孫云霞
摘? 要:基于離散時(shí)間狀態(tài)觀測(cè),研究帶Markov切換的隨機(jī)Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)穩(wěn)定的問(wèn)題.通過(guò)構(gòu)造? Lyapunov函數(shù),利用[Ito]微分公式、Borel-Cantellis引理及穩(wěn)定性分析理論,得到非線性和線性系統(tǒng)幾乎必然指數(shù)穩(wěn)定的充分條件.最后,通過(guò)一個(gè)例子驗(yàn)證所得結(jié)果的可行性.
關(guān)鍵詞:隨機(jī)Cohen-Grossberg神經(jīng)網(wǎng)絡(luò);Markov鏈;離散時(shí)間觀測(cè);Lyapunov函數(shù);幾乎必然指數(shù)穩(wěn)定
中圖分類號(hào):O231? ? ? ? ? ? ? ? ?DOI:10.16375/j.cnki.cn45-1395/t.2020.04.012
0? ? 引言
自Cohen[1]首次提出和研究Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)模型(簡(jiǎn)稱CGNNs)以來(lái),神經(jīng)網(wǎng)絡(luò)在模式識(shí)別、并行處理、聯(lián)想記憶、最優(yōu)化計(jì)算等方面得到了較深入研究.而系統(tǒng)在其運(yùn)行過(guò)程經(jīng)常會(huì)出現(xiàn)隨機(jī)突變現(xiàn)象,如外部環(huán)境的突然變化、大系統(tǒng)內(nèi)部各子系統(tǒng)間連接方式的改變等都會(huì)引起系統(tǒng)結(jié)構(gòu)的改變,而對(duì)于這種具有可變結(jié)構(gòu)的系統(tǒng)常常利用Markov切換[2-3]系統(tǒng)模型來(lái)刻畫(huà).另外,噪聲[4-5]能使一個(gè)不穩(wěn)定的系統(tǒng)穩(wěn)定,甚至使一個(gè)穩(wěn)定的系統(tǒng)更穩(wěn)定.由于在實(shí)際的系統(tǒng)中不可避免的存在噪聲的因素,因此,在很多的文獻(xiàn)中都考慮到它的影響.
眾所周知,傳統(tǒng)的反饋控制[σxt, rt, t]要求在所有的時(shí)間上對(duì)狀態(tài)是進(jìn)行連續(xù)觀測(cè)的,在經(jīng)濟(jì)上是非常昂貴的,實(shí)際上連續(xù)情況的觀測(cè)也可能無(wú)法實(shí)現(xiàn).所以,毛學(xué)榮[6]提出了在離散時(shí)間觀測(cè)的基礎(chǔ)上設(shè)計(jì)一個(gè)反饋控制[σxδt, rδt, t],使其控制更加合理和實(shí)用,其中,[δt=tττ],[τ>0],[tτ]表示[tτ]的整數(shù)部分.通過(guò)選擇正常數(shù)[τ],只需要觀測(cè)[0],[τ],[2τ],[…]. 離散時(shí)間間隔[τ]的上界非常小,對(duì)于如何選擇一段時(shí)間間隔,以使得反饋控制系統(tǒng)穩(wěn)定,可以參考文獻(xiàn)[6]. 然而,能否通過(guò)用[σxδt, rδt, t]替換,使帶Markov切換的隨機(jī)CGNNs具有幾乎必然指數(shù)穩(wěn)定性?這是本文研究的關(guān)鍵問(wèn)題.
近年來(lái),隨機(jī)CGNNs的穩(wěn)定性[7-9]研究吸引了大量學(xué)者,并取得了許多有意義的結(jié)果.目前關(guān)于帶Markov切換的隨機(jī)CGNNs的幾乎必然指數(shù)穩(wěn)定性[10]和基于離散時(shí)間狀態(tài)觀測(cè)的隨機(jī)系統(tǒng)的幾乎必然指數(shù)穩(wěn)定性[6, 11-12]均已研究.但是,基于離散時(shí)間狀態(tài)觀測(cè),研究帶Markov切換的隨機(jī)CGNNs的幾乎必然(a.s)指數(shù)穩(wěn)定性的卻不多見(jiàn).
本文在文獻(xiàn)[6, 11-12]的啟發(fā)下,研究如何通過(guò)加入一個(gè)反饋控制器使不穩(wěn)定的隨機(jī)CGNNs達(dá)到穩(wěn)定性的問(wèn)題.
1? ? 準(zhǔn)備知識(shí)
本文采用以下記號(hào):記[Ω, F, Ftt≥0, P]為含有滿足通常條件的代數(shù)流[Ftt≥0]的完備概率空間,令[Bt=B1t, B2t, …, BmtT]為定義于該空間上的[m]維標(biāo)準(zhǔn)布朗運(yùn)動(dòng).
4? ? 結(jié)論
本文利用[Ito]微分公式、Borel-Cantellis引理和穩(wěn)定性分析理論對(duì)基于布朗運(yùn)動(dòng)和離散時(shí)間狀態(tài)觀測(cè)的帶Markov切換的隨機(jī)CGNNs的幾乎必然指數(shù)穩(wěn)定性進(jìn)行了研究.最后得到判定其線性和非線性隨機(jī)CGNNs幾乎必然指數(shù)穩(wěn)定的充分性條件.同時(shí),對(duì)于間隔[τ],如果能指出一個(gè)較大的上界可以大大地降低控制成本.
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(責(zé)任編輯:羅小芬、黎? ?婭)