ( t ) δ A ( t ) 有界, f ( t , 0 , 0 , 0 ) = 0 x ( t ) = φ ( t ) ,于 δ t 0

由引理1,可得与系统(2.1)的解等价的积分系统的解:

x ( t ) = Y ( t , 0 ) φ ( 0 ) + 0 t Y ( t , s ) f ( s , x ( s δ ( s ) ) , 0 s h { τ , x ( τ δ ( τ ) ) , 0 τ g [ σ , x ( σ δ ( σ ) ) ] d σ } d τ ) d s (2.2)

主要结论:对于系统(2.1)而言,假设下列条件于 t 0 时成立:

1) Y ( t , s ) exp ( r ( t s ) ) ( t s 0 ) r 0

2) f ( t , x ( t δ ( t ) ) , 0 t h { s , x ( s δ ( s ) ) , 0 s g [ σ , x ( σ δ ( σ ) ) ] d σ } d s ) a ( t ) x ( t δ ( t ) ) + i = 1 I b i ( t ) x ( t δ ( t ) ) α i + 1 + j = 1 J c j ( t ) 0 t exp ( r ( t s ) ) d j ( s ) x ( s δ ( s ) ) d s + k = 1 K e k ( t ) 0 t exp ( r ( t s ) ) f k ( s ) x ( s δ ( s ) ) β k + 1 d s + i = 1 I b i ( t ) x ( t δ ( t ) ) α i 0 t exp ( r ( t s ) ) l = 1 L g l ( s ) 0 s h l ( σ ) x ( σ δ ( σ ) ) d σ d s + i = 1 I b i ( t ) x ( t δ ( t ) ) α i 0 t m = 1 M exp ( r ( t s ) ) p m ( s ) 0 s q m ( σ ) x ( σ δ ( σ ) ) γ m + 1 d σ d s

3) 其中 a ( t ) b i ( t ) ( i = 1 , 2 , , I ) c j ( t ) , d j ( t ) ( j = 1 , 2 , , J )

e k ( t ) , f k ( t ) ( k = 1 , 2 , , K ) g l ( t ) , h l ( t ) ( l = 1 , 2 , , L )

p m ( t ) , q m ( t ) ( m = 1 , 2 , , M ) 如 [6] 中定理 3.1.1 所设,且单调不增。

4) F ( t ) = M 2 a ( t ) + M 2 j = 1 J c j ( t ) 0 t d j ( t ) d t 0 + F ( s ) d s < +

G ( t ) = i = 1 I M 2 b i ( t ) e α i r t Щ α i α _ + k = 1 K M 2 e k ( t ) 0 t f k ( s ) e β k r s Щ β k α _ d s + i = 1 I M 2 b i ( t ) e α i r t N α i α _ 0 t l = 1 L g l ( s ) 0 s h l ( σ ) d σ

H ( t ) = m = 1 M M 2 p m ( t ) 0 t q m ( s ) e γ m r s Щ γ m α _ d s

E ( t ) = 1 α _ c α _ 0 t [ G ( τ ) + H ( τ ) ] exp ( α _ 0 τ F ( σ ) d σ ) d τ

E ( t ) > 0 ,且, [ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] > 0

0 G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s < 其中 Щ 如 [7] [8] [9] 中引理所设

则系统(2.1)的零解在 C 1 中一致Lipschitz渐近稳定。

证明:由系统(2.1)的等价系统(2.2):

x ( t ) = Y ( t , 0 ) φ ( 0 ) + 0 t Y ( t , s ) f ( s , x ( s δ ( s ) ) , 0 s h { τ , x ( τ δ ( τ ) ) , 0 τ g [ σ , x ( σ δ ( σ ) ) ] d σ } d τ ) d s

由条件1),2),得到:

x ( t ) φ ( 0 ) exp ( r t ) + 0 t exp ( r ( t s ) ) a ( s ) x ( s δ ( s ) ) d s + 0 t i = 1 I b i ( s ) exp ( r ( t s ) ) x ( s δ ( s ) ) α i + 1 d s + 0 t j = 1 J c j ( s ) exp ( r ( t s ) ) 0 s exp ( r ( s θ ) ) d j ( θ ) x ( θ δ ( θ ) ) d θ d s + 0 t k = 1 K e k ( s ) exp ( r ( t s ) ) 0 s exp ( r ( s θ ) ) f k ( θ ) x ( θ δ ( θ ) ) β k + 1 d θ d s

+ 0 t i = 1 I b i ( s ) x ( s δ ( s ) ) α i exp ( r ( t s ) ) × 0 s exp ( r ( s θ ) ) l = 1 L g l ( θ ) 0 θ h l ( σ ) x ( σ δ ( σ ) ) d σ d θ d s + 0 t i = 1 I b i ( s ) x ( s δ ( s ) ) α i exp ( r ( t s ) ) × 0 s m = 1 M exp ( r ( s θ ) ) p m ( θ ) 0 θ q m ( σ ) x ( σ δ ( σ ) ) γ m + 1 d σ d θ d s (2.3)

