# Asymptotical Stability of Riemann-Liouville Nonlinear Fractional Neutral Neural Networks with Time-Varying Delays

Authors: Erdal Korkmaz; Abdulhamit Ozdemir; Kenan Yildirim
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, the asymptotic stability of solutions is investigated for a class of nonlinear fractional neutral neural networks with time-dependent delays which are unbounded. By constructing the appropriate Lyapunov functional, sufficient conditions for asymptotic stability of neural networks are obtained with the help of LMI. An example is presented by using the LMI Toolbox to demonstrate the effectiveness of the obtained results.


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## Body

## 1. Introduction

Fractional calculus with a history of more than 300 years has been proved to be valuable tools in modeling many phenomena in the various fields of engineering, physics, control systems, diffusion, epidemic model, financial systems, and so on [1–3]. Fractional derivative, which is a generalization of the integer derivative, has different definitions such as Caputo, Riemann-Liouville, Grünwald-Letnikov, and Hadamard, which were defined by different researchers [3]. Qualitative properties such as stability, asymptotic stability, and exponential stability of the solutions of the equations with delay or without delay of phenomena modeled in various fields with fractional differential equations have been studied by many authors. In most of these analyses, sufficient conditions for qualitative properties have been obtained by using the Lyapunov-Krasovskii functional known as the energy function. Some of these studies used LMI because it is one of the most useful tools for showing the derivative of the Lyapunov functional to be less than zero [4]. In [5], making use of the stability theorem of fractional-order systems with multiple time delays, some fractional derivative inequalities, and comparison theorem, several sufficient criteria are established for confirming that the synchronization error of the concerned system can reach zero within a limited time. In [6–8], sufficient conditions are obtained for the stability of solutions of certain fourth-order differential equations by using Lyapunov’s second method. In [9–14], sufficient conditions were searched for qualitative properties such as stability, uniform stability, and asymptotic stability of solutions of fractional order delayed, undelayed, or neutral differential equations.The other popular topic of recent times is neural networks, known as a part of the human brain, which has been the subject of research for more than 1000 years. Artificial neural networks are an information processing technology inspired by the information processing technique of the human brain. Researchers can refer to the references and their references for detailed information on the work done on fractional order modeled neural networks. In [15–38], global asymptotic stability, global stability, and Mittag-Leffler stability analyses of solutions of neural network equations modeled in fractional order are reported, and valuable results are obtained.When the existing literature is examined, it is seen that the Riemann-Liouville derivative and Caputo derivative are mostly used among the definitions of fractional derivatives for fractional equations and systems of equations. A comparison between these two derivatives shows that the Caputo derivative is used more often because the initial conditions coincide with systems of integer order, which contributes to describing some well-understood properties of the physical process and makes it more applicable to real phenomena. On the other hand, the main advantage of the Riemann-Liouville derivative lies in the composition properties of the Riemann-Liouville fractional derivative and integral. Furthermore, the Riemann-Liouville derivative is a continuous operator of orderq [1–3].In this paper, our main goal is to build an appropriate Lyapunov functional to discuss the asymptotic stability of the Riemann-Liouville fractional-order neural networks with time-varying delays, using the mentioned advantage of the Riemann derivative. The acquired stability criteria are expressed as the matrix inequalities, which are also suitableand practicable to test the asymptotic stability of the addressed neural networks. Specifically, the original contribution of the present paper to literature is that by employing the Lyapunov functional method, sufficient conditions, for showing the asymptotic stability of solutions for a class of nonlinear fractional neutral neural networks with time-dependent unbounded delays, are derived. In order to indicate the validation of the obtained results, an example is presented. Obtained results show that conditions proved in this paper are sufficient for asymptotic stability of solutions of nonlinear fractional neutral neural networks with time-dependent unbounded delays.


## 2. Preliminaries

In this section, we introduce some fundamental definitions of fractional calculus together with important lemmas.Note thatn-dimensional Euclidean space is denoted by Rn. The set of entire n×n real matrices is indicated by Rn×n. The Euclidean norm of a real vector x is denoted by x. The spectral norm of matrix A is indicated by A. When A<0 (or A>0), the symmetric matrix A is negative definite (or positive definite).Definition 1.
The Riemann-Liouville fractional derivative and integral are described as the following, respectively,(1)t0Dtαxt=1Γn−αdndtn∫t0txst−sα+1−ndsn−1≤α<n,t0Dt−αxt=1Γα∫t0tt−sα−1xsdsα>0(see [3]).Lemma 2.
Ifα>β>0, then
(2)t0Dtβt0Dt−αxt=t0Dtβ−αxtholds for “sufficiently good” functions xt. Particularly, if xt is integrable, then this relation holds (see [2]).Lemma 3.
Ifxt∈Rn is a vector of differentiable function, then the following relationship satisfies
(3)12t0DtαxTtPxt≤xTtPt0Dtαxt,∀α∈0,1,∀t≥t0,where P∈Rn×n is a positive semidefinite, symmetric, square, and constant matrix (see [35]).Lemma 4.
For any positive definite matrixQ>0, scalar β>0, vector function f·:0,β⟶Rn such that the integrations concerned are well defined, the following inequality holds:
(4)∫0βfsdsTQ∫0βfsds≤β∫0βfTsQfsds(see [27]).


