# Stability Analysis Based on Caputo-Type Fractional-Order Quantum Neural Networks

Authors: Yumin Dong; Xiang Li; Wei Liao; Dong Hou
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, a quantum neural network with multilayer activation function is proposed by using multilayer Sigmoid function superposition and learning algorithm to adjust quantum interval. On this basis, the quasiuniform stability of fractional quantum neural networks with mixed delays is studied. According to the order of two different cases, the conditions of quasi uniform stability of networks are given by using the techniques of linear matrix inequality analysis, and the sufficiency of the conditions is proved. Finally, the feasibility of the conclusion is verified by experiments.


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## Body

## 1. Introduction

Fractional calculus is an arbitrary extension of integer calculus in order. It has strong advantages and wide application prospects in the fields of physics, chemistry, biology, economy, control, signal, and image processing. It has attracted extensive attention from scholars at home and abroad and has become one of the current research hotspots. In recent years, due to the continuous development of fractional differential equations, many researchers began to pay attention to the fractional-order theory, and the combination of fractional-order and neural network give full play to the advantages of fractional order. For example, literature [1–4] combined fractional order with neural network and achieved a good effect. Among them, Boroomand and Menhaj [4] presented the fractional-order Hopfield neural network model and studied its stability through the quasienergy function. [5–7] study on different fractional-order neural networks and explore the influence of different factors on fractional-order neural networks. This paper summarizes the synchronization problem of neural network [8–12]. Dominik et al. [13] considered discrete fractional-order artificial neural networks. Chaos and chaotic synchronization of fractional-order neural networks are proposed [14]. Literature [15, 16] explained and analyzed the dynamics of fractional-order neural networks. The fractional-order neural network was applied in different fields [17–21]. In recent years, the stability of fractional-order neural network system has become a research hotspot [22–31]. In reference [22], the stability and passivity of a memristor-based fractional-order competitive neural network (MBFOCNN) are analyzed by using Caputo’s fractional derivative. The effectiveness of the proposed results is finally verified by using analysis techniques and other computational tools. In reference [23], the problem of robust dissipation of Hopfield-type complex valued neural network (HTCVNN) model with time-varying delay and linear fractional uncertainty is studied, and many numerical models are designed to verify the results. In reference [24, 25], the global asymptotic stability of fractional quaternion numerical bidirectional associative memory neural networks (FQVBAMNNs) and fractional quaternion numerical memristic neural networks (FOQVMNNs) is studied. The effectiveness of the results is proved by using related methods. In reference [26, 27], the stability of fractional-order continuous time quaternion numerical leaky integral echo state neural network (NN) with multiple time-varying delays is studied, and the feasibility of the method is verified by numerical examples. In reference [28], the uniform stability of a fractional-order leaky integral echo state neural network (FOESN) with multiple delays is studied. The simulation results show the effectiveness of the method. Literature [32, 33] proposed the time-delay correlation study of Caputo fractional-order neural network. However, there are few studies on the behavior of fractional quantum neural networks with mixed delay. In this paper, a multilayer activation function quantum neural network model is presented, and the quasiuniform stability of fractional quantum neural networks with mixed delay is studied. It is proved by the formula and simulated by the numerical case.This article is organized as follows. In the second section, we give the structure of the multilayer activation function of the quantum neural network, based on which a fractional quantum neural network model with mixed delay is proposed. In the third section, it is proved that the fractional quantum neural network system with mixed time delay is quasiuniformly stable by corresponding definitions and lemmas. In the fourth section, a concrete example is given to verify the validity and applicability of the given results.


