A NEW CRITERION OF ASYMPTOTIC STABILITY FOR HOPFIELD NEURAL NETWORKS WITH TIME-VARYING DELAY Текст научной статьи по специальности «Компьютерные и информационные науки»

П ■ J. ■ ■ 2021. T. 25. № 6. C. 753-761

iPolytech Journal 2021;25(6):753 761-issn 2782^004 (print)

POWER ENGINEERING

Original article

https://doi.org/10.21285/1814-3520-2021-6-753-761

A new criterion of asymptotic stability for Hopfield neural networks with time-varying delay

Weiru Guo1H, Fang Liu2

1 9

l,2Central South University, Changsha, People's Republic of China

1 weiruguo@csu.edu.cn, http://orcid.org/0000-0002-7390-0400

2 csuliufang@csu.edu.cn, http://orcid.org/0000-0003-0750-8344

Abstract. The objective of this paper is to analyze the stability of Hopfield neural networks with time-varying delay. For the system to operate in a steady state, it is important to guarantee the stability of Hopfield neural networks with time-varying delay. The Lyapunov-Krasovsky functional method is the main method for investigating the stability of time-delayed systems. On the basis of this method, the stability of Hopfield neural networks with time-varying delay is analysed. It is known that due to such factors as communication time, limited switching speed of various active devices, time delays often arise in various technical systems, which significantly degrade the performance of the system, which can in turn lead to a complete loss of stability. In this regard, a Lyapunov-Krasovsky type delay-product functional was constructed in the paper, which allows more information about the time delay and reduces the conservatism of the method. Then a generalized integral inequality based on the free matrix was used. A new criterion for asymptotic stability of Hop-field neural networks with time-varying delay, which has less conservatism, was formulated. The effectiveness of the proposed method is illustrated. Thus an asymptotic stability criterion for Hopfield neural networks with time-varying delay was formulated and justified. The expanded Lyapunov-Krasovsky functional is constructed on the basis of delay and quadratic multiplicative functional, and the derivative of the functional is defined by a matrix integral inequality with free weights. The effectiveness of the method is illustrated by a model example.

Keywords: hopfield neural networks, asymptotical stability, LKF method, time-varying delay For citation: Guo Weiru, Liu Fang. A new criterion of asymptotic stability for Hopfield neural networks with time-varying delay. iPolytech Journal. 2021;25(6):753-761. (In Russ.). https://doi.org/10.21285/1814-3520-2021-6-753-761.

ЭНЕРГЕТИКА

Научная статья УДК 518.5:517.977.58

Новый критерий асимптотической устойчивости нейронных сетей Хопфилда с переменным запаздыванием

Вэйжу Го1н, Фан Лю2

1 2

• Центральный Южный университет, г. Чанша, Китайская Народная Республика

1 weiruguo@csu.edu.cn, http://orcid.org/0000-0002-7390-0400

2 csuliufang@csu.edu.cn, http://orcid.org/0000-0003-0750-8344

Резюме. Цель - анализ устойчивости нейронных сетей Хопфилда с изменяющейся во времени задержкой. Для того чтобы система могла работать в устойчивом состоянии, важно гарантировать устойчивость нейронных сетей Хопфилда с изменяющейся во времени задержкой. Метод функционала Ляпунова-Красовского является основным методом исследования устойчивости систем с временной задержкой. На основе данного метода в работе анализируется устойчивость нейронных сетей Хопфилда с изменяющейся во времени задержкой. Известно, что из-за таких факторов, как время связи, ограниченная скорость переключения различных активных устройств, в различных технических системах часто возникают временные задержки, которые существенно ухудшают работу системы, что может в свою очередь приводить к полной потере устойчивости. В связи с этим в работе был построен функционал Ляпунова-Красовского типа «delay-product», что позволяет использовать больше информации о временной задержке и уменьшать консерватизм метода. Затем было использовано обобщенное инте-

© Guo Weiru, Liu Fang, 2021 https://vestirgtu.elpub.ru -

гральное неравенство на основе свободной матрицы. Сформулирован новый критерий асимптотической устойчивости нейронных сетей Хопфилда с изменяющейся во времени задержкой, который обладает меньшим консерватизмом. Проиллюстрирована эффективность предложенного метода. Таким образом, в работе сформулирован и обоснован критерий асимптотической устойчивости для нейронных сетей Хопфилда с изменяющейся во времени задержкой. При этом расширенный функционал Ляпунова-Красовского строится на основе запаздывания и квадратичного мультипликативного функционала, а производная функционала определяется матричным интегральным неравенством со свободными весами. Эффективность метода иллюстрируется на модельном примере.

