不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定

張景發(fā)

(華南理工大學(xué) 數(shù)學(xué)學(xué)院, 廣東 廣州, 510640)

不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定

張景發(fā)

(華南理工大學(xué) 數(shù)學(xué)學(xué)院, 廣東 廣州, 510640)

為了探討帶有時變時滯的不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定問題, 通過構(gòu)造一個合適的李雅普諾夫函數(shù), 利用李雅普諾夫函數(shù)法、隨機分析法及線性矩陣不等式得到了不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定的充分條件, 驗證了已知條件滿足引理, 表明帶時滯的隨機系統(tǒng)在時滯小于某個上界時，帶時變時滯的不確定隨機神經(jīng)網(wǎng)絡(luò)是幾乎必然指數(shù)穩(wěn)定的。所給出的判據(jù)是由線性矩陣不等式表示的, 該判據(jù)是否有解可以通過Matlab工具箱快速地得到解決。

隨機神經(jīng)網(wǎng)絡(luò); 幾乎必然指數(shù)穩(wěn)定; 李雅普諾夫函數(shù); 線性矩陣不等式; 時滯

過去幾十年以來, 神經(jīng)網(wǎng)絡(luò)這一領(lǐng)域不斷地被發(fā)展、研究, 獲得了大量的研究成果, 并廣泛地運用于組合優(yōu)化、信號過程、模式識別等領(lǐng)域[1–3]。這些領(lǐng)域的運用都緊緊地依賴于神經(jīng)網(wǎng)絡(luò)的動力學(xué)行為,其中穩(wěn)定性是神經(jīng)網(wǎng)絡(luò)動力學(xué)行為中最重要的性質(zhì)[4–11]。由于系統(tǒng)建模、測量誤差、系統(tǒng)線性化等原因,神經(jīng)網(wǎng)絡(luò)常常受到參數(shù)不確定和隨機干擾因素的影響, 導(dǎo)致系統(tǒng)動力學(xué)行為不理想或系統(tǒng)性能差。文獻[12–19]研究了參數(shù)不確定因素或者隨機干擾因素。另外, Deng等[20]研究了帶有馬爾科夫鏈的隨機微分方程的幾乎必然指數(shù)穩(wěn)定性, Guo等[21]研究了一類時滯系統(tǒng)的幾乎必然指數(shù)穩(wěn)定。上述文獻只是研究了不確定隨機神經(jīng)網(wǎng)絡(luò)的魯棒穩(wěn)定性及漸近穩(wěn)定等問題, 但是對于帶有不確定性和時滯的隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定的研究文獻比較少。本文研究一類帶有區(qū)間實變時滯的不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定。

1 預(yù)備知識

一般地, (Ω, Γ,{Γt}t≥0,ρ)表示全概率空間, 濾子流{Γt}t≥0包含所有的零測子集且右連續(xù)。ω(t)=[ω1(t),ω2(t),…,ωm(t )]為定義在全概率空間上的m維布朗運動。Rn表示n維歐幾里德空間,表示歐幾里德空間的范數(shù)。Rn×n表示n×n維的實矩陣集合, In為nn維單位矩陣。對于對稱矩陣A∈Rn×n,λmax(A)、λmin(A)分別表示矩陣A的最大特征值及最小特征值。AT表示矩陣A的轉(zhuǎn)置。對于對稱正定矩陣X和Y, X-Y＞0表示對稱正定的。對稱矩陣中符號*表示對稱位置的元素的轉(zhuǎn)置, diag{M1, M2,…,Mn}表示對角矩陣的簡寫?？紤]一般的隨機微分方程

這里x(t)∈Rn表示神經(jīng)元的狀態(tài)向量, τ1(t)、τ2(t)是2個不同的神經(jīng)元時變時滯,是m維布朗運動, f(x(t), t)是恰當(dāng)維的向量函數(shù), g(x(t), t)是恰當(dāng)維的矩陣函數(shù)。為簡單起見, 假設(shè)f(0)=0和g(0)=0, 因此可知x(t)≡0是方程(1)的平凡解。