## 2. Model Composition and Preparation

### 2.1. Quantum Neural Network

Quantum neural network belongs to the feed-forward type of neural network [34, 35]. Compared with the traditional feed-forward type of neural network, the neurons in the hidden layer of quantum neural network refer to the idea of quantum state superposition in the quantum theory and carry out the linear superposition of several Sigmoid functions, which is called the multilayer activation function. Traditional activation functions can only represent two states and orders of magnitude. When quantized, a hidden layer neuron can represent more states and orders of magnitude.Each Sigmoid function superimposed has a different quantum interval. By adjusting the quantum interval, the data of different classes can be mapped to different orders of magnitude or steps, so that the classification can have more degrees of freedom. The quantum interval of the quantum neural network can be obtained by training. The uncertainty in the sampled data can be obtained and quantified by a quantum neural network with an appropriate learning algorithm.Figure1 shows a traditional three-layer feedforward neural network. Assume that the input layer I has n nodes, the output layer O has k nodes, and the number of nodes in the hidden layer H is m. Adjacent layer nodes are fully interconnected, and nodes of the same layer are not connected. The node output function in the hidden layer is
(1)Hr=fWTX−θr=1,2,⋯,u.Figure 1
Three-layer feedforward neural network graph structure.The output function of the node in the output layer is(2)Oi=fVTH−hj=1,2,⋯,n.In the formula,f adopts Sigmoid function, and W is the connection weight vector between each neuron in the input layer and each neuron in the hidden layer. V is the connection weight vector between each neuron in the hidden layer and each neuron in the output layer; θ is the threshold of the hidden layer, and h is the threshold of the unit of the output layer.Quantum neural networks with multiple excitation functions:(3)Hr=1n∑s=1nfUWTX−θs.In the formula:fx=1/1+exp−x, W is the network weight vector; X is the network input vector; U is the slope; WTX is the input excitation of the quantum neuron; θs is the quantum interval s=1,2,⋯,n.The learning of quantum neural network can be divided into two steps: (1) adjusting the weight to make the input data correspond to different class spaces; (2) adjust the quantum interval of quantum neurons in the hidden layer to reflect the uncertainty of data. The BP algorithm is used to adjust the weight. Once the network weight is obtained, the quantum interval can be adjusted by an appropriate algorithm [36]. The idea of the algorithm is to minimize the output change of the hidden layer neurons in the quantum neural network based on the same kind of sample data.Assume that for classCm, the output of the ith hidden layer neuron changes as:
(4)ei,m2=∑xk∑xk∈CmOi,m−Oi,k2,in the formula: Oi, k represents the output of the ith neuron in the hidden layer when the network input vector is xk;
(5)Oi,m=1Cm∑xixk∈CmOi,k.Cm in the formula represents the cardinality of class Cm. It can be seen that ei,m2 is a function of the quantum interval θs. By taking the derivative of θss=1,⋯,n on both sides of Equation (4) and finding the minimum value of ei,m2, the variation formula of θi,s (i.e., layer S of the ith neuron in the hidden layer) can be obtained.
(6)Δθi,s=ZUnS∑m=1xkk0∑xk∈CmOi,m−Oi,k∗Vi,m,s−Vi,k,s,(7)Vi,m,s=1Cm∑xk∑xk∈CmVi,k,s.In formula (6), Z is the learning rate; k0 is the number of nodes in the output layer, namely, the total number of classes; k is the number of quantum interval layers; xk: xk∈Cm represents all samples belonging to the Cm class.Among them:(8)Vi,k,s=Oi,k,s∗1−Oi,k,s.In the formula:Oi,k,s=sigU∗WTxk−θs represents the output of the sth quantum layer of the ith hidden layer neuron when the input vector is xks=1,2,⋯,n.


### 2.2. Caputo-Type Fractional Derivative Definition

In definition,fs is a continuous function over R, for any β>0; the β-order Caputo-type derivative of 0,fs is defined as:
(9)D0,sβfs=D0,s−n−βdndtnfs=1Γn−β∫0ss−τβ−1fnτdτ.The following corollary can be drawn:(1)
When0<β<1, D0,sβfs=1/Γ1−β∫0ss−τβ−1fτdτ(2)
Whenfs is a constant function, D0sβfs=0(3)
D−βDβfs=fs−∑−fm, β>0,especially,when 0<β<1, when fs(4)
is a one-dimensional function,D−βDβfs=fs−f0.(5)
Ifα and γ are two constants, then D−βαfs+γht=αD−βfs+γD−βht,β≥0