## 3. Main Results

Now, we present the stability of solutions of nonlinear fractional neutral neural networks having time-varying delays. We further consider linear matrix inequality to determine sufficient conditions on the asymptotical stability of solutions of these systems.Let the fractional nonlinear neutral neural system be given by the following:(5)0Dtαxt=Axt+B1f1t,xt+B2f2t,xt−τ1t+C0Dtαxt−τ2t+B3∫t−τ3ttf3s,xsds,where 0<α<1 is a real number, x=x1,x2,⋯,xnT∈Rn is the state vector, A,B1,B2,B3,C∈Rn×n are known constant matrices, for all t>t0, τ1t,τ2t,τ3t>0 are time-varying delays. fjt,xj=1,2,3 are vector-valued time-varying nonlinear functions with fjt,0=0 and satisfy the following Lipschitz condition for all t,x,t,x^∈R×Rn:
(6)fjt,x−fjt,x^≤ajMjx−x^,j=1,2,3,where Mj are constant matrices with appropriate dimension and aj are positive numbers. Consequently, from (6), we have
(7)fjt,x≤ajMjx,j=1,2,3.Theorem 5.
The trivial solution of system (5) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε, and there exist positive and symmetric definite matrices P,R, and S such that the following LMI satisfies:
(8)K=K11K12K13K14K15K12TK22K23K24K25K13TK23TK33K34K35K14TK24TK34TK44K45K15TK25TK35TK45TK55<0,where
(9)K11=ATP+PA+ATR+mSA+a12M1TM1+a22M2TM2+ε2a32M3TM3,K12=PB1+ATR+mSB1,K13=PB2+ATR+mSB2,K14=PC+ATR+mSC,K15=PB3+ATR+mSB3,K22=B1TR+mSB1−I,K23=B1TR+mSB2,K24=B1TR+mSC,K25=B1TR+mSB3,K33=B2TR+mSB2−1−d1I,K34=B2TR+mSC,K35=B2TR+mSB3,K44=CTR+mSC−1−d2R,K45=CTR+mSB3,K55=B3TR+mSB3−1−d3I,m and ε are positive constants and I unit matrix.Proof.
Let the Lyapunov-Krasovskii functional be defined by(10)Vt=t0Dtα−1xTtPxt+∫−τ2t0t0Dtαxt+sTRt0Dtαxt+sds+a22∫t−τ1ttxTsM2TM2xsds+∫t−mt∫θtt0Dtαxt+sTSt0Dsαxsdsdθ+εa32∫−τ3t0∫t+stxTηM3TM3xηdηds.
SinceP,R, and S matrices are positive definite, the Lyapunov-Krasovskii functional Vt is positive definite.
From (7) inequality,
(11)fjTt,xtfjt,xt≤aj2Mjx2=aj2xTtMjTMjxt,j=1,2,3,can be written. Hence, according to Lemma 4 and (11), for all fixed t, we can get
(12)−εa32∫t−τ3ttxTsM3TM3xsds≤−τ3ta32∫t−τ3ttxTsM3TM3xsds≤−τ3t∫t−τ3ttf3Ts,xsf3s,xsds≤−∫t−τ3ttf3s,xsdsTI∫t−τ3ttf3s,xsds.
From (12) and Lemmas 2 and 3, the derivative of Vt is obtained along the trajectories of system (5) as follows:
(13)V̇t=t0DtαxTtPxt+a12xTtM1TM1xt−a12xTtM1TM1xt+t0DtαxtTRt0Dtαxt−1−τ2′tt0Dtαxt−τ2tTRt0Dtαxt−τ2t+a22xTtM2TM2xt−1−τ1′ta22xTt−τ1tM2TM2xt−τ1t+mt0DtαxtTSt0Dtαxt−∫t−mtt0DsαxtTSt0Dsαxtds+εa32τ3txTtM3TM3xt−εa321−τ3′t∫t−τ3ttxTsM3TM3xsds≤2xTtPDtaxt+a12xTtM1TM1xt+a22xTtM2TM2xt+ε2a32xTtM3TM3xt+t0DtαxtTR+mSt0Dtαxt−1−d1f2Tt,xt−τ1tIf2t,xt−τ1t−f1Tt,xtIf1t,xt−1−d2t0Dtαxt−τ2tTRt0Dtαxt−τ2t−1−d3∫t−τ3ttf3s,xsdsTI∫t−τ3ttf3s,xsds=xTtATP+PA+a12M1TM1+a22M2TM2+ε2a32M3TM3+ATR+mSAxt+2xTtPB1f1t,xt+2xTtPB2f2t,xt−τ1t+2xTtPCt0Dtαxt−τ2t+2xTtPB3∫t−τ3ttf3s,xsds+xTtATR+mSB1f1t,xt+xTtATR+mSB2f2t,xt−τ1t+xTtATR+mSCt0Dtαxt−τ2t+xTtATR+mSB3∫t−τ3ttf3s,xsds+f1t,xtTB1TR+mSAxt+f1t,xtTB1TR+mSB1f1t,xt+f1t,xtTB1TR+mSB2f2t,xt−τ1t+f1t,xtTB1TR+mSCt0Dtαxt−τ2t+f1t,xtTB1TR+mSB3∫t−τ3ttf3s,xsds+f2t,xt−τ1tTB2TR+mSAxt+f2t,xt−τ1tTB2TR+mSB1f1t,xt+f2t,xt−τ1tTB2TR+mSB2f2t,xt−τ1t+f2t,xt−τ1tTB2TR+mSCt0Dtαxt−τ2t+f2t,xt−τ1tTB2TR+mSB3∫t−τ3ttf3s,xsds+t0Dtαxt−τ2tTCTR+mSAxt+t0Dtαxt−τ2tTCTR+mSB1f1t,xt+t0Dtαxt−τ2tTCTR+mSB2f2t,xt−τ1t+t0Dtαxt−τ2tTCTR+mSCt0Dtαxt−τ2t+t0Dtαxt−τ2tTCTR+mSB3∫t−τ3ttf3s,xsds+∫t−τ3ttf3s,xsdsTB3TR+mSAxt+∫t−τ3ttf3s,xsdsTB3TR+mSB1f1t,xt∫t−τ3ttf3s,xsdsTB3TR+mSB2f2t,xt−τ1t+∫t−τ3ttf3s,xsdsTB3TR+mSCt0Dtαxt−τ2t+∫t−τ3ttf3s,xsdsTB3TR+mSB3∫t−τ3ttf3s,xsds−f1Tt,xtIf1t,xt−1−d1f2Tt,xt−τ1tIf2t,xt−τ1t−1−d2t0Dtαxt−τ2tTRt0Dtαxt−τ2t−1−d3∫t−τ3ttf3s,xsdsTI∫t−τ3ttf3s,xsds.
Therefore, we can write(14)V̇t≤ξTKξ,where
(15)K=K11K12K13K14K15K12TK22K23K24K25K13TK23TK33K34K35K14TK24TK34TK44K45K15TK25TK35TK45TK55,K11=ATP+PA+ATR+mSA+a12M1TM1+a22M2TM2+ε2a32M3TM3,K12=PB1+ATR+mSB1,K13=PB2+ATR+mSB2,K14=PC+ATR+mSC,K15=PB3+ATR+mSB3,K22=B1TR+mSB1−I,K23=B1TR+mSB2,K24=B1TR+mSC,K25=B1TR+mSB3,K33=B2TR+mSB2−1−d1I,K34=B2TR+mSC,K35=B2TR+mSB3,K44=CTR+mSC−1−d2R,K45=CTR+mSB3,K55=B3TR+mSB3−1−d3I,ξ=xTt,f1Tt,xt,f2Tt,xt−τ1t,t0Dtαxt−τ2tT∫t−τ3ttf3s,xsdsTT.
From axiom (8) of the theorem V̇t is negative definite. The trivial solution of system (5) is asymptotically stable. This completes the proof.Theorem 6.
The trivial solution of system (5) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist positive and symmetric define matrices P,Q2 and R such that the following LMI satisfies:
(16)L=L11L12L13L14L15L12TL22L23L24L25L13TL23TL33L34L35L14TL24TL34TL44L45L15TL25TL35TL45TL55<0,where