假設(shè)1 設(shè)隨機微分方程(1)的f、g是Borel可測, 并且存在2個非負常數(shù)K1、K2使得對于任意x、,t≥0,有。

定義1 如果方程(1)的解對于所有的滿足, a.s.則隨機微分方程(1)的平凡解是幾乎必然指數(shù)穩(wěn)定的。設(shè)C2,1(Rn×[t0,+∞];R+)表示Rn×[t0,+∞]上關(guān)于x兩次連續(xù)可微, 關(guān)于t一次連續(xù)可微的所有非負函數(shù)V(x(t), t)的全體。對任意的(x,t)∈Rn×[t0,+∞], 則對V(x(t), t)有如下伊藤微分公式:

本文考慮如下形式的不確定隨機神經(jīng)網(wǎng)絡(luò)

假設(shè)2 系統(tǒng)參數(shù)不確定矩陣ΔA(t),ΔW0(t),ΔC1(t),ΔC2(t)滿足假設(shè)3 激活函數(shù)f(x(t))每個分量fi(?)是連續(xù)函數(shù), 存在實數(shù), 使得對于任意的xi∈R 有, 定義對角矩陣

引理1[21]假設(shè)存在函數(shù)V(x(t), t)∈C2,1(Rn×[t0,∞];R+)和常數(shù)p＞0, c1＞0, c2＞0, c3≥0使得對于任意的x≠0, 當(dāng)t≥t0時,

則隨機微分方程(1)的平凡解是幾乎必然指數(shù)穩(wěn)定。

引理2[22]對于適當(dāng)維的矩陣Ψ11, Ψ12,Ψ22, 且滿足當(dāng)且僅當(dāng)。

引理3[23]對于任意向量x, y∈Rn, 矩陣P∈Rn×n且P＞0, 那么2xTy≤xTP-1x+yTPy成立。

引理4[24]假設(shè)M, N, X是適當(dāng)維的實矩陣, 且矩陣X滿足X=XT, 則對于所有F(t)滿足FT(t)?F(t)≤I, 有X+MF(t )N+NTFT(t )MT＜0,當(dāng)且僅當(dāng)存在標(biāo)量ε＞0使得X+ε-1MMT+εNTN＜0。

引理5[21]假設(shè)引理1和假設(shè)1成立, 那么存在一個正數(shù)τ?使得對于任意初值, 當(dāng)τ＜τ?時隨機微分方程(3)的解有該引理表明隨機微分方程(1)的平凡解是幾乎必然指數(shù)穩(wěn)定時, 隨機時滯微分方程(2)的平凡解也是幾乎必然指數(shù)穩(wěn)定的。

2 幾乎必然指數(shù)穩(wěn)定判據(jù)

通過引理2, 本文找到一個合適的李雅普諾夫函數(shù), 并利用LMIs的方式得到如下定理。

定理1 如果存在正定矩陣P, 正定對角矩陣D=diag(d1, d2,…,dn)以及實數(shù)α≥0, 使得

3 總結(jié)

本文通過構(gòu)造一個正定的李雅普諾夫函數(shù), 利用李雅普諾夫函數(shù)法、隨機分析方法及線性矩陣等式得到了不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定的充分條件, 然后根據(jù)定理得出帶時變時滯的不確定隨機神經(jīng)網(wǎng)絡(luò)的幾乎必然指數(shù)穩(wěn)定。

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(責(zé)任編校: 劉剛毅)

The almost sure exponential stability of uncertain stochastic neural networks

Zhang Jingfa
(School of Mathematics, South China University of Technology, Guangzhou 510640, China)

By constructing a appropriate Lyapunov function, and utilizing Lyapunov functional method, stochastic analysis method and linear matrix inequalities, the sufficient conditions for the almost sure exponential stability of uncertain stochastic neural networks are derived, which can verify the given conditions are satisfied with the lemma.The result shows that the same stability of systems with time-varying delays when time-varying delys are less than a upper bound. The given conditions are in the form of linear matrix inequalities, so it is easy to get its solution by Matlab Toolbox when checking the stability conditions of the systems.

stochastic neural networks; almost sure exponential stability; Lyapunov functions; time-varying delays

TP 183

: A

1672–6146(2017)03–0013–07

10.3969/j.issn.1672–6146.2017.03.004

張景發(fā), 502462024@qq.com。

: 2017–02–22