### 2.3. Fractional-Order Quantum Neural Network Model

Suppose the following two conclusions are true:(1)
If vectorx=xi and matrix A¯=aij, we define the Euclidean norm x of vector x to be x=∑xi. The matrix norm of the matrix A¯ is defined as A¯=max1≤i≤n∑aij. In this paper, we set C=C¯,A=A¯,B=B¯, and M=M¯(2)
The excitation functionsFx, Gx, and Hx of the fractional quantum neural network with mixed delay both satisfy the Lipschiz condition, that is, for any u,v∈R,u≠v, there exists a corresponding real number F,G,H>0, such that Fu−Fv≤Fu−v,Gu−Gv≤Gu−vThe fractional quantum neural network model with mixed time delay is shown below:(10)Dβxit=−cixit+∑aijfjxjt+∑bijgjxjt−τ+∑mij∫t−σthixiμdμ+Ii,xit=ψit,t∈−γ,0,γ∈maxτ,0,is converted to:
(11)Dβxt=−C¯xt+A¯Fxt+B¯Gxt−τ+M¯∫t−σtHμdμ+I,xt=ψt,t∈−γ,0,γ∈maxτ,σ.Among them,0<β<1,i=1,2,⋯,n, n represents the number of neurons in a fractional quantum neural network with mixed delay, and xt=x1t,x2t,⋯,xnt∈R is the state vector of the neuron at time t.Fxt=f1x1t,f2x2t,⋯,fnxnt,Gxt=g1x1t,g2x2t,⋯,gnxnt and Hxt=h1x1t,h2x2t,⋯,hnxntT are the activation function of fractional quantum neural network; C¯=diagci>0, A¯=aii, B¯=bij, and M¯=mij are all constant matrices; ci>0 represents the rate of the isolated resting state of the first neuron in the fractional-order quantum neural network in the state of unconnected and without external additional voltage difference; aj, bij, and mij represent the weight of the connection between the jth neuron and the ith neuron; τj and σi represent the transmission delay of the jth neuron along the axon; and I=I1t,I2t,⋯,IntT represents the external input and deviation of the neuron.Set the initial conditions of the system, usually assumingψis∈C−γ,0R,i∈N+, and the norm on C is defined as ψ=supψs.Assume thatxt and yt are two different solutions whose initial values of model (11) are ψ∈C and ϕ∈C, respectively, where ψ0=ϕ0=0; let φ=ψ−φ, xt=yt=et=e1t,e2t,⋯,entT can be obtained:
(12)Dβet=-C-t+A-Fxt-Fyt+B-Gxt-τ-Gyt-τ+M-∫t-σtHxμ-Hyμ·dμet=φt,t∈-γ,0,γ∈maxτ,σ,where φ∈C,φ0=0 is the initial condition of model (12), φ=sups∈−γ,0φs.


