雷國(guó)梁,岳田,宋曉秋

(1.湖北汽車(chē)工業(yè)學(xué)院理學(xué)院,湖北十堰442002;2.中國(guó)礦業(yè)大學(xué)理學(xué)院,江蘇徐州221008)

Banach空間中線性離散時(shí)間系統(tǒng)的一致多項(xiàng)式膨脹性

雷國(guó)梁1,岳田1,宋曉秋2

(1.湖北汽車(chē)工業(yè)學(xué)院理學(xué)院,湖北十堰442002;2.中國(guó)礦業(yè)大學(xué)理學(xué)院,江蘇徐州221008)

給出了Banach空間中線性離散時(shí)間系統(tǒng)一致多項(xiàng)式膨脹性的概念,并討論了其離散特征.借助Lyapunov函數(shù)給出了線性離散時(shí)間系統(tǒng)滿足一致多項(xiàng)式膨脹的充要條件.所得結(jié)論將一致指數(shù)穩(wěn)定性、指數(shù)膨脹性及多項(xiàng)式穩(wěn)定性中的若干經(jīng)典結(jié)論推廣到了一致多項(xiàng)式膨脹性的情形.

線性離散時(shí)間系統(tǒng);一致多項(xiàng)式膨脹性;Lyapunov函數(shù)

1 引言

近年來(lái),關(guān)于有限或無(wú)限維Banach空間中演化方程解的漸近行為研究取得了突破性的進(jìn)展,尤其對(duì)于演化型算子的指數(shù)型漸近行為做了大量研究,獲得了非常豐富的成果[113].如文獻(xiàn)[1]利用Banach函數(shù)空間及序列空間給出了Banach空間中斜積半流一致指數(shù)不穩(wěn)定的充要條件;文獻(xiàn)[2-3]分別討論了Banach空間中演化族與斜積流的一致指數(shù)不穩(wěn)定性;文獻(xiàn)[4]利用完全容許性給出了線性斜積流一致指數(shù)膨脹的離散和連續(xù)型特征;文獻(xiàn)[7-8]分別對(duì)Banach空間中演化族及斜演化半流的弱指數(shù)膨脹性進(jìn)行了相關(guān)討論,給出了滿足相應(yīng)概念的若干充要條件.對(duì)于線性離散時(shí)間系統(tǒng)解的指數(shù)型漸近行為,文獻(xiàn)[9-11]分別給出了其滿足指數(shù)穩(wěn)定、指數(shù)二分、指數(shù)三分的若干離散特征.

自從Barreira L,Valls C[14]提出多項(xiàng)式漸近概念以來(lái),關(guān)于演化方程多項(xiàng)式的漸近行為的研究已成為一個(gè)熱點(diǎn)問(wèn)題,但目前大都局限于多項(xiàng)式穩(wěn)定性方面的研究.如文獻(xiàn)[15]針對(duì)演化算子提出了一致多項(xiàng)式穩(wěn)定、Barreira-Valls意義下多項(xiàng)式穩(wěn)定、非一致多項(xiàng)式穩(wěn)定三個(gè)概念,并分別給出了相應(yīng)的積分特征,同時(shí)指出了三者之間的聯(lián)系與區(qū)別;文獻(xiàn)[16]重點(diǎn)討論了演化算子在Barreira-Valls意義下呈多項(xiàng)式穩(wěn)定的相關(guān)特征.與多項(xiàng)式穩(wěn)定性相比,演化方程多項(xiàng)式膨脹性研究相對(duì)較少,本文將在上述文獻(xiàn)的基礎(chǔ)上,給出Banach空間中線性離散時(shí)間系統(tǒng)一致多項(xiàng)式膨脹性概念,利用指數(shù)型漸近行為研究方法,給出其相應(yīng)的充要條件.且作為應(yīng)用,借助Lyapunov函數(shù)給出線性離散時(shí)間系統(tǒng)一致多項(xiàng)式膨脹性的特征.

2 預(yù)備知識(shí)

設(shè)X為一實(shí)或復(fù)的Banach空間,記空間X上的范數(shù)及作用其上的有界線性算子全體B(X)中的范數(shù)為||·||.記

I為X上的恒等算子.

考慮如下線性離散時(shí)間系統(tǒng):

其中A:N→B(X)是B(X)中的一個(gè)序列.則系統(tǒng)(1)的每個(gè)解x={xn}將可表示為:

其中,定義映射:

由此可以驗(yàn)證如下關(guān)系成立:

特別地,當(dāng)(1)式為一個(gè)自治系統(tǒng),即

則有

定義2.1稱(chēng)系統(tǒng)(1)為一致多項(xiàng)式膨脹的,如果存在兩個(gè)常數(shù)N≥1及α>1,使得

注2.1顯然,系統(tǒng)(1)是一致多項(xiàng)式膨脹的當(dāng)且僅當(dāng)存在兩個(gè)常數(shù)N≥1及α>1,使得

但現(xiàn)在，我發(fā)現(xiàn)我錯(cuò)了，卻不知錯(cuò)在哪里。錯(cuò)在我當(dāng)初的選擇嗎？錯(cuò)在我沒(méi)有教學(xué)能力嗎？錯(cuò)在我的教學(xué)水平越來(lái)越低嗎？

例2.1設(shè)X=R,考慮映射:

則

進(jìn)而

故在定義2.1中取N=α=2,即可得系統(tǒng)(1)是一致多項(xiàng)式膨脹的.