于是有:

x ( t ) e r t φ ( 0 ) 0 t a ( s ) x ( s δ ( s ) ) e r s d s + 0 t i = 1 I b i ( s ) x ( s δ ( s ) ) α i + 1 e r s d s + 0 t j = 1 J c j ( s ) e r s 0 s e r ( s θ ) d j ( θ ) x ( θ δ ( θ ) ) d θ d s + 0 t k = 1 K e k ( s ) e r s 0 s e r ( s θ ) f k ( θ ) x ( θ δ ( θ ) ) β k + 1 d θ d s + 0 t i = 1 I b i ( s ) x ( s δ ( s ) ) α i e r s 0 s e r ( s θ ) l = 1 L g l ( θ ) 0 θ h l ( σ ) x ( σ δ ( σ ) ) d σ d θ d s + 0 t i = 1 I b i ( s ) x ( s δ ( s ) ) α i e r s 0 s e r ( s θ ) m = 1 M p m ( θ ) 0 θ q m ( σ ) x ( σ δ ( σ ) ) γ m + 1 d σ d θ d s (2.4)

u ( t ) = x ( t ) e r t ,由 e r t = e r ( t δ ( t ) ) e r δ ( t ) ,注意到 δ ( t ) δ ,故可设 sup 0 t < + e r δ ( t ) = M 1 < + ,由此可得:

u ( t ) φ ( 0 ) + M 1 0 t a ( s ) u ( s δ ( s ) ) d s + 0 t i = 1 I M 1 α i + 1 b i ( s ) e α i r s u α i + 1 ( s δ ( s ) ) d s + M 1 0 t j = 1 J c j ( s ) 0 s d j ( θ ) u ( θ δ ( θ ) ) d θ d s + 0 t k = 1 K M 1 β k + 1 e k ( s ) 0 s e β k r θ f k ( θ ) u β k + 1 ( θ δ ( θ ) ) d θ d s + 0 t i = 1 I M 1 α i + 1 b i ( s ) u α i ( s δ ( s ) ) e α i r s 0 s l = 1 L g l ( θ ) 0 θ h l ( σ ) u ( σ δ ( σ ) ) d σ d θ d s + 0 t i = 1 I M 1 α i + 1 b i ( s ) u α i ( s δ ( s ) ) e α i r s 0 s m = 1 M p m ( θ ) 0 θ M 1 γ m e γ m r σ q m ( σ ) u γ m + 1 ( σ δ ( σ ) ) d σ d θ d s (2.5)

令: φ 1 = sup δ t 0 φ ( t ) φ 2 = sup δ t 0 φ ( t ) α i + 1

φ 3 = sup δ t 0 φ ( t ) β k + 1 φ 4 = sup δ t 0 φ ( t ) γ m + 1

φ = max { φ 1 , φ 2 , φ 3 , φ 4 , φ 2 φ 4 }

令: u ( t ) = { max { φ 1 , max ( u ( ξ ) ) } , 0 ξ t φ 1 δ t 0

显然 u ( t ) 单调不减,且由 u ( t ) 的定义,有 u ( t δ ( t ) ) φ 1 ,因此:

u ( t ) φ + M 2 0 t a ( s ) u ( s ) d s + 0 t i = 1 I M 2 b i ( s ) e α i r s u ( s ) α i + 1 d s + M 2 0 t j = 1 J c j ( s ) 0 s d j ( θ ) u ( θ ) d θ d s + 0 t k = 1 K M 2 e k ( s ) 0 s e β k r θ f k ( θ ) u ( θ ) β k + 1 d θ d s + 0 t i = 1 I M 2 b i ( s ) u ( s ) α i e α i r s 0 s l = 1 L g l ( θ ) 0 θ h l ( σ ) u ( σ ) d σ d θ d s + 0 t i = 1 I M 2 b i ( s ) u ( s ) α i e α i r s 0 s m = 1 M p m ( θ ) 0 θ e γ m r σ q m ( σ ) u ( σ ) γ m + 1 d σ d θ d s (2.6)

其中 M 2 = max { M 1 , M 1 α i + 1 , M 1 β k + 1 , M 1 γ m + 1 , M 1 α i + γ m + 1 }

注意到定理条件,由引理可得;

u ( t ) c exp ( 0 t F ( s ) d s ) [ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] 1 α _ 1

其中: c = φ ,注意到 u ( t ) 的定义,可得:

x ( t ) φ e r t exp ( 0 t F ( s ) d s ) [ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] 1 α _ 1

注意到定理的条件,有如下事实:

r < 0 0 + F ( s ) d s < + E ( t ) > 0 α _ 1

[ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] > 0 ,且有:

[ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] < 1 ,于是令:

M ( t , 0 ) = e r t exp ( 0 t F ( s ) d s ) [ 1 ( α _ 1 ) c α _ 0 t G ( s ) E 1 α _ ( s ) exp ( α _ 0 s F ( τ ) d τ ) d s ] 1 α _ 1 e r t exp ( 0 t F ( s ) d s )

于是,对一切 t 0 时,有 lim t M ( t , t 0 ) = 0 一致的成立。于是,可得结论成立。

基金项目

内蒙古自治区高等学校科学研究项目(NJZY16141,NJZY17064)。

文章引用

霍 冉,王晓丽. 一类非线性时滞微分系统的稳定性
The Stabilities for a Class of Nonlinear Differential Systems with Time-Delay[J]. 应用数学进展, 2019, 08(11): 1845-1851. https://doi.org/10.12677/AAM.2019.811214

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  10. NOTES

    *通讯作者。

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