## 2.1. Quantum Neural Network

Quantum neural network belongs to the feed-forward type of neural network [34, 35]. Compared with the traditional feed-forward type of neural network, the neurons in the hidden layer of quantum neural network refer to the idea of quantum state superposition in the quantum theory and carry out the linear superposition of several Sigmoid functions, which is called the multilayer activation function. Traditional activation functions can only represent two states and orders of magnitude. When quantized, a hidden layer neuron can represent more states and orders of magnitude.Each Sigmoid function superimposed has a different quantum interval. By adjusting the quantum interval, the data of different classes can be mapped to different orders of magnitude or steps, so that the classification can have more degrees of freedom. The quantum interval of the quantum neural network can be obtained by training. The uncertainty in the sampled data can be obtained and quantified by a quantum neural network with an appropriate learning algorithm.Figure1 shows a traditional three-layer feedforward neural network. Assume that the input layer I has n nodes, the output layer O has k nodes, and the number of nodes in the hidden layer H is m. Adjacent layer nodes are fully interconnected, and nodes of the same layer are not connected. The node output function in the hidden layer is
(1)Hr=fWTX−θr=1,2,⋯,u.Figure 1
Three-layer feedforward neural network graph structure.The output function of the node in the output layer is(2)Oi=fVTH−hj=1,2,⋯,n.In the formula,f adopts Sigmoid function, and W is the connection weight vector between each neuron in the input layer and each neuron in the hidden layer. V is the connection weight vector between each neuron in the hidden layer and each neuron in the output layer; θ is the threshold of the hidden layer, and h is the threshold of the unit of the output layer.Quantum neural networks with multiple excitation functions:(3)Hr=1n∑s=1nfUWTX−θs.In the formula:fx=1/1+exp−x, W is the network weight vector; X is the network input vector; U is the slope; WTX is the input excitation of the quantum neuron; θs is the quantum interval s=1,2,⋯,n.The learning of quantum neural network can be divided into two steps: (1) adjusting the weight to make the input data correspond to different class spaces; (2) adjust the quantum interval of quantum neurons in the hidden layer to reflect the uncertainty of data. The BP algorithm is used to adjust the weight. Once the network weight is obtained, the quantum interval can be adjusted by an appropriate algorithm [36]. The idea of the algorithm is to minimize the output change of the hidden layer neurons in the quantum neural network based on the same kind of sample data.Assume that for classCm, the output of the ith hidden layer neuron changes as:
(4)ei,m2=∑xk∑xk∈CmOi,m−Oi,k2,in the formula: Oi, k represents the output of the ith neuron in the hidden layer when the network input vector is xk;
(5)Oi,m=1Cm∑xixk∈CmOi,k.Cm in the formula represents the cardinality of class Cm. It can be seen that ei,m2 is a function of the quantum interval θs. By taking the derivative of θss=1,⋯,n on both sides of Equation (4) and finding the minimum value of ei,m2, the variation formula of θi,s (i.e., layer S of the ith neuron in the hidden layer) can be obtained.
(6)Δθi,s=ZUnS∑m=1xkk0∑xk∈CmOi,m−Oi,k∗Vi,m,s−Vi,k,s,(7)Vi,m,s=1Cm∑xk∑xk∈CmVi,k,s.In formula (6), Z is the learning rate; k0 is the number of nodes in the output layer, namely, the total number of classes; k is the number of quantum interval layers; xk: xk∈Cm represents all samples belonging to the Cm class.Among them:(8)Vi,k,s=Oi,k,s∗1−Oi,k,s.In the formula:Oi,k,s=sigU∗WTxk−θs represents the output of the sth quantum layer of the ith hidden layer neuron when the input vector is xks=1,2,⋯,n.


## 2.2. Caputo-Type Fractional Derivative Definition

In definition,fs is a continuous function over R, for any β>0; the β-order Caputo-type derivative of 0,fs is defined as:
(9)D0,sβfs=D0,s−n−βdndtnfs=1Γn−β∫0ss−τβ−1fnτdτ.The following corollary can be drawn:(1)
When0<β<1, D0,sβfs=1/Γ1−β∫0ss−τβ−1fτdτ(2)
Whenfs is a constant function, D0sβfs=0(3)
D−βDβfs=fs−∑−fm, β>0,especially,when 0<β<1, when fs(4)
is a one-dimensional function,D−βDβfs=fs−f0.(5)
Ifα and γ are two constants, then D−βαfs+γht=αD−βfs+γD−βht,β≥0