定義2.2稱(chēng)映射L:?×X→R為系統(tǒng)(1)的一致多項(xiàng)式膨脹Lyapunov函數(shù),如果存在常數(shù)μ>1,使得

3 主要結(jié)論

定理3.1系統(tǒng)(1)是一致多項(xiàng)式膨脹的當(dāng)且僅當(dāng)存在常數(shù)M≥1,ω>0,L≥1及l(fā)>0,使得?(m,n,p,x)∈T×X,有如下兩個(gè)不等式成立:

證明必要性.(a)式顯然,下證(b)式.

由此可得,

充分性.當(dāng)m>2n時(shí),則由條件可得,

進(jìn)而

當(dāng)2n≥m≥n時(shí),則由(a)式可得,

綜上可知系統(tǒng)(1)是一致多項(xiàng)式膨脹的.

注3.1若線性離散時(shí)間系統(tǒng)(1)滿足(a)式,則稱(chēng)其是多項(xiàng)式衰退的.

定理3.2具有多項(xiàng)式衰退的線性離散時(shí)間系統(tǒng)(1)是一致多項(xiàng)式膨脹的當(dāng)且僅當(dāng)存在一致多項(xiàng)式膨脹Lyapunov函數(shù)及常數(shù)K≥1,使得

證明必要性.定義

其中常數(shù)l由定義2.1給出.下證它為一致多項(xiàng)式膨脹Lyapunov函數(shù).

其中(m,n,p,x)∈T×X,m>n.

由定理3.1知,存在常數(shù)L≥1,使得

充分性.當(dāng)n=p時(shí),則由已知條件及一致多項(xiàng)式膨脹Lyapunov函數(shù)的定義可知,存在K≥1和μ>1,使得

進(jìn)而由定理3.1可得系統(tǒng)(1)是一致多項(xiàng)式膨脹的.

[1] Megan M,Sasu A L,Sasu B.Exponential instability of linear skew-product semi fl ows in terms of Banach function spaces[J].Result Math.,2004,45:309-318.

[2] Megan M,Sasu A L,Sasu B.Banach function spaces and exponential instability of evolution families[J]. Arch.Math.,2003,39:277-286.

[3] Megan M,Sasu A L,Sasu B.Exponential stability and exponential instability for linear skew-product fl ows[J].Math.Bohem.,2004,129:225-243.

[4] Megan M,Sasu A L,Sasu B.Perron conditions for uniform exponential expansiveness of linear skew-product fl ows[J].Monatsh.Math.,2003,138:145-157.

[5] Megan M,Sasu B,Sasu A L.Exponential expansiveness and complete admissibility for evolution families[J]. Czech.Math.J.,2004,54:1485-1493.

[6] Sasu B.New criteria for exponential expansiveness of variational di ff erence equations[J].J.Math.Anal. Appl.,2007,327:287-297.

[7] Yue T,Song X Q,Li D Q.On weak exponential expansiveness of evolution families in Banach spaces[J]. Sci.World J.,2013,2013(Article ID 284630):1-6.

[8] Yue T,Song X Q,Li D Q.On weak exponential expansiveness of skew-evolution semi fl ows in Banach spaces[J].J.Inequal.Appl.,2014,2014(Article 165):1-11.

[9] Popa I L,Ceau?su T,Megan M.On exponential stability for linear discrete-time systems in Banach spaces[J]. Comput.Math.Appl.,2012,63:1497-1503.

[10] Popa I L,Megan M,Ceau?su T.Exponential dichotomies for linear discrete-time systems in Banach spaces[J]. Appl.Anal.Discrete Math.,2012,6:140-155.

[11] Song X Q,Yue T,Li D Q.Nonuniform exponential trichotomy for linear discrete-time systems in Banach spaces[J].J.Func.Space Appl.,2013,2013(Article ID 645250):1-6.

[12] 岳田.幾類(lèi)演化型算子漸近行為的理論研究[D].徐州:中國(guó)礦業(yè)大學(xué),2014.

[13] 岳田,宋曉秋,李志剛.n次積分C半群與抽象Cauchy問(wèn)題的指數(shù)穩(wěn)定性[J].河南大學(xué)學(xué)報(bào):自然科學(xué)版, 2014,44(1):15-19.

[14] Barreira L,Valls C.Polynomial growth rates[J].Nonlinear Anal.,2009,71:5208-5219.

[15] Megan M,Ceau?su T,Rˇamneant?u M L.Polynomial stability of evolution operators in Banach spaces[J]. Opuscula Math.,2011,31:279-288.

[16] Megan M,Ceau?su T,Minda A A.On Barreira-Valls polynomial stability of evolution operators in Banach spaces[J].Electron.J.Qual.Theo.Di ff er.Equat.,2011,33:1-10.

On uniform polynomial expansiveness of linear discrete-time systems in Banach spaces

Lei Guoliang1,Yue Tian1,Song Xiaoqiu2
(1.School of Science,Hubei University of Automotive Technology,Shiyan442002,China; 2.College of Science,China University of Mining and Technology,Xuzhou221008,China)

In this paper we study uniform polynomial expansiveness concept for linear discrete-time systems in Banach spaces.Our main objective is to give discrete characterizations for uniform polynomial expansiveness. As for applications we obtain a necessary and sufficient condition in terms of Lyapunov functions.Well-known results for uniform exponential stability,exponential expansiveness and polynomial stability are extended to the case of uniform polynomial expansiveness.

linear discrete-time systems,uniform polynomial expansiveness,Lyapunov function

O231

A

1008-5513(2014)05-0485-06

10.3969/j.issn.1008-5513.2014.05.008

2014-08-04.

國(guó)家自然科學(xué)基金(51374199);中央高?；究蒲袠I(yè)務(wù)費(fèi)專(zhuān)項(xiàng)資金(2012LWB53).

雷國(guó)梁(1971-),碩士,講師,研究方向：應(yīng)用數(shù)學(xué).

2010 MSC:34D20,34D05