# A Hierarchical Optimal Control of Uncertain and Time-Varying Knowledge Dissemination Model in Complex Network

Authors: Dan Xia; Jian Shen; Jun Mei; Si Zhang
Journal: Discrete Dynamics in Nature and Society
 (2023)

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## Abstract

In this paper, considering the multifactor influence of the knowledge dissemination process, a new ignorant-knower-spreader-forgetter (IKSF) knowledge dissemination model is proposed, which considers internalization mechanism, degradation mechanism, communication, and willingness, as well as time-varying and uncertain parameters. First, we prove that the knowledge loss equilibrium of the model is globally asymptotically stable when the basic reproduction numberR0<1, and knowledge is permanent when R0>1. Next, improving the willingness of knower individuals and reducing the knowledge degradation function of spreader individuals can make the best effect of propagation; a hierarchical control strategy is designed. At the upper layer, an effective optimal control mechanism of the IKSF knowledge dissemination model is studied to provide optimal control action and minimization costs. At the lower layer, to guarantee robustness control performance and track the control targets, an intervention optimal guaranteed cost control strategy for the IKSF knowledge dissemination system with uncertain parameters is studied. Converting the controller’s design problem into a minimization problem with linear matrix inequalities, not only the impact of uncertain parameters is reduced but also the propagation effect of the knowledge dissemination model is guaranteed. Simulation results confirm our method.