Ключевые слова: нейронные сети Хопфилда, асимптотическая устойчивость, метод функционала Ляпуно-ва-Красовского

Для цитирования: Го Вэйжу, Лю Фан. Новый критерий асимптотической устойчивости нейронных сетей Хопфилда с переменным запаздыванием // iPolytech Journal. 2021. Т. 25. № 6. С. 753-761. https://doi.org/10.21285/1814-3520-2021-6-753-761.

INTRODUCTION

It is well known that neural networks have many applications in the area of signal processing, pattern recognition etc. Hopfield neural network is one kind of neural networks given by J. J. Hopfield in 1982. In recent years, Hopfield neural networks (HNNs) has attracted an increasing attention since it has found many applications in classification of patterns, associative memory, image treatment, solving optimization problems and other areas [1 -6]. Especially, HNNs have been widely applicated in power system. For example, HNNs can solve the economic dispatch, which is a typical optimal problem in power system, and give a proper dispatch bringing great economic benefits. What's more, in the electric power network planning, HNNs can be used to select each load node's in-degree and direction of in-degree and the structure of distribution can be also decided. The problems of power flow can be solved by HNNs in the same time. However, the time-varying delay usually exists in the HNNs and it has a negative influence on system performance. It means the existence of time-varying delay will make the performance of the system to be worse and even make it instable. Therefore, the stability analysis of HNNs is a hot topic.

In the existing literature, the stability analysis of HNNs is often based on time-invariant delays or based on simple Lyapunov-Krasovskii functional in [7-12]. However, the conclusions are conservative due to the less information about delays in the LKF. Therefore, this article will analyze the stability of HNNs with timevarying delays based on the augmented Lyapunov-Krasovskii functional with delay-

product-type terms. [7] constructed a simple LKF for HNNs with time-invariant time delays, and obtained the stability criterion by using the time-delay segmentation method. The more time-delay segments, the lower the conservativeness, but the computational complexity also increased. [8] constructed an augmented LKF with more information about delays for HNNs with time-varying delays, and analyze its stability by the free weight matrix method. It still has room to decrease the conservativeness in terms of LKF construction and processing of functional derivatives. In terms of LKF construction, augmented LKF that contain more information of delays has been used widely. For example, [13] proposed two new LKFs with delay-product-type terms. The relationship between time delay and quadratic terms is changed from simple addition to multiplication. It can effectively reduce the conservatism of the conclusion. On the other hand, the derivative processing aspect of the functional is mainly changed from the free weight matrix method [14] to the integral inequality method, such as Jensen inequality [15], Wirtinger inequality [16], B-L inequality [17] , auxiliary function-based integral inequalities [18] and so on. The generalized free matrix integral inequality proposed in [19], which is based on Legendre series. By introducing some free matrices, the integral term can be estimated more tightly. Besides the two aspects mentioned above, how to find the condition that guarantees the negative definiteness of the derivative of LKFs is also important, especially when the derivative is a quadratic function with respect to the time-varying delay. The sufficient condition reported in [20] is commonly used but recently a

2021;25(6):753-761

relaxed quadratic function negativedetermination lemma is proposed in [21]. In this lemma, an adjustable parameter is introduced which provides potential to reduce the conservatism without much computational complexity.

In this paper, a suitable LKF is constructed based on the delay and quadratic multiplication LKF, and the derivative of the LKF is estimated by the integral inequality method and a relaxed quadratic function negative-determination lemma is employed to obtain the asymptotic stability criteria of HNNs. Finally, a numerical example is given to demonstrate the advantages and effectiveness of the proposed method.

Notation: The notation M" denotes the n-dimensional Euclidean space; p> o means that the matrix P is positive definite; I and 0 represent an appropriately dimensioned identify matrix and zero matrix respectively; * stands for the symmetric term in the symmetric matrix; the transpose and the inverse of a matrix are denoted by the superscripts T and -1; Sym{ X} = X + XT.

PROBLEM FORMULATION

Consider the following Hopfield neural networks(HNNs):

y(t) = -Mt) + Bg{y(t-T)) + u,

(1)

wheredzDzdenotes the neuron state vector;

h = [7/j,h2,...,hn]T e!" is a constant input

vector; A = diag {a, a2a }> 0 is a diagonal

matrix; B is the delayed connection weight matrix;

g(y(*)) = [g(y,(-)),g(y2(•)),...,g(y„('))f e

is the neuron activation function; z is the time delay.

