周之薇，宋瑞麗

變時(shí)滯反饋控制的混合中立型隨機(jī)延遲微分方程的穩(wěn)定性

*周之薇，宋瑞麗

（南京財(cái)經(jīng)大學(xué)應(yīng)用數(shù)學(xué)學(xué)院，江蘇，南京 210023）

0 引言

隨機(jī)微分方程（SDDE）描述了隨機(jī)系統(tǒng)不僅依賴于現(xiàn)在的狀態(tài)，同時(shí)也依賴于過(guò)去的狀態(tài)。許多科研人員研究了隨機(jī)微分方程的穩(wěn)定性和有界性[1-4]。而連續(xù)時(shí)間的馬氏鏈?zhǔn)怯糜诿枋鲭S機(jī)系統(tǒng)突然性的結(jié)構(gòu)變化。Mao在文獻(xiàn)[5]中研究了帶有馬氏鏈的隨機(jī)微分方程，這樣的隨機(jī)系統(tǒng)又被稱之為混合的隨機(jī)延遲微分方程?；旌想S機(jī)延遲微分方程的有界性和穩(wěn)定性可以參考文獻(xiàn)[6-14]。而混合中立型隨機(jī)延遲微分方程（NSDDEs）是用來(lái)描述一類(lèi)混合隨機(jī)微分方程依賴于過(guò)去一段時(shí)間狀態(tài)的變化率。混合中立型隨機(jī)延遲微分方程的穩(wěn)定性相關(guān)研究可以參閱文獻(xiàn)[15-19]。

但是，有一些隨機(jī)系統(tǒng)是不穩(wěn)定的。最經(jīng)典的做法是在不穩(wěn)定的隨機(jī)系統(tǒng)中加入反饋控制函數(shù)，從而使隨機(jī)系統(tǒng)變得穩(wěn)定。文獻(xiàn)[20-25]研究不穩(wěn)定隨機(jī)系統(tǒng)穩(wěn)定化的問(wèn)題。文獻(xiàn)[20]研究了延遲反饋控制的混合中立型隨機(jī)延遲微分方程的穩(wěn)定性問(wèn)題，其中漂移系數(shù)和擴(kuò)散系數(shù)滿足線性增長(zhǎng)條件。文獻(xiàn)[24]研究了延遲反饋控制的混合中立型隨機(jī)延遲微分方程在多項(xiàng)式增長(zhǎng)條件下的穩(wěn)定化問(wèn)題。以上文獻(xiàn)所提及的時(shí)滯都是常數(shù)。而變時(shí)滯的隨機(jī)系統(tǒng)并不拘泥于一個(gè)常數(shù)，可以推廣到函數(shù)，這將會(huì)帶來(lái)更多的實(shí)用價(jià)值，所以有必要去研究在變時(shí)滯反饋控制且具有變時(shí)滯的混合中立型隨機(jī)延遲微分方程在多項(xiàng)式增長(zhǎng)條件下的穩(wěn)定化問(wèn)題。

1 模型描述和假設(shè)

考慮以下NSDDE，

初值：

基于穩(wěn)定性問(wèn)題的研究，

假設(shè)

提出以下假設(shè):

假設(shè)1（局部李普希茲條件）

假設(shè)2（多項(xiàng)式增長(zhǎng)條件）

(9)

引用文獻(xiàn)[19]的結(jié)論作為引理。

2 主要結(jié)論

整理式（12），可以得到：

可以得到以下的結(jié)論。

其中

定理1 在假設(shè)1，2，3，4，6，7成立的條件下，假設(shè)

對(duì)于任意的初值(2)，系統(tǒng)(3)的解滿足

并且有

證明

從假設(shè)7，可以得到

聯(lián)立(19)和(20)式，可以推出

其中，

由于

因此，

將(22)式代入(21)式中，可得

其中，

其中，

由富比尼定理可以得到

由于

那么，由假設(shè)4，可以推出

由于

因此，

(26)

其中，

從(23)式，得到：

同樣可以從(13)和(25)式中可以推出

由(17)式可以得到此推論。

從(27)式可以得到，

由假設(shè)4，可以推斷出

3 例子

為了說(shuō)明結(jié)論的有效性，舉例如下。

考慮一維變時(shí)滯混合中立型隨機(jī)延遲微分方程：

此時(shí)，令

顯然，不滿足線性增長(zhǎng)條件。

使用Euler-Maruyama方法對(duì)隨機(jī)系統(tǒng)進(jìn)行離散化處理，用Matlab模擬以上隨機(jī)系統(tǒng)。

圖1 馬氏鏈的樣本路徑

圖2 當(dāng)時(shí)系統(tǒng)(33)的樣本路徑（使用了Euler-Maruyama的方法且步長(zhǎng)為0.01）

本研究的目的是設(shè)計(jì)一個(gè)反饋控制使不穩(wěn)定的系統(tǒng)(33)變得穩(wěn)定。

使用混合的NSDDE(33)，定義控制函數(shù)：

由控制函數(shù)控制的系統(tǒng)(3)則具有以下形式

接下來(lái)，證明假設(shè)6，定義

可以得到

從而有

由定理1，可以推斷出系統(tǒng)(36)的解滿足

綜上，可以看出在加了反饋控制函數(shù)后的隨機(jī)系統(tǒng)的解變得穩(wěn)定，即證得結(jié)論的有效性。

圖3 當(dāng)時(shí)系統(tǒng)(36)的樣本路徑(使用了Euler-Maruyama的方法，樣本數(shù)1000且步長(zhǎng)為0.01)

4 結(jié)論

[1] Mao X. Attraction, stability and boundedness for stochastic differential delay equations[J]. Nonlinear Analysis:Theory, Methods & Applications,2001,47(7):4795-4806.