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## Body

## 1. Introduction

People acquire knowledge mainly to improve their own quality, but knowledge can not only improve economic production but also promote social prosperity. Knowledge is power. With the advent of the information age, knowledge can be flexibly used in our hands to create what we want; at this time, knowledge dissemination is essential. For example, knowledge dissemination plays an important role in social innovation and competition [1–3]. However, the process of knowledge transfer is very difficult [4, 5] to describe in the real world [6]. Therefore, how to describe an accurate transmission process, in reality, has been puzzling to researchers. To solve this problem, many researchers have introduced complex networks. Therefore, how to analyze complex networks becomes particularly important. Jun Hu reviews the latest state estimation schemes for complex dynamical networks, especially those over the networked environment [7]. Rai et al. provide a brief overview of complex networks, as well as measurements of several key related concepts, the structure and social impact of complex networks, and more [8].In any system, the mechanism plays a fundamental role. In the ideal state, with a suitable mechanism, even when the external conditions are uncertain changes, the system can automatically respond quickly and adjust the original strategy and measures to achieve the optimization goal. Mechanism originally refers to how a machine works, but it can mean different things in different fields. In the area of knowledge dissemination, the mechanism represents the internal law of knowledge dissemination. There are a lot of researchers who have studied this in depth; for example, Cao et al. [9] took the forgetting mechanism into account and developed a completely new model. Rozewski and Jankowski [10] integrated self-learning ability into the model and simulated it with a new method. Furthermore, Wang et al. found in the course of their research that human self-learning ability can spread knowledge more widely [11]. Chen and Chung [12] found that regular review can indeed preserve knowledge longer. According to the different ways of knowledge existence, Liao and Yi [13] used an internalization mechanism to analyze knowledge transmission in complex networks. So far, many internal mechanisms of knowledge dissemination have been considered, and they are incorporated into the model from the abstract space, which makes the model more vivid and concrete. However, the mechanisms do not exist in isolation. To work, different mechanisms must echo and complement each other. In recent years, many researchers have been studying multiple mechanisms. For example, in the field of biology, Warren et al. [14] found that the invasion of plant species in woodlands is jointly affected by multiple mechanisms. In the field of sociology, Tanner [15] found that language production is influenced by multiple mechanisms.Knowledge transmission is a complex behavior; most of the existing studies only focus on the influence of a certain mechanism in the dissemination process, such as the dissemination route [16], self-learning [11], forgetting mechanism, forgetting level [9], forgetting review [17], and internalization mechanisms [13]. However, in real life, knowledge dissemination is affected by various mechanisms. For example, knowledge receivers can acquire knowledge in various ways (self-learning, contact, and communication), and knowledge disseminators will also lose knowledge due to forgetting or losing interest. It may happen in real life. Therefore, it is necessary to comprehensively consider the process of knowledge dissemination in complex networks through various mechanisms. We investigate whether the interaction of multiple mechanisms affects the performance of knowledge dissemination according to the existing mechanisms. Specifically, an ignorant-knower-spreader-forgetter (IKSF) knowledge propagation model is proposed, which integrates existing knowledge propagation mechanisms and analyzes the dynamics in scale-free networks.The uncertainty of the system is generally divided into two kinds, namely, the uncertainty of the parameters and the uncertainty of the unknown function. The uncertainty of the parameters mainly means that the parameters will change over time, and the range of change is known, but the change in the system is unknown, for example, in engineering applications, air resistance, and coefficient of friction. The uncertainty created by these parameters generally does not affect the structure of the system. However, the uncertainty of unknown functions may lead to uncertainty in the system structure because they are affected by the external environment, resulting in their own structure not being determined. In uncertain systems, nonlinear systems tend to have uncertainties. This is because the models’ people designs are often different from the real models. In engineering applications, there will be many external interferences or errors that cause uncertainty to occur. Many researchers today are studying the uncertainty of nonlinear systems. Wang et al. [18] studied the tracking control of uncertain nonlinear systems under neural networks and proposed a new adaptive critical framework. Souzanchi-K and Akbarzadeh-T [19] studied the impedance control of emotional learning in the brain with time delay, integrating emotional learning into an uncertain system. Haddad and Mirkin [20] applied adaptive tracking for uncertain nonlinear systems to urban traffic.However, in the field of knowledge dissemination, the default parameters of the previous knowledge dissemination model are constant, but in real life, their values changed over time. When there are uncertain parameters in the system, whether the system is still stable needs to enter an in-depth study. To remedy this defect, we introduce a feedback control strategy to solve the influence of system parameter disturbance on system stability [21]. The introduction of control will also bring a specific cost and the investment of a large number of charges may not be worth the loss, but the optimal control technology can effectively solve this problem. Optimal control [22] is simply to optimize the system by adjusting parameters on the premise of accurately describing the system. Optimal control has been applied in many fields, such as simulated annealing algorithms, genetic algorithms, predictive control, and chaotic optimal control. In epidemiology, optimal control techniques can be used to reduce viral infections [23–25]. Different from epidemiology, the optimal control in knowledge dissemination [26] is mainly to speed up the dissemination of knowledge while reducing control costs. To deal with uncertain parameters, the guaranteed cost control (GCC) technology [21] can not only solve the stability of the control system but also make the control system have a good performance level. The proposal of GCC solves many practical problems, and in recent years, it has also been deeply studied [27–31]. For example, Chen et al. [31] added cost control to the gene regulation network to make it more precise. Li et al. [30] imposed cost control on the high-speed travel of the train to make its description more realistic. In the study, we found that knowledge dissemination also has the problem of cost control, which is because the current communication efficiency often relies on a large amount of cost input to improve. People often do not consider the cost for the sake of high efficiency, which is effective but not the best choice.There are other recent works on the robust control of uncertain systems using the LMI approach. Turki et al. [32] adopted a state-feedback controller to robustly control the motion of a 1-DoF double-side impact mechanical oscillator subject to norm-bounded parametric uncertainties. Gritli [33] studied the robust calming problem of a class of continuous-time nonlinear systems and applied it to a simple helicopter model. He also addressed the problem of static output feedback (SOF) stabilization for continuous-time linear systems subject to norm-bounded parameter uncertainties using the LMI approach [34]. Badri and Sojoodi investigated the robust stability and stabilization analysis of interval fractional-order systems with time-varying delay [35]. Moradi et al. proposed an offline robust model predictive control (RMPC) method based on LMIs to solve vehicle suspension design problems [36]. Nodozi and Rahmani proposed a LMI approach to mixed-integer model predictive control (MPC) of uncertain hybrid systems with binary and real-valued control inputs [37]. These studies have great implications for our application in the field of knowledge dissemination.Through the above description, this paper integrates the existing mechanism, studies the IKSF propagation model under multiple mechanisms, also considers the impact of uncertain parameters on the system, and optimizes the control.The innovations of this paper are as follows:(1)
The model incorporates self-learning mechanisms, forgetting mechanisms self-learning ability, communication, willingness, internalization, forgetting level, etc. At the same time, the prerequisites for the dissemination of knowledge are derived.(2)
The time-varying and uncertain parameters are taken into account in the IKSF model.(3)
Using the optimal control theory to improve the above given model, the purpose of hierarchical optimal control is to speed up the spread of knowledge with the minimum resources consumed. The optimal guaranteed cost control technology is employed to guarantee the stability and robust performance level of the IKSF model under uncertain parameters.Other contents in this article are as follows. In Section2, we mainly describe the propagation mechanism of the IKSF model. In Section 3, we conduct theoretical research on the IKSF model, including the conditions for judging whether knowledge can be disseminated, and the study of the equilibrium point. Section 4 mainly discusses optimal control strategies, namely, optimal control and optimal cost control. Section 5 primarily validates the conclusions we made in the previous sections, including the simulation of the results of the validation theory, the study of the influence of different parameters on the four groups of people, and the application of optimal control and optimal cost control of uncertain parameters. Finally, the summary is contained in Section 6.