Assumption 1: The neuron activation function in system (1) for and x^y

satisfies the following condition:

g,( x) - gt ( x)

0 <■

x - y

< Lj, j = 1,2,..., n, (2)

Let x = [x*,x2,...,x*]T be the equilibrium point of system (1). By the transformation _y(.) = *(•)-/,we can simplify the equation and the HNNs (1) is rewritten as:

x(t) = -Ax(t) + Bf (x(t - r))

(3)

where

f,(X,(.)) = gi(xi(.) + x* ) -gj{x, ),i = 1,2,...,*. (4)

that

By Assumption (1) and (4), it is easy to verify

f ( x )

0 < fj(j < Lj, f (0) =

= 0, Vx. e!j = 1,2,...,«.

(5)

The time-varying delay satisfies the following condition:

0<T<h,0<i<jU.

(5)

Lemma 1: Given a positive integer [19] N, an positive definite symmetric matrix iiel", M, eM" '"(/ = 0,1,2) and a vector ^er, for

any continuous differentiable function, the following inequality holds:

Ja

<£ [2CxiMkz+^cKR-lMka

t=n 2k +1

where

[ xr (ß) xr (a)]r,

[ xr (ß) xr (a) -1 $ ß-a ... -r1 ß-a

4 7 -7 ], N = 0

= l[ 7 (-1)i+17 X>k7■■■ 'O ], N > 0

N = 0

where L (j = 1,2,...,n) are positive constants.

rß

$ = I Lk (u )x(u)du

Ja

N

-(2i +1)(1 - (-1)*+г), i < к

0,

Lk (s) - (-1)к]С

l-0

(-1)

Г к Y к+1 ^

V У

i > к +1

l

s-а

V у.

Р-а.

Lemma 2: For a quadratic function [21]

f(y) = a2y2 + a,y+fl0, f(y) < 0 holds for y e[a,fij if the following holds for any given k g [0,1]:

f (а) < 0

f (Р) < 0

-к2a2 (Р-а)2 + f (а) < 0 ' -(1 -к)2 а2(Р-а)2 + f (Р) < 0

Lemma 3: Assuming that [22] (5) holds, then Jfi[f (s)-f (s)]ds <[fi-a][f (fi)- f (a)],i = 1,2,...,n.

ASYMPTOTIC STABILITY CRITERIA

In this section, a new delay-dependent asymptotic stability criterion for HNNs with time-varying delay is derived.

Theorem 1: Given a fixed kg[0,1], h , ^,

ju2, system (3) is asymptotically stable if there

exist symmetric matrix pt > 0(/ = 1,2,3),

q. > 0(i = 1,2,3,4), r > 0, matrix

A = diag [\,X2,...,\)> 0,

St = diag{sfl,Si2,...,sm}>0 (i = 1,2) and any matrix Mi,M, (i = 0,1,2), such that the following holds

Mi - col {Mi 0, M„, Mi 2};

M1 = col {M,, (1 - k)Mx , kM2 };

R, = diag{R, (1 - k)R, kR},

R - diag{R,3R, 5R};

e -[0„x(/-1)n , ' ,0„x(12-/)n Li -1,2,'",12;

nx(12-i)n -I

es --Ae1 + Be1

Y(t, f) = Т,(т, f) + Y 2(f) + Y 3(f) + £ Y,.;

i= 4

r1(t) = sym{nt1p1n21}

+тП[2^2П12 +rSym{ n[2 P2 П 22 ( -t nf3 Ръ п13 +rSym {п[з Ръ п23};

T2(t) = e(Q1e1-(l-t)eT2Q1e2 + + (\-T)e'2Q2e2-elQ2e,:

Y4 - Sym {e^Aes};

2

Y5 -heTsRes + 2Sym{gTEM + g^iETlkM2k};

к-0

Y(h, ,) hMT

* -hR

< 0;

yй.^+ОДЯ)-^ ш;

* -hR,

(6)

<0, (7)

where l -1,2; c -1,2; h - 0; h - h; , - 0 ; ^ - к; -1 - к; T - h - г;