## 2.3. Fractional-Order Quantum Neural Network Model

Suppose the following two conclusions are true:(1)
If vectorx=xi and matrix A¯=aij, we define the Euclidean norm x of vector x to be x=∑xi. The matrix norm of the matrix A¯ is defined as A¯=max1≤i≤n∑aij. In this paper, we set C=C¯,A=A¯,B=B¯, and M=M¯(2)
The excitation functionsFx, Gx, and Hx of the fractional quantum neural network with mixed delay both satisfy the Lipschiz condition, that is, for any u,v∈R,u≠v, there exists a corresponding real number F,G,H>0, such that Fu−Fv≤Fu−v,Gu−Gv≤Gu−vThe fractional quantum neural network model with mixed time delay is shown below:(10)Dβxit=−cixit+∑aijfjxjt+∑bijgjxjt−τ+∑mij∫t−σthixiμdμ+Ii,xit=ψit,t∈−γ,0,γ∈maxτ,0,is converted to:
(11)Dβxt=−C¯xt+A¯Fxt+B¯Gxt−τ+M¯∫t−σtHμdμ+I,xt=ψt,t∈−γ,0,γ∈maxτ,σ.Among them,0<β<1,i=1,2,⋯,n, n represents the number of neurons in a fractional quantum neural network with mixed delay, and xt=x1t,x2t,⋯,xnt∈R is the state vector of the neuron at time t.Fxt=f1x1t,f2x2t,⋯,fnxnt,Gxt=g1x1t,g2x2t,⋯,gnxnt and Hxt=h1x1t,h2x2t,⋯,hnxntT are the activation function of fractional quantum neural network; C¯=diagci>0, A¯=aii, B¯=bij, and M¯=mij are all constant matrices; ci>0 represents the rate of the isolated resting state of the first neuron in the fractional-order quantum neural network in the state of unconnected and without external additional voltage difference; aj, bij, and mij represent the weight of the connection between the jth neuron and the ith neuron; τj and σi represent the transmission delay of the jth neuron along the axon; and I=I1t,I2t,⋯,IntT represents the external input and deviation of the neuron.Set the initial conditions of the system, usually assumingψis∈C−γ,0R,i∈N+, and the norm on C is defined as ψ=supψs.Assume thatxt and yt are two different solutions whose initial values of model (11) are ψ∈C and ϕ∈C, respectively, where ψ0=ϕ0=0; let φ=ψ−φ, xt=yt=et=e1t,e2t,⋯,entT can be obtained:
(12)Dβet=-C-t+A-Fxt-Fyt+B-Gxt-τ-Gyt-τ+M-∫t-σtHxμ-Hyμ·dμet=φt,t∈-γ,0,γ∈maxτ,σ,where φ∈C,φ0=0 is the initial condition of model (12), φ=sups∈−γ,0φs.