## 2. Description

The design of network topology and the interaction rules [10] that drive knowledge diffusion are two important dimensions of knowledge diffusion. Generally speaking, a knowledge transmission model starts with ignorant and spreader. Ignorant individuals are learning individuals who resemble susceptible populations in the mode of viral transmission. Spreader individuals are communicators who possess the knowledge and share it with their neighbors. Since knowledge exists in different ways, the ignorant individual absorbs and digests the knowledge first after being spread by knowledge, whether he/she is willing to spread it or not. Individuals who become the third type after absorbing knowledge are called knowers. Knower refers to an individual who possesses the knowledge and has the ability to share knowledge. The acquisition of knowledge can bring benefits to people [38]; it can be supposed that ignorant people like to learn knowledge, but knowers may be motivated by external conditions to share knowledge. Spreader individuals may forget knowledge or lose interest in knowledge in the interference of themselves or external factors and then degenerate into forgetters; forgetters can regain knowledge through self-learning and communication.Next, we will describe in detail the propagation mechanism of the IKSF model. When an ignorant individual contacts a spreader individual, the ignorant individual will change to the state of the knower individual with probabilityλ1. Ignorant individuals can acquire knowledge through instinctive self-learning abilities; μ1 represents the average self-learning ability of ignorant individuals. The knower shares knowledge with probability δ in response to an external stimulus. Since knower individuals will not remember their knowledge forever unless they use knowledge, if knower individuals may lose their willingness to share knowledge, they will degenerate into ignorant individuals with probability ε. The probability that a knowledge disseminator will degenerate into an ignorant individual due to himself/herself or external factors is γ. At the same time, the forgetter individual can regain knowledge through interaction with the neighbors or self-learning and return to the state of the knower individual. Let λ2 represent the probability of the forgetter individual’s secondary interaction and use μ2 to represent the self-learning rate of the forgetter individual. The specific details of the IKSF model are shown in Figure 1. In general, all parameters defining a procedure are nonnegative and λ1≠μ1,λ2≠μ2.Figure 1
Structure of the IKSF knowledge transmission process.Suppose the system is on a scale-free network with degree distributionpk, and the average network degree is k=∑kpk,k=1,2,…,kmax. Let Ikt,Kkt,Skt,Fkt denote the density of ignorant, knower, spreader, and forgetter individuals with degrees k at time t, respectively. The probability that an ignorant person is infected by the people around him/her at time t is Θt, and Θt is represented as(1)Θt=∑kkpkSktk,k=1,2,…,kmax,where kmax represents the maximum degree of the node.Applying the mean field theory, the IKSF model can be expressed as follows:(2)dIktdt=−λ1k+μ1IktΘt+εKkt,dKktdt=λ1k+μ1IktΘt+λ2k+μ2Fkt−δKkt−εKkt,dSktdt=δKkt−γSkt,dFktdt=γSkt−λ2k+μ2Fkt,where 0≤Ikt,Kkt,Skt,Fkt≤1.For the above model, we analyze the multimechanism fusion of the knowledge transfer process and model it, considering the internalization mechanism, degradation mechanism, communication, willingness, and self-learning, respectively.Remark 1.
In the existing research study, knowledge diffusion models mainly focus on the influence of a certain mechanism, such as self-learning [11], internalization [13], forgetting [9], and review [17]. However, they did not study these mechanisms in a single model. Knowledge dissemination is affected by these mechanisms simultaneously. Therefore, we should fully consider the impact of multiple mechanisms on the model.