Y6 - Sym{e\QAes + efLS1e10

-eT S e + eTLS e - eTS e }•

a2 = iNTxP2Nx +Sym{NTlP2U22} -tN2P3N2 + Sym {Л^Р3П23}

?

d - coi{ex, e, e}; d - col{e4, e5}, d - col{e6, e };

<

2021;25(6):753-761

d4 = col{es,(l-i)es,e9};

d5 = col{ex - (1 - i)e2, e1 - (1 - f)e4 - ie5};

d6=co!{(\-i)e2-e3,(l- f)e2 -e6 + te7};

nn = col {d\, zd2, rd3}, n21 = col{d4, d5, d6 };

V2(t) = JV (s)Qx(s)ds; V3(t) =f i2T(s)Q,l2(s)ds;

Jt — T

V4(t) = 224 J j f (s)ds;

ni2 = col {dx, Td2 }, n22 = col {d4, d5}; ni3 = col {di3 Td3}, n23 = col {d4, d6};

ns = co1 {es, eio} n4 = co1 {^ eii}; n5 = co1 {e9 , e3 , ei2 } , E10 = [1, —I, 0 0] ;

En = [I,I, —2I,0], E12 =[I, —1,0, —6I];

gi = co1 {e2, ^ e6, —e6 + 2e};

g2 = co1 {i e2, e4, —e4 + 2e5};

N = col {0,0,0, d2}, N = col {0,0,0, d3 };

£(t) = col{x(t), x(t — t), x(t — h),

Jt X<£)ds,Jt JtX(f)

Jt-T t Jt-TJ6 T

Tt—T x(s) V—Tit—T x(s)

Jt-h t •'t-h J6 T

x(t— T),x(t — h),

dsd6,

J t— XH ds, Jt-TJ—T xm dsd6

Jt-h T Jt-h J6 T x(t— T),x(t — h),

f(x(t)),f(x(t-r)),f(x(t-h))}.

Proof: Consider the following LKF candidate:

V (t) = 2V (t),

(8)

i=i

where

V (t) = tf (t )pn (t) + Ti£ (t )P2n2 (t) -+ (h—T)nT (t )Pn3 (t);

v5(t) = J° ^ x (s)Rx(s)dscld\

and

n (t) = col{x(t), x(t — t), x(t — h),

Jt xis) ds, Jt J^ dsd6,

Jt—t T Jt—tJ6 t

Jt—T xis) ds, Jt—T Jt—T xM dsd6};

Jt—h T ^t—h J6 T

n (t) = col{x(t), x(t — t), x(t — h).

x(s) ^ V fix(s)

T

J' xis) ds, J' J'

Jt—TT ' Jt—TJ6

dsd6};

n (t) = col{x(t), x(t — t), x(t — h),

J'—T x(£) ds, J'—T J'—T x^ dsd6}

Jt—h T Jt—h J6 T

and p > 0(/ = 1,2,3), Q > 0(/ = 1,2,3,4), R > 0, A > 0 which shows V(t) >e\\x||2 for a

sufficient small s> 0.

Then calculating the derivatives of V(t) defined in (9) and the derivatives of

V (t) (i = 1,2,...,5) are given by

V, (t) = 2/71 (Oift (0 + tr,l {t)P2ri2 (0 + +2r/72r (t)P2r/2 (t) - vif, (OP3//3 (0 + (9) (OP3//3 (t) = f (OT, (r, f

V2(t) = /(t)Qlx(s)-xT(t-T)(Ql-Q2)

x(t -T)-02x(t-h) = f (OKöa - e2 0,^2 (10)

+el01e1-elQ1e^(t)=f(t)T1(im.

V3 (0 = л I (т>ь (О - лI О - T)(Q3 - Q4)

ъ(t -г) - (t - h)Qtf2(t - h) -е (t ){nT Q3 П3-nT Q3 П4

=ew зшо-

V4(t) = 2f(x(t))hx(t)

(11)

= e(t)Sym{eTwAes}Z(t) -eT (t )Y ).