(17)L11=PA+ATP+a12M1TM1+a22M2TM2+ε2a32M3TM3+Q2+mATRA,L12=PB1+mATRB1,L13=PB2+mATRB2,L14=−ATPC,L15=PB3+mATRB3,L22=mB1TRB1−I,L23=mB1TRB2,L24=−B1TPC,L25=mB1TRB3,L33=mB2TRB2−1−d1I,L34=−B2TPC,L35=mB2TRB3,L44=−1−d2Q2,L45=−CTPB3,L55=mB3TRB3−1−d3I,m and ε are positive constants and I unit matrix.Proof.
Let the Lyapunov-Krasovskii function be defined by(18)Vt=t0Dtα−1xt−Cxt−τ2tTPxt−Cxt−τ2t+∫t−τ2ttxTsQ2xsds+a22∫t−τ1ttxTsM2TM2xsds+εa32∫−τ3t0∫t+stxTηM3TM3xηdηds+∫t−mt∫θtt0Dsαxs−Cxs−τ2sTRt0Dsαxs−Cxs−τ2sdsdθ.
SinceP,Q2, and R matrices are positive definite, the functional Vt is positive definite. From (12) and Lemmas 2 and 3, the derivative of Vt is obtained along the trajectories of system (5) as follows:
(19)V̇t=t0Dtαxt−Cxt−τ2tTPxt−Cxt−τ2t+a12xTtM1TM1xt−a12xTtM1TM1xt+a22xTtM2TM2xt−1−τ1′ta22xTt−τ1tM2TM2xt−τ1t+xTtQ2xt−1−τ2′txTt−τ2tQ2xt−τ2t+εa32τ3txTtM3TM3xt−εa321−τ3′t∫t−τ3ttxTsM3TM3xsds+mt0Dtαxt−Cxt−τ2tTRt0Dtαxt−Cxt−τ2t−∫t−mtt0Dsαxt−Cxt−τ2tTRt0Dsαxt−Cxt−τ2tds≤2xt−Cxt−τ2tTPt0Dtαxt−Cxt−τ2t+a12xTtM1TM1xt+a22xTtM2TM2xt+ε2a32xTtM3TM3xt+xTtQ2xt+mt0Dtαxt−Cxt−τ2tTRt0Dtαxt−Cxt−τ2tT−f1Tt,xtIf1t,xt−1−d1f2Tt,xt−τ1tIf2t,xt−τ1t−1−d2xTt−τ2tQ2xt−τ2t−1−d3∫t−τ3ttf3s,xsdsTI∫t−τ3ttf3s,xsds=xTtPA+ATP+a12M1TM1+a22M2TM2+ε2a32M3TM3+mATRA+Q2xt−2xTt−τ2tCTPAxt+2xTtPB1f1t,xt−2xTt−τ2tCTPB1f1t,xt+2xTtPB2f2t,xt−τ1t−2xTt−τ2tCTPB2f2t,xt−τ1t+2xTtPB3∫t−τ3ttf3s,xsds−2xTt−τ2tCTPB3∫t−τ3ttf3s,xsds+mxTtATRB1f1t,xt+mxTtATRB2f2t,xt−τ1t+mxTtATRB3∫t−τ3ttf3s,xsds+mf1Tt,xtB1TRAxt+mf1Tt,xtB1TRB1f1t,xt+mf1Tt,xtB1TRB2f2t,xt−τ1t+mf1Tt,xtB1TRB3∫t−τ3ttf3s,xsds+mf2Tt,xt−τ1tB2TRAxt+mf2Tt,xt−τ1tB2TRB1f1t,xt+mf2Tt,xt−τ1tB2TRB2f2t,xt−τ1t+mf2Tt,xt−τ1tB2TRB3∫t−τ3ttf3s,xsds+m∫t−τ3ttf3s,xsdsTB3TRAxt+m∫t−τ3ttf3s,xsdsTB3TRB1f1t,xt+m∫t−τ3ttf3s,xsdsTB3TRB2f2t,xt−τ1t+m∫t−τ3ttf3s,xsdsTB3TRB3∫t−τ3ttf3s,xsds−f1Tt,xtIf1t,xt−1−d1f2Tt,xt−τ1tIf2t,xt−τ1t−1−d2xTt−τ2tQ2xt−τ2t−1−d3∫t−τ3ttf3s,xsdsTI∫t−τ3ttf3s,xsds.
Therefore, we can write(20)V̇t≤ξTLξ,where
(21)L=L11L12L13L14L15L12TL22L23L24L25L13TL23TL33L34L35L14TL24TL34TL44L45L15TL25TL35TL45TL55,L11=PA+ATP+a12M1TM1+a22M2TM2+ε2a32M3TM3+Q2+mATRA,L12=PB1+mATRB1,L13=PB2+mATRB2,L14=−ATPC,L15=PB3+mATRB3,L22=mB1TRB1−I,L23=mB1TRB2,L24=−B1TPC,L25=mB1TRB3,L33=mB2TRB2−1−d1I,L34=−B2TPC,L35=mB2TRB3,L44=−1−d2Q2,L45=−CTPB3,L55=mB3TRB3−1−d3I,ξ=xTt,f1Tt,xt,f2Tt,xt−τ1t,xTt−τ2t,∫t−τ3ttf3s,xsdsTT.
From the axiom (16) of the theorem V̇t is negative definite. The trivial solution of system (5) is asymptotically stable. This completes the proof.