## 3. Theoretical Analysis

### 3.1. Equilibrium Point and Basic Reproduction Number

Theorem 1.
There are two equilibrium points for system (2); if R0<1, the system has a knowledge loss equilibrium point E0=I10,K10,S10,F10,…,Ikmax0,Kkmax0,Skmax0,Fkmax0, where Ik0=1,Kk0=0,Sk0=0,Fk0=0; if R0>1, there is a knowledge-endemic equilibrium in the system E∗=I1∗,K1∗,S1∗,F1∗,…,Ikmax∗,Kkmax∗,Skmax∗,Fkmax∗, where(3)Ik∗=λ2k+μ2εγλ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Kk∗=λ1k+μ1λ2k+μ2γΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Sk∗=λ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Fk∗=λ1k+μ1γδΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,and Θ∗t=∑kkpkSkt/k is the only positive root of the following equation:(4)fΘ∗=Θ∗−∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ.Proof.
SubstitutingE0=1,0,0,0,…,1,0,0,0 into equation (2), it is easy to see that E0 is the solution to equation (2). Next to prove the equilibrium of E∗, according to the stability condition of system (2), we obtain(5)−λ1k+μ1IktΘt+εKkt=0,λ1k+μ1IktΘt+λ2k+μ2Fkt−δKkt−εKkt=0,δKkt−γSkt=0,γSkt−λ2k+μ2Fkt=0.
The solution in (5) can be obtained as(6)Ik∗=λ2k+μ2εγλ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Kk∗=λ1k+μ1λ2k+μ2γΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Sk∗=λ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Fk∗=λ1k+μ1γδΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ.
Substituting the third equation of (3) into (1), Θ∗ can be converted to the following form:(7)Θ∗=∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ.
It can be seen thatHTMLtranslationfailed is the solution of system (2); next, we discuss the nontrivial solution of Θ∗∈0,1 in system (2).
Defining(8)fΘ∗=Θ∗−∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,it is easy to obtain(9)dfΘ∗dΘ∗=1−1k∑kpkABδBγ+δ+γδAΘ∗+Bεδ−Bγ+δ+γδA2BδΘ∗Bγ+δ+γδAΘ∗+Bεγ2,where A=λ1k+μ1andB=λ2k+μ2.
It is easy to see thatdf2Θ∗/dΘ∗2>0 and satisfied f0=0, while(10)f1=1−∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ>0,k=1,2,…,kmax.
To allow the existence of an extraordinary solution, the inequality(11)dfΘ∗dΘ∗=1−1k∑kpkAδεγ<0,must be satisfied.
Therefore,(12)R0=δεγλ1k2k−1+μ11−1k.
To sum up, whenR0<1, Θ∗ has no nontrivial solution at (0, 1) (see Figure 2(a)). When R0>1, Θ∗ has only one nontrivial solution at (0, 1) (see Figure 2(b)).
Then, we have the following: whenR0<1, system (2) has a knowledge-free equilibrium E0=1,0,0,0,…,1,0,0,0; when R0>1, system (2) has a knowledge-endemic equilibrium E∗=I1∗,K1∗,S1∗,F1∗,…,Ikmax∗,Kkmax∗,Skmax∗,Fkmax∗.Figure 2
Diagram of solutions forfΘ in different cases.
(a)(b)