(12)

VAt) = hxT(t)Rx(t)- Г xT (t)Rx(t)ds. (13)

Jt-h

Based on lemma 1, Г xT {t)Rx{t)ds with

Jt-h

R > 0 can be estimated as:

- [' / (t)Rx(t)ds = - Г T xT (t)Rx(t)ds -

4=0

2k + 1

2

+Y[2d(t KM е)

к-0

+^T(t)MTuR-lMJ(t)]

2к +1

= ? (t) Sym {gTEM + gTXM2к} + Y5 W(t).

where

1 rt-t

Q (t) - col{x(t - t), x(t - h),— I x(s)ds

T it-h

Therefore,

v5(t)<e(t){Y5+Y5(T)}aty (14)

For matrices St =diag {sfl, si2,..., sn}>0 (i = 1,2), the following hold by using lemma 3:

0 < 2 xT (t)LSf (x(t)) - 2fT (x(t)) Sf (x(t)) + 2 xT (t -t) LSf (x(t -t) ) -2fT (x(t -t)) Sf (x(t -t)) = ? (t)Y«£(t).

(15)

It follows from (10)-(16) that

v(t)<e(t)(Y(T)+r5)at)=v(T). (16)

It is found that cr(r,r) is a quadratic function. Thus, based on lemma 2, the following holds:

^(0,0) < 0 m(h, 0) < 0

-к2a2h2 + ^(0,0) < 0

(17)

-(1 -к)2a2h +rn(h,0) < 0

ш(0,,) < 0

tu(h,,) < 0

-к2a2h2 + rn(0,,) < 0

-(1 - к)2 a2h2 + rn(h,,) < 0

(18)

1 rt-T 2 rt-T rt-T

x(s)ds + 37 I I x(s)dsdu};

T it-h T 2 it-his

2 rt-T Г t-T

1 г

Q (t) - col{x(t), x(t -t),— I x(s)ds,

T it-T

It follows that (18),(19) ^ (7),(8) by using Schur complement. This completes the proof. NUMERICAL EXAMPLES

In this section, a numerical example is given to demonstrate the effectiveness and advantages of the proposed method.

Example 1: Consider the following HNN (3) with time-varying delay (6):

1 rt 2 rt rt

— I x(s)ds + — I I x(u )duds}.

T i t-T T2 it-t i s

2 i' Г

T" J t-г Js

"0.8 0 " "0.1 0.3"

A - , B -

0 5.3 0.9 0.1

<

2021;25(6):753-761

L =

1 0 0 1

, ju = 0.9.

By the Theorem 1 of this paper, the maximum delay h that guarantees the asymptotic stability of HNN (3) is 51.3012 while it is 14.660 in [8]. So our result is less conservative.

» \ » \ \ — x,(t) *2(t>

\ * \ \

N

\

/ 1 j

1

5

Time (s)

10

State trajectories of the system of Example 1 Траектории состояний системы примера 1

Setting

x(0) = col{2, - 1}т = 0.9 sin t + 50.4012 f (x) = col{tanh (x (t)), tanh (x2 (t))} .

The responses of the HNN (3) with a time-varying delay when ju = 0.9, h = 51.3012 are shown in the figure above. The result indicates that the system is stable at its equilibrium points, which verifies the effectiveness of the proposed method.

CONCLUSION

In this paper, an augmented functional is constructed based on the delay and quadratic multiplicative functional, and the derivative of the functional is defined by the free-weight matrix integral inequality. We choose the relaxed quadratic function negative-determination lemma to deal with the quadratic function and obtain the stability criterion. Finally, a numerical example is given to prove the effectiveness and advantages of the proposed method in this paper.

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16. Seuret A., Gouaisbaut F. Wirtinger-based integral inequality: application to time-delay systems. Automatica. 2013;49(9):2860-2866.

https://doi.org/10.1016/j.automatica.2013.05.030.

17. Seuret A., Gouaisbaut F. Hierarchy of LMI conditions for the stability analysis of time-delay systems. Systems & Control Letters. 2015;81:1-7. https://doi.org/10.1016/j.sysconle.2015.03.007.

18. Park PooGyeon, Lee Won Il, Lee Seok Young. Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems. Journal

of the Franklin Institute. 2015;352(4):1378-1396. https://doi.org/10.1016/j.jfranklin.2015.01.004.

19. Zeng Hong-Bing, Liu Xiao-Gui, Wang Wei. A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems Applied Mathematics and Computation. 2019;354:1-8. https://doi.org/10.1016/j.amc.2019.02.009.

20. Kim Jin-Hoon. Further improvement of Jensen inequality and application to stability of time-delayed systems. Automatica. 2016;64:121-125. https://doi.org/10.1016/j.automatica.2015.08.025.

21. Zhang Chuan-Ke, Long Fei, He Yong, Yao Wei, Jiang Lin, Wu Min. A relaxed quadratic function negative-determination lemma and its application to time-delay systems. Automatica. 2020;113:108764. https://doi.org/10.1016/j.automatica.2019.108764.