## 4. Corollary

In this section, some results for the asymptotic stability of the solutions of various variations of the fractional neutral neural network (5) are expressed as a convex optimization problem. Two examples are given to show that the results obtained are applicable. The effectiveness of the results obtained with the help of these examples is discussed.Corollary 7.
The trivial solution of system (5) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12, a22, and a32 numbers and symmetric matrices P,R, and Ssuch that the following convex optimization problem on a12 number and P,R, and S matrices is solvable:
(22)minimize−a12subjecttoP>0,R>0,S>0,K<0,where K matrix defined by (8), m and ε are positive constants and I unit matrix.Corollary 8.
The trivial solution of system (5) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12, a22, and a32 numbers and symmetric matrices P,Q2, and R such that the following convex optimization problem on a12 number and P,Q2, and R matrices is solvable:
(23)minimize−a12subjecttoP>0,Q2>0,R>0,L<0,where L matrix defined by (16), m and ε are positive constants and I unit matrix.
Iff2t,xt−τ1t=xt−τ1t, then the nonlinear fractional neutral neural system (5) can be written as the following:
(24)t0Dtαxt=Axt+B1f1t,xt+B2xt−τ1t+C0Dtαxt−τ2t+B3∫t−τ3ttf3s,xsds,where 0<α<1 is a real number, x=x1,x2,⋯,xnT∈Rn is the state vector, A,B1,B2,B3,C∈Rn×n are known constant matrices, for all t>t0, τ1t,τ2t,τ3t>0 are time-varying delays. fjt,xj=1,3 are vector-valued time-varying nonlinear functions with fjt,0=0 and satisfy the following Lipschitz condition for all t,x∈R×Rn:
(25)fjt,x≤ajMjx,where Mj are constant matrices with appropriate dimension and aj are positive numbers.Corollary 9.
The trivial solution of system (24) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12 and a32 numbers and symmetric matrices P,Q1,R, and S such that the following convex optimization problem on a12 number and P,Q1,R, and S matrices is solvable:
(26)minimize−a12subjecttoP>0,Q1>0,R>0,S>0,Δ=Δ11Δ12Δ13Δ14Δ15Δ12TΔ22Δ23Δ24Δ25Δ13TΔ23TΔ33Δ34Δ35Δ14TΔ24TΔ34TΔ44Δ45Δ15TΔ25TΔ35TΔ45TΔ55<0,where
(27)Δ11=ATP+PA+ATR+mSA+a12M1TM1+Q1+ε2a32M3TM3,Δ12=PB1+ATR+mSB1,Δ13=PB2+ATR+mSB2,Δ14=PC+ATR+mSC,Δ15=PB3+ATR+mSB3,Δ22=B1TR+mSB1−I,Δ23=B1TR+mSB2,Δ24=B1TR+mSC,Δ25=B1TR+mSB3,Δ33=B2TR+mSB2−1−d1Q1,Δ34=B2TR+mSC,Δ35=B2TR+mSB3,Δ44=CTR+mSC−1−d2R,Δ45=CTR+mSB3,Δ55=B3TR+mSB3−1−d3I,m and ε are positive constants and I unit matrix.Corollary 10.
The trivial solution of system (24) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12 and a32 numbers and symmetric matrices P,Q1,Q2, and R such that the following convex optimization problem on a12 number and P,Q1,Q2, and R matrices is solvable:
(28)minimize−a12subjecttoP>0,Q1>0,Q2>0,R>0,Z=Z11Z12Z13Z14Z15Z12TZ22Z23Z24Z25Z13TZ23TZ33Z34Z35Z14TZ24TZ34TZ44Z45Z15TZ25TZ35TZ45TZ55<0,where
(29)Z11=PA+ATP+a12M1TM1+ε2a32M3TM3+Q1+Q2+mATRA,Z12=PB1+mATRB1,Z13=PB2+mATRB2,Z14=−ATPC,Z15=PB3+mATRB3,Z22=mB1TRB1−I,Z23=mB1TRB2,Z24=−B1TPC,Z25=mB1TRB3,Z33=mB2TRB2−1−d1Q1,Z34=−B2TPC,Z35=mB2TRB3,Z44=−1−d2Q2,Z45=−CTPB3,Z55=mB3TRB3−1−d3I,m and ε are positive constants and I unit matrix.
Iff3t,xt=xt, then the nonlinear fractional neutral neural system (5) can be written as the following:
(30)t0Dtαxt=Axt+B1f1t,xt+B2f2t,xt−τ1t+Ct0Dtαxt−τ2t+B3∫t−τ3ttxsds,where 0<α<1 is a real number, x=x1,x2,⋯,xnT∈Rn is the state vector, A,B1,B2,B3,C∈Rn×n are known constant matrices, for all t>t0, τ1t,τ2t,τ3t>0 are time-varying delays. fjt,xj=1,2 is vector-valued time-varying nonlinear function with fjt,0=0 and satisfies the following Lipschitz condition for all t,x∈R×Rn:
(31)fjt,x≤ajMjx,where Mj are constant matrices with appropriate dimension and aj are positive numbers.Corollary 11.
The trivial solution of system (30) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12 and a22 numbers and symmetric matrices P,Q3,R, and S such that the following convex optimization problem on a12 number and P,Q3,R, and S matrices is solvable:
(32)minimize−a12subjecttoP>0,Q3>0,R>0,S>0,O=O11O12O13O14O15O12TO22O23O24O25O13TO23TO33O34O35O14TO24TO34TO44O45O15TO25TO35TO45TO55<0,where
(33)O11=ATP+PA+ATR+mSA+a12M1TM1+a22M2TM2+ε2Q3,O12=PB1+ATR+mSB1,O13=PB2+ATR+mSB2,O14=PC+ATR+mSC,O15=PB3+ATR+mSB3,O22=B1TR+mSB1−I,O23=B1TR+mSB2,O24=B1TR+mSC,O25=B1TR+mSB3,O33=B2TR+mSB2−1−d1I,O34=B2TR+mSC,O35=B2TR+mSB3,O44=CTR+mSC−1−d2R,O45=CTR+mSB3,O55=B3TR+mSB3−1−d3Q3,m and ε are positive constants and I unit matrix.Corollary 12.
The trivial solution of system (30) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12 and a22 numbers and symmetric matrices P,Q2,Q3, and R such that the following convex optimization problem on a12 number and P,Q2,Q3, and R matrices is solvable:
(34)minimize−a12subjecttoP>0,Q2>0,Q3>0,R>0,Π=Π11Π12Π13Π14Π15Π12TΠ22Π23Π24Π25Π13TΠ23TΠ33Π34Π35Π14TΠ24TΠ34TΠ44Π45Π15TΠ25TΠ35TΠ45TΠ55<0,where
(35)Π11=PA+ATP+a12M1TM1+a22M2TM2+ε2Q3+Q2+mATRA,Π12=PB1+mATRB1,Π13=PB2+mATRB2,Π14=−ATPC,Π15=PB3+mATRB3,Π22=mB1TRB1−I,Π23=mB1TRB2,Π24=−B1TPC,Π25=mB1TRB3,Π33=mB2TRB2−1−d1I,Π34=−B2TPC,Π35=mB2TRB3,Π44=−1−d2Q2,Π45=−CTPB3,Π55=mB3TRB3−1−d3Q3,m and ε are positive constants and I unit matrix.
Iff2t,xt−τ1t=xt−τ1t and f3t,xt=xt, then the nonlinear fractional neutral neural system (5) can be written as the following:
(36)t0Dtαxt=Axt+B1f1t,xt+B2xt−τ1t+Ct0Dtαxt−τ2t+B3∫t−τ3ttxsds,where 0<α<1 is a real number, x=x1,x2,⋯,xnT∈Rn is the state vector, A,B1,B2,B3,C∈Rn×n are known constant matrices, for all t>t0, τ1t,τ2t,τ3t>0 are time-varying delays. f1t,x is vector-valued time-varying nonlinear function with f1t,0=0 and satisfies the following Lipschitz condition for all t,x∈R×Rn:
(37)f1t,x≤a1M1x,where M1 is a constant matrix with appropriate dimension and a1 is a positive number.Corollary 13.
The trivial solution of system (36) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12 number and symmetric matrices P,Q1,Q3,R, and S such that the following convex optimization problem on a12 number and P,Q1,Q3,R, and S matrices is solvable:
(38)minimize−a12subjecttoP>0,Q1>0,Q3>0,R>0,S>0,Σ=Σ11Σ12Σ13Σ14Σ15Σ12TΣ22Σ23Σ24Σ25Σ13TΣ23TΣ33Σ34Σ35Σ14TΣ24TΣ34TΣ44Σ45Σ15TΣ25TΣ35TΣ45TΣ55<0,where
(39)Σ11=ATP+PA+ATR+mSA+a12M1TM1+Q1+ε2Q3,Σ12=PB1+ATR+mSB1,Σ13=PB2+ATR+mSB2,Σ14=PC+ATR+mSC,Σ15=PB3+ATR+mSB3,Σ22=B1TR+mSB1−I,Σ23=B1TR+mSB2,Σ24=B1TR+mSC,Σ25=B1TR+mSB3,Σ33=B2TR+mSB2−1−d1Q1,Σ34=B2TR+mSC,Σ35=B2TR+mSB3,Σ44=CTR+mSC−1−d2R,Σ45=CTR+mSB3,Σ55=B3TR+mSB3−1−d3Q3,m and ε are positive constants and I unit matrix.Corollary 14.
The trivial solution of system (36) is asymptotically stable, if C<1, for all t>t0, τi′t≤di<1i=1,2,3, τ3t≤ε and there exist a12 number and symmetric matrices P,Q1,Q2,Q3, and R such that the following convex optimization problem on a12 number and P,Q1,Q2,Q3, and R matrices is solvable:
(40)minimize−a12subjecttoP>0,Q1>0,Q2>0,Q3>0,R>0,Y=Y11Y12Y13Y14Y15Y12TY22Y23Y24Y25Y13TY23TY33Y34Y35Y14TY24TY34TY44Y45Y15TY25TY35TY45TY55<0,where
(41)Y11=PA+ATP+a12M1TM1+ε2Q3+Q1+Q2+mATRA,Y12=PB1+mATRB1,Y13=PB2+mATRB2,Y14=−ATPC,Y15=PB3+mATRB3,Y22=mB1TRB1−I,Y23=mB1TRB2,Y24=−B1TPC,Y25=mB1TRB3,Y33=mB2TRB2−1−d1Q1,Y34=−B2TPC,Y35=mB2TRB3,Y44=−1−d2Q2,Y45=−CTPB3,Y55=mB3TRB3−1−d3Q3,m and ε are positive constants and I unit matrix.