### 3.2. Stability Analysis

Theorem 2.
IfR0<1, then the knowledge loss equilibrium E0 of system (2) is globally asymptotically stable.Proof.
For a sufficiently smallε1>0, there exists t1>0 such that when t>t1, there is Ikt≤Ik0+ε1, where Ik0=1.
The second formula of system (2) has dKkt/dt≤λ1k+μ1ΘtIk0+ε1−εK+kt,t⟶∞.
Forε2>0 small enough, there exists t2>0 such that when t>t2, Kkt≤λ1k+μ1ΘtIk0+ε1/ε+ε2.
Whent>maxt1,t2, we have Skt/dt≤δλ1k+μ1ΘtIk0+ε1/ε+ε2δ−γSkt.
This is enough to prove that the positive solution of the following auxiliary system tends to zero ast tends to infinity:(13)dS˜ktdt=δλ1k+μ1ΘtIk0+ε1ε+ε2δ−γSkt,where Θ˜t=∑kkpkS˜kt/k.
LetS˜k0=Sk0; there is S˜k0≤Sk0.
The Lyapunov function is defined as follows:Lt=∑kkpkS˜kt/k.
Lt is derived along system (13) to obtain(14)L′t=1k∑kkpkδλ1k+μ1ΘtIk0+ε1ε+ε2δ−γ≤γΘ˜t1k∑kkpkδλ1k+μ11+ε1ε−1+ε2δ1−1k=γΘ˜tR01+ε1−1+ε2δ1−1k.
ForR0<1, when t⟶∞, ε1,ε2 is small enough to make L′t<0. Guaranteed dLt/dt≤0 for all S˜kt≥0 and S˜kt=0 only if dLt/dt=0. It can be known from the comparison theorem that when t tends to infinity, the solution of system (2) tends to 0, that is,(15)limt⟶+∞Skt=0.
The proof of Theorem2 is completed.Theorem 3.
IfR0>1, knowledge will always be preserved; i.e., there is a constant ζ>0 such that(16)limt⟶+∞infSt=limt⟶+∞inf∑kkpkSkt>ζ.Proof.
Using the results in [39], we can get the knowledge is permanent; Liu and Zhang [40] provide the relevant theorem.
Defining(17)X=I1,K1,S1,F1,…,Ikmax,Kkmax,Skmax,Fkmax:Ik,Kk,Sk,Fk≥0,Ik+Kk+Sk+Fk=1,X0=I1,K1,S1,F1,…,Ikmax,Kkmax,Skmax,Fkmax∈X:∑kpkSk>0,∂X0=X\X0,where k=1,2,..,kmax, X is the feasible region of system (2).
Clearly,X is positively invariant to system (2). From the third equation of system (2), ∑kpkSkt′≥−γ∑kpkSkt is easily obtained.
Because of∑kkpkSkt>0, it can be seen that ∑kpkSkt≥∑kpkSk0e−γt>0 indicates that X0 is positive and invariant, and there exists a compact set B where all solutions initiated by system (2) in X go into B and exist forever; for this set B, this condition is easily verified in [39].
Defining the following set,(18)M∂=Ik0,Kk0,Sk0,Fk0:Ikt,Kkt,Skt,Fkt∈∂X0,k=1,2,…,kmax,t≥0,Ω=∪ωy,y∈M∂,where ωy is the limit set of model (2).
According toM∂, we rewrite the system:(19)Ikt=εKkt,Kkt=λ2k+μ2Fkt−δKkt−εKkt,Skt=δKkt−γSkt,Fkt=γSkt−λ2k+μ2Fkt.
It is easy to show that system (19) has a unique equilibrium E0 on X. Hence, E0 is the only equilibrium of system (2) in M∂. Since (19) is a linear system, it is easy to prove that E0 is globally asymptotically stable. Hence, Ω=E0, where E0 is a cover of Ω, and E0 is isolated acyclic. Next, we proved that E0 is the weak repulsion force of X0, that is,(20)limt⟶+∞supdistIkt,Kkt,Skt,Fkt,E0>0,where Ikt,Kkt,Skt,Fkt,E0 is any solution with the initial value on X0. According to Lemma 3.5 in [41], it is only necessary to prove WsE0∩X0=∅, where WsE0 is the stable manifold of E0. By contradiction, assuming it is incorrect, then there is a solution Ikt,Kkt,Skt,Fkt,E0∈X0, such that(21)Ikt⟶1,Kkt⟶0,Skt⟶0,Fkt⟶0,ast⟶∞.
ForR0=δ/εγλ1k2/k−1+μ11−1/k>1, we choose a sufficiently small δ>0,η>0, such that(22)δεγλ1k2k−1+μ11−1k1−σ−η>1.
From (21), we know that there is t>t3, such that(23)1−σ<Ikt<1+σ,0≤Kkt<σ,0≤Skt<σ,0≤Fkt<σ,k=1,2,…,kmax.
From the second equation of system (2), we can obtain(24)dKktdt≥λ1k+μ1ΘtIkt+λ2k+μ2Fkt−εKkt.
Whent⟶∞, from (21), we obtain(25)dKktdt≥λ1k+μ1ΘtIkt−εKkt.
Fort>t4, there exists t4>0 such that(26)Kkt≥λ1k+μ1ε−ε3ΘtIkt,where ε3 is an arbitrary constant.
LetWt=∑kkpkSkt; we have(27)W′t=∑kkpkδλ1k+μ1ε−ε3ΘtIkt−γSkt≥∑kkpkSktδλ1k+μ1ε−ε31k∑kkpk1−σ−γ=γ∑kkpkSktR01−η−1−1−1kδε31−σγ=γ∑kkpkSktR01−σ−1−η=ψWt.
For allt>maxt3,t4, there is(28)ψ=γδεγλ1k2k−1+μ11−1k1−σ−η,where(29)η=1−1kδε31−σγ.
Therefore, whent⟶∞, Wt⟶∞, which contradicts the assumption; hence, the theorem holds.