22. Zhang Jinhui, Shi Peng, Qiu Jiqing. Novel robust stability criteria for uncertain stochastic Hopfield neural networks with time-varying delays. Nonlinear Analysis: Real World Applications. 2007;8(4):1349-1357. https://doi.org/10.1016Zj.nonrwa.2006.06.010.

23. Karamov D. N., Sidorov D. N., Muftahov I. R., Zhukov A. V., Liu F. Optimization of isolated power systems with renewables and storage batteries based on nonlinear Volterra models for the specially protected natural area of lake Baikal. Journal of Physics: Conference Series. 2021;1847(1):12037.

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16. Seuret A., Gouaisbaut F. Wirtinger-based integral inequality: application to time-delay systems // Automatica. 2013. Vol. 49. Iss. 9. P. 2860-2866. https://doi.org/10.1016/j.automatica.2013.05.030.

17. Seuret A., Gouaisbaut F. Hierarchy of LMI conditions for the stability analysis of time-delay systems // Systems & Control Letters. 2015. Vol. 81. P. 1-7. https://doi.org/10.1016/j.sysconle.2015.03.007.

18. Park PooGyeon, Lee Won Il, Lee Seok Young. Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems // Journal of the Franklin Institute. 2015. Vol. 352. Iss. 4. P. 1378-1396.

https://doi.org/10.1016/j.jfranklin.2015.01.004.

19. Zeng Hong-Bing, Liu Xiao-Gui, Wang Wei. A generalized free-matrix-based integral inequality for stability

INFORMATION ABOUT THE AUTHORS

Weiru Guo,

PhD student,

School of Automation

Central South University

Changsha 410083, Minzhu Building, People's

Republic of China

Fang Liu,

full professor,

School of Automation

Central South University

Changsha 410083, Minzhu Building, People's

Republic of China

Contribution of the authors

The authors contributed equally to this article.

Conflict of interests

The authors declare no conflicts of interests.

The final manuscript has been read and approved by all the co-authors.

Information about the article

The article was submitted 08.11.2021; approved after reviewing 02.12.2021; accepted for publication 29.12.2021.

analysis of time-varying delay systems // Applied Mathematics and Computation. 2019. Vol. 354. P. 1-8. https://doi.org/10.1016/j.amc.2019.02.009.

20. Kim Jin-Hoon. Further improvement of Jensen inequality and application to stability of time-delayed systems // Automatica. 2016. Vol. 64. P. 121-125. https://doi.org/10.1016/j.automatica.2015.08.025.

21. Zhang Chuan-Ke, Long Fei, He Yong, Yao Wei, Jiang Lin, Wu Min. A relaxed quadratic function negativedetermination lemma and its application to time-delay systems // Automatica. 2020. Vol. 113. P. 108764. https://doi.org/10.1016/j.automatica.2019.108764.

22. Zhang Jinhui, Shi Peng, Qiu Jiqing. Novel robust stability criteria for uncertain stochastic Hopfield neural networks with time-varying delays // Nonlinear Analysis: Real World Applications. 2007. Vol. 8. Iss. 4. P. 1349-1357. https://doi.org/10.1016/j.nonrwa.2006.06.010.

23. Karamov D. N., Sidorov D. N., Muftahov I. R., Zhukov A. V., Liu F. Optimization of isolated power systems with renewables and storage batteries based on nonlinear Volterra models for the specially protected natural area of lake Baikal // Journal of Physics: Conference Series. 2021. Vol. 1847. Iss.1. P. 12037.

ИНФОРМАЦИЯ ОБ АВТОРАХ

Вэйжу Го,

аспирант,

Школа автоматизации, Центральный Южный университет, 410083, г. Чанша, зд. Миньчжу, Китайская Народная Республика

Фан Лю,

профессор, Школа автоматизации, Центральный Южный университет, 410083, г. Чанша, зд. Миньчжу, Китайская Народная Республика

Вклад авторов

Все авторы сделали эквивалентный вклад в подготовку публикации.

Конфликт интересов

Авторы заявляют об отсутствии конфликта интересов.

Все авторы прочитали и одобрили окончательный вариант рукописи.

Информация о статье

Статья поступила в редакцию 08.11.2021; одобрена после рецензирования 02.12.2021; принята к публикации 29.12.2021.