## 5. Numerical Examples

Example 1.
Let the nonlinear fractional neutral neural system be given by(42)t0Dtαxt=Axt+B1f1t,xt+B2f2t,xt−τ1t+Ct0Dtαxt−τ2t+B3∫t−τ3ttf3s,xsds,where
(43)A=−40534−20112−20,B1=B2=B3=500050005,C=0.10000.20000.3,f1t,xt=a1sinx2tsinx1tsinx3t,f2t,xt−τ1t=0.4e−0.5tsinx1t−τ1te−0.3tsinx2t−τ1te−0.4tsinx3t−τ1t,f3t,xt=0.612sinx1t+x3tsinx2t12sinx1t−x3t,α∈1,0, τ1t=0.4t+0.2sint, τ2t=0.2t+0.3cost, and τ3t=0.5+0.04sint.
Forf1t,xt, we have
(44)f1t,xt2=a12sin2x2t+sin2x1t+sin2x3t≤a12x12t+x22t+x32t=a12xTtM1TM1xt,where M1=I3. For f2t,xt−τ1t, we have
(45)f2t,xt−τ1t2=0.16e−tsin2x2t−τ1t+e−0.6tsin2x1t−τ1t+e−0.8tsin2x3t−τ1t≤0.16x12t+x22t+x32t=0.16xTt−τ1tM2TM2xt−τ1t,where M2=I3. For f3t,xt, we have
(46)f3t,xt2=0.3612sin2x1t+x3t+sin2x1t+12sin2x1t−x3t≤0.3612x1t+x3t2+x22t+12x1t−x3t2=0.36x12t+x22t+x32t=0.36xTtM3TM3xt,where M3=I3.
Let us choosed1=0.6, d2=0.5, d3=0.04, ε=0.54, m=0.001. Let the P,Q1,Q2,Q3,R,S matrices be defined by
(47)P=p×I3,Q1=q1×I3,Q2=q2×I3,Q3=q3×I3,R=r×I3,S=s×I3,for positive numbers p,q1,q2,q3,r,s. amax is the solution of the convex optimization problem in the corresponding result; λmax is the largest eigenvalue of the matrix in the corresponding result, which must be negative definite.The solutions to the convex optimization problems in the conclusions obtained for the variations of the (42) system are presented in Table 1 to make it easier to compare the conclusions. Matlab-LMI Toolbox has been used to calculate the numerical solutions in this table.Table 1
Numerical solutions for Example1.
Corollariespq1q2q3rsamaxλmaxCorollary70.1249———0.00400.02831.3360−7.7393×10−7Corollary80.1515—0.5198—0.0817—1.3832−0.0059Corollary90.13371.1361——0.00430.01410.9778−4.7656×10−7Corollary100.16761.21120.5721—0.0467—1.0479−4.4837×10−5Corollary110.1326——1.34430.00430.01251.2761−1.1118×10−6Corollary120.1626—0.55691.42560.0641—1.3285−2.0478×10−4Corollary130.17151.4609—1.74410.00560.00880.9214−1.2880×10−6Corollary140.22251.61030.75811.92570.0473—1.0168−4.6914×10−5The tolerable bounds (amax) of the results obtained from Theorem 6 are greater than the tolerable bounds of the results obtained from Theorem 5, as shown in Table 1.Example 2.
Let the nonlinear fractional neutral neural system be given by(48)t0Dtαxt=Axt+B1f1t,xt+B2f2t,xt−τ1t+Ct0Dtαxt−τ2t+B3∫t−τ3ttf3s,xsds,where