## 3.1. Equilibrium Point and Basic Reproduction Number

Theorem 1.
There are two equilibrium points for system (2); if R0<1, the system has a knowledge loss equilibrium point E0=I10,K10,S10,F10,…,Ikmax0,Kkmax0,Skmax0,Fkmax0, where Ik0=1,Kk0=0,Sk0=0,Fk0=0; if R0>1, there is a knowledge-endemic equilibrium in the system E∗=I1∗,K1∗,S1∗,F1∗,…,Ikmax∗,Kkmax∗,Skmax∗,Fkmax∗, where(3)Ik∗=λ2k+μ2εγλ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Kk∗=λ1k+μ1λ2k+μ2γΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Sk∗=λ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Fk∗=λ1k+μ1γδΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,and Θ∗t=∑kkpkSkt/k is the only positive root of the following equation:(4)fΘ∗=Θ∗−∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ.Proof.
SubstitutingE0=1,0,0,0,…,1,0,0,0 into equation (2), it is easy to see that E0 is the solution to equation (2). Next to prove the equilibrium of E∗, according to the stability condition of system (2), we obtain(5)−λ1k+μ1IktΘt+εKkt=0,λ1k+μ1IktΘt+λ2k+μ2Fkt−δKkt−εKkt=0,δKkt−γSkt=0,γSkt−λ2k+μ2Fkt=0.
The solution in (5) can be obtained as(6)Ik∗=λ2k+μ2εγλ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Kk∗=λ1k+μ1λ2k+μ2γΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Sk∗=λ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,Fk∗=λ1k+μ1γδΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ.
Substituting the third equation of (3) into (1), Θ∗ can be converted to the following form:(7)Θ∗=∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ.
It can be seen thatHTMLtranslationfailed is the solution of system (2); next, we discuss the nontrivial solution of Θ∗∈0,1 in system (2).
Defining(8)fΘ∗=Θ∗−∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ,it is easy to obtain(9)dfΘ∗dΘ∗=1−1k∑kpkABδBγ+δ+γδAΘ∗+Bεδ−Bγ+δ+γδA2BδΘ∗Bγ+δ+γδAΘ∗+Bεγ2,where A=λ1k+μ1andB=λ2k+μ2.
It is easy to see thatdf2Θ∗/dΘ∗2>0 and satisfied f0=0, while(10)f1=1−∑kkpkkλ1k+μ1λ2k+μ2δΘ∗λ2k+μ2γ+δ+γδλ1k+μ1Θ∗+λ2k+μ2εγ>0,k=1,2,…,kmax.
To allow the existence of an extraordinary solution, the inequality(11)dfΘ∗dΘ∗=1−1k∑kpkAδεγ<0,must be satisfied.
Therefore,(12)R0=δεγλ1k2k−1+μ11−1k.
To sum up, whenR0<1, Θ∗ has no nontrivial solution at (0, 1) (see Figure 2(a)). When R0>1, Θ∗ has only one nontrivial solution at (0, 1) (see Figure 2(b)).
Then, we have the following: whenR0<1, system (2) has a knowledge-free equilibrium E0=1,0,0,0,…,1,0,0,0; when R0>1, system (2) has a knowledge-endemic equilibrium E∗=I1∗,K1∗,S1∗,F1∗,…,Ikmax∗,Kkmax∗,Skmax∗,Fkmax∗.Figure 2
Diagram of solutions forfΘ in different cases.
(a)(b)