[2] Yin W, Ren Y. Asymptotical boundedness and stability for stochastic differential equations with delaydriven by g-brownian motion[J]. Applied Mathematics Letters, 2017,74:121-126.

[3] Bao Z, Tang J, Shen Y, et al. Equivalence of pth moment stability between stochastic differential delay equations and their numerical methods[J]. Statistics & Probability Letters, 2021,168:108952.

[4] Hu W, Zhu Q. Stability analysis of impulsive stochastic delayed differential systems with unbounded delays[J]. Systems & Control Letters, 2020,136:104606.

[5] Mao X, Yuan C. Stochastic Differential Equations with Markovian Switching[M], London: Imperial College Press,2006.

[6] Zhang T. The stability with a general decay of stochastic delay differential equations with markovian switching[J]. Applied Mathematics and Computation, 2019,359:294-307.

[7] Du N H, Dang N H, Dieu N T. On stability in distribution of stochastic differential delay equations with markovian switching[J]. Systems & Control Letters, 2014,65:43-49.

[8] Li B, Li D, Xu D. Stability analysis for impulsive stochastic delay differential equations with markovian switching[J]. Journal of the Franklin Institute, 2013,350(7):1848-1864.

[9] Fei C, Shen M, Fei W, et al. Stability of highly nonlinear hybrid stochastic integro-differential delay equations[J]. Nonlinear Analysis: Hybrid Systems, 2019,31:180-199.

[10] Rathinasamy K B A. Mean-square stability of milstein method for linear hybrid stochastic delay integro, differential equations[J]. Nonlinear Analysis: Hybrid Systems, 2008(2):1256-1263.

[11] Hu L, Mao X, Shen Y. Stability and boundedness of nonlinear hybrid stochastic differential delay equations[J]. Systems & Control Letters, 2013,62 (2):178-187.

[12] Fei W, Hu L, Mao X , et al. Delay dependent stability of highly nonlinear hybrid stochastic systems[J].Automatica, 2017,82: 165-170.

[13] Lygeros J, Mao X, Yuan C. Stochastic hybrid delay population dynamics [M]. Springer:Heidelberg, 2006:436-450.

[14] Fei W, Hu L, Mao X, et al. Generalized criteria on delay-dependent stability of highly nonlinear hybrid stochastic systems[J]. International Journal of Robust and Nonlinear Control, 2019(5): 1201-1215.

[15] Wu A, You S, Mao W, et al. On exponential stability of hybrid neutral stochastic differential delay equations with different structures[J]. Nonlinear Analysis: Hybrid Systems,2021, 39:100971.

[16] Shen M, Fei W, Mao X, et al. Stability of highly nonlinear neutral stochastic differential delay equations[J]. Systems &Control Letters, 2018, 115: 1-8.

[17] Li X, Mao X. A note on almost sure asymptotic stability of neutral stochastic delay differential equationswith markovian switching[J]. Automatica, 2012, 48(9):2329-2334.

[18] Mao X, Shen Y, Yuan C. Almost surely asymptotic stability of neutral stochastic differential delay equations with markovian switching[J]. Stochastic Processes and their Applications, 2008,118 (8): 1385-1406.

[19] Shen M, Fei C, Fei W, et al. Boundedness and stability of highly nonlinear hybrid neutral stochastic systems with multiple delays[J]. Science China Information Sciences, 2019, 62 (10): 202205.

[20] Li X, Mao X. Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control[J]. Automatica, 2020, 112:108657.

[21] Mao X, Lam J, Huang L. Stabilisation of hybrid stochastic differential equations by delay feedback control[J]. Systems & Control Letters, 2008, 57(11):927-935.

[22] Wang P, Feng J, Su H. Stabilization of stochastic delayed networks with markovian switching and hybrid nonlinear coupling via aperiodically intermittent control[J]. Nonlinear Analysis:Hybrid Systems, 2019, 32:115-130.

[23] Chen W, Xu S, Zou Y. Stabilization of hybrid neutralstochastic differential delay equations by delay feedback control[J]. Systems & Control Letters, 2016, 88:1-13.

[24] Shen M, Fei C, Fei W, et al. Stabilisation by delay feedback control for highly nonlinear neutral stochastic differential equations[J]. Systems & Control Letters, 2020, 137:104645.

[25] Song R, Wang B, Zhu Q. Delay-dependent stability of nonlinear hybrid neutral stochastic differential equations with multiple delays[J]. International Journal of Robust and Nonlinear Control, 2021, 31(1):250-267.

STABILIZATION OF THE HYBRID NEUTRAL STOCHASTIC DIFFERENTIAL EQUATIONS CONTROLLED BY THE TIME-VARYING DELAY FEEDBACK

*ZHOU Zhi-wei，SONG Rui-li

（Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, China）

There are many research results on the quantization of quantum entanglement, but many of the existing entanglement measures are still difficult to calculate. In the paper “Entanglement measures based on observable correlations”, LUO Shun-long proposed an observable correlation measure of bipartite quantum states based on mutual information, and obtained a class of entanglement measures of bipartite quantum states. In this paper, we generalize the entanglement measure of the bipartite system to the multipartite composite quantum system, and prove that it satisfies the necessary physical conditions of entanglement measure.

multipartite quantum system; quantum states; observable correlations; mutual information; entanglement measure

1674-8085（2022）03-0006-09

O175

A

10.3969/j.issn.1674-8085.2022.03.002

2021-12-01；

2022-01-25

國(guó)家自然科學(xué)基金項(xiàng)目(61773217)

*周之薇(1995-)，女，陜西西安人，碩士生，主要從事概率論與數(shù)理統(tǒng)計(jì)的研究(E-mail:412481190@qq.com).