(49)A=−65512−25131−80,B1=100020003,B2=400050006,B3=700080009,C=0.10000.20000.1,f1t,xt=a112sinx1t+x2tsinx3t12sinx1t−x2t,f2t−τ1t=0.5e−0.6tsinx1t−τ1te−0.5tsinx2t−τ1te−0.8tsinx3t−τ1t,f3t,xt=0.1512sinx1t+x3tsinx2t+x3t12sinx1t+x2t,α∈1,0, τ1t=30+0.2t+0.1cost, τ2t=10+0.3t+0.2sint, and τ3t=0.1+0.03sint. For f1t,xt, we have
(50)f1t,xt2=a1212sin2x1t+x2t+sin2x3t+12sin2x1t−x2t≤a1212x1t+x2t2+x32t+12x1t−x2t2=a12x12t+x22t+x32t=a12xTtM1TM1xt,where M1=I3. For f2t,xt−τ1t, we have
(51)f2t,xt−τ1t2=0.25e−1.2tsin2x2t−τ1t+e−tsin2x1t−τ1t+e−1.6tsin2x3t−τ1t≤0.25x12t+x22t+x32t=0.25xTt−τ1tM2TM2xt−τ1t,where M2=I3. For f3t,xt, we have
(52)f3t,xt2=0.0225sin2x1t+x3t+sin2x2t+x3t+sin2x1t+x2t=0.0225x1t+x3t2+x2t+x3t2+x1t+x2t2=0.09xTtM3TM3xt,where
(53)M3=461616164616161646.
Let us choosed1=0.3, d2=0.5, d3=0.03, ε=0.13, m=0.001. Let the P,Q1,Q2,Q3,R,S matrices be defined by
(54)P=p×I3,Q1=q1×I3,Q2=q2×I3,Q3=q3×I3,R=r×I3,S=s×I3,for positive numbers p,q1,q2,q3,r,s. amax is the solution of the convex optimization problem in the corresponding result; λmax is the largest eigenvalue of the matrix in the corresponding result, which must be negative definite.The solutions to the convex optimization problems in the conclusions obtained for the variations of the (48) system are presented in Table 2 to make it easier to compare the conclusions. Matlab-LMI Toolbox has been used to calculate the numerical solutions in this table.Table 2
Numerical solutions for Example2.
Corollariespq1q2q3rsamaxλmaxCorollary70.2034———0.00370.05912.1212−6.6719×10−6Corollary80.2347—0.4821—0.2406—2.2420−0.0010Corollary90.22621.4181——0.00420.02641.9674−1.8412×10−6Corollary100.26491.50660.5382—0.1582—2.1105−1.1406×10−4Corollary110.5133——33.28000.00930.09293.3488−1.6887×10−6Corollary120.5976—1.207536.03900.2756—3.5497−1.8091×10−4Corollary133.577822.4603—277.03930.06080.27337.6140−2.2252×10−6Corollary144.380525.19458.9098265.55473.0956—8.3108−1.1824×10−4The tolerable bounds (amax) of the results obtained from Theorem 6 are greater than the tolerable bounds of the results obtained from Theorem 5, as shown in Table 2.


## 6. Conclusion

The asymptotic stability of solutions for a class of nonlinear fractional neutral neural systems with time-dependent delays is obtained using the Lyapunov-Krasovskii method with linear matrix equation inequality. The obtained sufficient conditions are expressed in terms of LMI to find the less conservative criteria.



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*Source: 6832472-2022-09-17.xml*