## 3. Main Result

Relevant definition:Lemma 1.
Ifxs∈Cn0,+∞ and n−1<α,β<n∈Z+ then
(13)D−αD−βxs=D−α+βxs,α,β>0,DβD−βxs=xs,β≥0,D−βDβxs=xs−∑m=0n−1smm!xm,β≥0.Lemma 2 (Hölder inequality).
Suppose that the real numberp,q>1, and p, q satisfies 1/p+1/q=1, if f·p, h·q is a measurable function in space, and f,g:E⟶R satisfies ∫Efxdx<∞, ∫Egxdx<∞, then f·h· is also a measurable function and satisfies
(14)∫Efxhxdx≤∫Efxpdx1/p∫Ehxqdx1/q.
In particular, whenp=q=2, it is the inequality that we usually see. That is
(15)∫Efxhxdx≤∫Efx2dx1/2∫Ehx2dx1/2.Lemma 3.
Letk∈N,x1,x2,⋯,xk be a nonnegative real number, then it can be obtained for any
(16)∑i=1kxiη≤kη−1∑i=1kxiη.Lemma 4 (Gronwall inequality).
Ifxt,ft,gt≥0 is a continuous function on 0,T, T<∞ and satisfies the following inequality
(17)xt≤ft+∫0tgμxμdμ,t∈0,T.
Then, we can get(18)xt≤ft+∫0tgμfμexp∫μtgvdvdμ,t∈0,T.In special cases, ifft is a nonincreasing function, you can get
(19)xt≤ftexp∫0tgvdv,t∈0,T.Definition 5.
The initial time of the fractional quantum neural network system (11) with mixed delay is set to t0. For any ξ>0, there are two constants δ and T,0<δ<ξ,T>0, so that for any t∈J=t0,t0+T, when et0<δ has et<ξ, then the system (11) is called quasiuniformly stable.Theorem 6.
When the orderβ∈0.5,1 of the fractional quantum neural network system (11) with mixed delay is established, if the assumptions 2.3.1 and 2.3.2 are true and
(20)P+Qe2t+Wtewt+2tP1−e−2+Wtt2+Wt+Q1−e−WttWt<ξδt∈J,is true, where
(21)P=5−5M2H2γ2Γ2β−1Γ2β4β,N=5M2H22βΓβ,Q=5M2H2γ2Γ2β−1Γ2β4β+5B2G2Γ2β−11−e−2γΓ2β4β,L=10Γ2β−1C+AF2+B2G2e−2γΓ2β4β,Wt=L+Nt2β1−e−2t.
Then, the system (11) is quasiuniformly stable.Proof.
We set the initial timet0=0 of the error system (12), the initial condition is e0=φ0, and the expression of the solution of the error system can be obtained from Lemma 1 as
(22)et=φ0+D−β−C¯et+A¯Fxt−Fyt+B¯Gxt−τ−Gyt−τ+M¯∫t−σtHxμ−Hyμdμ=φ0+1Γβ∫0tt−μβ−1−C¯eμ+A¯Fxμ−Fyμ+B¯Gxμ−τ−Gyμ−τ+M¯∫t−δtHxs−Hysdsdμ.
From the hypotheses 1 and 2 and the basic properties of the norm, we can get(23)et≤φ0+1Γβ∫0tt−μβ−1Ceμ+AFeμ+BGeμ−τ+∫μ−σμMHesdsdμ≤φ0+1Γβ∫0tt−μβ−1C+AFeμdμ+1Γβ∫0tt−μβ−1BGeμ−τdμ+1Γβ∫0tt−μβ−1∫−σtMHesdsdμ≤φ+1Γβ∫0tt−μβ−1C+AFeμdμ+1Γβ∫0tt−μβ−1BGeμ−τdμMHtββΓβ∫0teμdμ+MHσφΓβ∫0tt−μβ−1dμ.
According to the Cauchy-Schwartz inequality in Lemma2, we know
(24)et≤φ+1Γβ∫0tt−μ2β−2e2μdμ1/2∫0tC+AF2∥eμ∥2e−2μdμ1/2+1Γβ∫0tt−μ2β−2e2μdμ1/2∫0tB2G2∥eμ−τ∥2e−2μdμ1/2+MHtββΓβ∫0te2μdμ1/2∫0t∥eμ∥2e−2μdμ1/2+MHσφΓβ∫0te2μt−μ2β−2dμ1/2∫0te−2μdμ1/2.
Bring(25)∫0tt−μ2β−2e2μdμ=∫0tz2β−2e2t−zdz=e2t∫0tz2β−2e−2zdz=e2t22β−1∫02tμ2β−2e−μdμ<2e2t4βΓ2β−1,into Equation (24) to get
(26)et≤φ+1Γβ2Γ2β−1e2t4β1/2∫0tC+AF2eμ2e−2μdμ1/2+1Γβ2Γ2β−1e2t4β1/2∫0tB2G2eμ−τ2e−2μdμ1/2+MHtββΓβe2t−121/2∫0teμ2e−2μdμ1/2+MHσφΓβ2Γ2β−1e2t4β1/21−e−2t21/2.
In Lemma3, let k=5,η=2, we can get
(27)et2≤5−5M2H2γ2Γ2β−1Γ2β4β+5M2H2σ2Γ2β−1Γ2β4β+5B2G2Γ2β−11−e2γΓ2β4βe2tφ2+10Γ2β−1C+AF2+B2G2e−2γe2tΓ2β4β+5M2H2t2βe2t−12βΓβ∫0teμ2e−2μdμ,P=5−5M2H2γ2Γ2β−1Γ2β4β,N=5M2H22βΓβ,Q=5M2H2γ2Γ2β−1Γ2β4β+5B2G2Γ2β−11−e−2γΓ2β4β,L=10Γ2β−1C+AF2+B2G2e−2γΓ2β4β,get
(28)et2e−2t≤Pe−2t+Qφ2+L+Nt2β1−e−2t∫0teμ2e−2μdμ.
Using Gronwall inequality and lettingWt=L+Nt2β1−e−2t, get
(29)et2e−2t≤Pe−2t+Q+WtetwtP1−e−2+Wtt2+Wt+Q1−e−WttWtφ2,so
(30)et2≤P+Qe2t+Wtewt+2tP1−e−2+Wtt2+Wt+Q1−e−WttWtφ2,that is
(31)et≤P+Qe2t+Wtewt+2tP1−e−2+Wtt2+Wt+Q1−e−WttWtφ.☐It can be seen that whenφ<δ,et<ξ is easy to know from Theorem 6. From Definition 5, it can be concluded that the fractional quantum neural system (11) with mixed time delay is quasiuniformly stable.Theorem 7.
If the orderβ∈0,0.5 of fractional-order quantum neural network system (11) with mixed delay is true, assuming that 1 and 2 are true and
(32)P~+Q~eqt+W~tP~ew~t+qt−1q+W~t+W~tQ~ew~t+qt1−e−WttW~tq<ξδt∈J,is true, where
(33)P~=5q−1,Q~=5q−1BqGqE~1−e−qγq+5q−1MqHqγqE~,L~=5q−1E~C+AFq+5q−1BqGqe−qγE~,E~=Γpβ−p+1Γpβppβ−p+1q/p,W~t=L+N~tqβ,N~=5q−1MqHqβqΓqβ,then the system (11) is quasiuniformly stable.Proof from Theorem6, we get
(34)et≤φ+1Γβ∫0tt−μβ−1C+AFeμdμ+1Γβ∫0tt−μβ−1BGeμ−τdμ+1Γβ∫0tt−μβ−1∫μ−σμMHesdsdμ,let p=1+β and p=1+1/β, obviously p,q>1, from Hölder’s inequality we can get
(35)et≤φ+1Γβ∫0tt−μpβ−pepμdμ1/p∫0tC+AFqeμqe−qμdμ1/q+1Γβ∫0tt−μpβ−pepμdμ1/p∫0tBqGq∥eμ−τ∥qe−qμdμ1/q+MHtβeptβΓβ∫0t∥eμ∥qe−qμdμ1/q+MHσ∥φ∥Γβ∫0tepμt−μpβ−pdμ1/q,because
(36)∫0tt−μpβ−pepμdμ=∫0tzpβ−pept−zdz=ept∫0tzpβ−pe−pzdz=eptppβ−p+1∫0ptμpβ−pe−μdμ<eptppβ−p+1Γpβ−p+1,so
(37)et≤φ+eptΓpβ−p+1Γpβppβ−p+11/p∫0tC+AFqeμqe−qμdμ1/q+eptΓpβ−p+1Γpβppβ−p+11/p∫0tBqGqeμ−τqe−qμdμ1/q+MHtβeptβΓβ∫0teμqe−qμdμ1/q+MHσφeptΓpβ−p+1Γpβppβ−p+11/p.In Lemma3, let k=5,η=q, we can get
(38)etq≤5q−1φq+5q−1eptΓpβ−p+1Γpβppβ−p+1q/p∫0tC+AFqeμqe−qμdμ+5q−1eptΓpβ−p+1Γpβppβ−p+1q/p∫0tBqGqeμ−τqe−qμdμ+5q−1MqHqtqβeptβqΓqβ∫0teμqe−qμdμ+5q−1MqHqσqφqeptΓpβ−p+1Γpβppβ−p+1q/p.Let(39)E~=Γpβ−p+1Γpβppβ−p+1q/p,get
(40)etq≤5q−1+5q−1MqHqγqE~eqt+5q−1BqGqE~1−e−qγeqtqφq+5q−1E~C+AFqeqt+5q−1MqHqtqβeqtβqΓqβ+5q−1BqGqe−qγE~eqt∫0teμqe−qμdμ,then let
(41)P~=5q−1,N~=5q−1MqHqβqΓqβ,Q~=5q−1BqGqE~1−e−qγq+5q−1MqHqγqE~,L~=5q−1E~C+AFq+5q−1BqGqe−qγE~,then
(42)etqe−qt≤P~e−qt+Qφq+L~+N~tqβ∫0teμqe−qμdμ.Use Gronwall inequality and makeW~t=L+N~tqβ, and get
(43)etq≤P~+Q~eqt+W~tP~ew~t+qt−1q+W~t+W~tQ~ew~t+qt1−e−WttW~tφq,which is
(44)et≤P~+Q~eqt+W~tP~ew~t+qt−1q+W~t+W~tQ~ew~t+qt1−e−WttW~tqφ.It can be seen that whenφ<δ, it is easy to know et<ξ from Theorem 6. From Definition 5, it can be obtained that the fractional quantum neural system (11) with mixed time delay is quasiuniformly stable.


## 4. Illustration

In this part, we give a specific example to verify the validity and applicability of the given results.(45)Dβx2t=−0.1x2t+0.1f1x1t−0.2f2x2t−0.2g1x1t−τ−0.lg2x2t−τ+∫t−σt0.1h1x1μ+0.2h2x2μdμ,Dβx1t=−0.1x1t+0.2f1x1t−0.1f2x2t−0.5g1x1t−τ−0.lg2x2t−τ+∫t−σt0.4h1x1μ−0.1h2x2μdμ.The activation function in the above formula is:fixit=gixit=hixit=sigmoidx,i=1,2⋯, F=H=G=1.ByB¯=−0.5−0.1−0.2−0.1, A¯=0.2−0.10.1−0.2, C¯=0.1000.1, and M¯=0.4−0.10.10.2, inferred B=0.7, A=0.3, C=0.1, and M=0.5.In this experiment, the experimental data show that by controlling the corresponding parameters, we can study the influence of another parameter on the trajectory ofx1 and x2 with different initial values. We set the parameters δ=0.1,σ=0.05,t0=0. Set the parameters when β=0.7, find L=2.7994, Q=0.2278, N=0.6878, Wt=3.0884, P=4.9938, and find T=0.6690 from the following inequality:
(46)P+Qe2t+WteWt+2tP1−e−Wt+2tWt+2+Q1−e−WttWt<εδ.Whenβ=0.3, find L~=83.7659,Q~=9.2269,N~=16.9440,W~t=84.1261,P~=213.7471,E~=2.4949. Find T=0.0522 from the following inequality.Figure2 is for σ=0.1, β=0.7, and x1t=0.5. For different τ values τ=0.1,0.5,0.8, the corresponding trajectory of x1. Figure 3 is for σ=0.1, β=0.7, and x2t=0.5. For different τ values τ=0.1,0.5,0.8, the corresponding trajectory of x2. It can be seen that the state trajectories of x1 and x2 converge to the equilibrium point.Figure 2
The trajectory graph ofx1 for different τ.Figure 3
The trajectory graph ofx2 for different τ.Figure4 shows the trajectory of x1 for τ=1, β=0.7, and the initial value x1(t) =0.5, for different values of σ=0.05, 0.2, 0.3. Figure 5 shows the trajectory of x1 for τ=1, β=0.7, the initial value x2(t) =0.5, for different values of σ=0.05,0.2,0.3 the trajectory of x2. It can be seen that the state trajectories of x1 and x2 converge to the equilibrium point.Figure 4
The trajectory graph ofx1 for different σ.Figure 5
The trajectory graph ofx2 for different σ.Figure6 shows the trajectory of x1 when β=0.7, σ=0.05, and τ=0.1, and the initial value x1t takes different values. Figure 7 shows the trajectory of x2 when β=0.7, σ=0.05, and τ=0.1, and the initial value x2t takes different values. It can be seen that the state trajectories of x1 and x2 converge to the equilibrium point.Figure 6
For the trajectory graph of different initial valuesx1.Figure 7
For the trajectory graph of different initial valuesx2.Figure8 shows the trajectory of x1 when β=0.3, σ=0.05, τ=0.1, and the initial value x1t takes different values. Figure 9 shows the trajectory of x2 when β=0.3, σ=0.05, and τ=0.1, and the initial value x2t takes different values. It can be seen that the state trajectories of x1 and x2 converge to the equilibrium point.Figure 8
For the trajectory graph of different initial valuesx1.Figure 9
For the trajectory graph of different initial valuesx2.


## 5. Conclusions

This paper uses the linear superposition of multilayer activation functions, uses learning algorithms to adjust quantum intervals and other operations to quantize the neural network, and proposes a quantum neural network model with multilayer activation functions. On this basis, the quasiuniform stability of fractional quantum neural networks with mixed time delays is studied. Whenβ belongs to different ranges, the sufficient conditions for the quasiuniform stability of the fractional quantum neural network system with mixed time delay are, respectively, discussed. Using the corresponding theorem, the proof of the theoretical result is given. Finally, through numerical simulation, the feasibility of the conclusions obtained in this paper is verified.



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*Source: 3820092-2021-06-28.xml*