## 3.2. Stability Analysis

Theorem 2.
IfR0<1, then the knowledge loss equilibrium E0 of system (2) is globally asymptotically stable.Proof.
For a sufficiently smallε1>0, there exists t1>0 such that when t>t1, there is Ikt≤Ik0+ε1, where Ik0=1.
The second formula of system (2) has dKkt/dt≤λ1k+μ1ΘtIk0+ε1−εK+kt,t⟶∞.
Forε2>0 small enough, there exists t2>0 such that when t>t2, Kkt≤λ1k+μ1ΘtIk0+ε1/ε+ε2.
Whent>maxt1,t2, we have Skt/dt≤δλ1k+μ1ΘtIk0+ε1/ε+ε2δ−γSkt.
This is enough to prove that the positive solution of the following auxiliary system tends to zero ast tends to infinity:(13)dS˜ktdt=δλ1k+μ1ΘtIk0+ε1ε+ε2δ−γSkt,where Θ˜t=∑kkpkS˜kt/k.
LetS˜k0=Sk0; there is S˜k0≤Sk0.
The Lyapunov function is defined as follows:Lt=∑kkpkS˜kt/k.
Lt is derived along system (13) to obtain(14)L′t=1k∑kkpkδλ1k+μ1ΘtIk0+ε1ε+ε2δ−γ≤γΘ˜t1k∑kkpkδλ1k+μ11+ε1ε−1+ε2δ1−1k=γΘ˜tR01+ε1−1+ε2δ1−1k.
ForR0<1, when t⟶∞, ε1,ε2 is small enough to make L′t<0. Guaranteed dLt/dt≤0 for all S˜kt≥0 and S˜kt=0 only if dLt/dt=0. It can be known from the comparison theorem that when t tends to infinity, the solution of system (2) tends to 0, that is,(15)limt⟶+∞Skt=0.
The proof of Theorem2 is completed.Theorem 3.
IfR0>1, knowledge will always be preserved; i.e., there is a constant ζ>0 such that(16)limt⟶+∞infSt=limt⟶+∞inf∑kkpkSkt>ζ.Proof.
Using the results in [39], we can get the knowledge is permanent; Liu and Zhang [40] provide the relevant theorem.
Defining(17)X=I1,K1,S1,F1,…,Ikmax,Kkmax,Skmax,Fkmax:Ik,Kk,Sk,Fk≥0,Ik+Kk+Sk+Fk=1,X0=I1,K1,S1,F1,…,Ikmax,Kkmax,Skmax,Fkmax∈X:∑kpkSk>0,∂X0=X\X0,where k=1,2,..,kmax, X is the feasible region of system (2).
Clearly,X is positively invariant to system (2). From the third equation of system (2), ∑kpkSkt′≥−γ∑kpkSkt is easily obtained.
Because of∑kkpkSkt>0, it can be seen that ∑kpkSkt≥∑kpkSk0e−γt>0 indicates that X0 is positive and invariant, and there exists a compact set B where all solutions initiated by system (2) in X go into B and exist forever; for this set B, this condition is easily verified in [39].
Defining the following set,(18)M∂=Ik0,Kk0,Sk0,Fk0:Ikt,Kkt,Skt,Fkt∈∂X0,k=1,2,…,kmax,t≥0,Ω=∪ωy,y∈M∂,where ωy is the limit set of model (2).
According toM∂, we rewrite the system:(19)Ikt=εKkt,Kkt=λ2k+μ2Fkt−δKkt−εKkt,Skt=δKkt−γSkt,Fkt=γSkt−λ2k+μ2Fkt.
It is easy to show that system (19) has a unique equilibrium E0 on X. Hence, E0 is the only equilibrium of system (2) in M∂. Since (19) is a linear system, it is easy to prove that E0 is globally asymptotically stable. Hence, Ω=E0, where E0 is a cover of Ω, and E0 is isolated acyclic. Next, we proved that E0 is the weak repulsion force of X0, that is,(20)limt⟶+∞supdistIkt,Kkt,Skt,Fkt,E0>0,where Ikt,Kkt,Skt,Fkt,E0 is any solution with the initial value on X0. According to Lemma 3.5 in [41], it is only necessary to prove WsE0∩X0=∅, where WsE0 is the stable manifold of E0. By contradiction, assuming it is incorrect, then there is a solution Ikt,Kkt,Skt,Fkt,E0∈X0, such that(21)Ikt⟶1,Kkt⟶0,Skt⟶0,Fkt⟶0,ast⟶∞.
ForR0=δ/εγλ1k2/k−1+μ11−1/k>1, we choose a sufficiently small δ>0,η>0, such that(22)δεγλ1k2k−1+μ11−1k1−σ−η>1.
From (21), we know that there is t>t3, such that(23)1−σ<Ikt<1+σ,0≤Kkt<σ,0≤Skt<σ,0≤Fkt<σ,k=1,2,…,kmax.
From the second equation of system (2), we can obtain(24)dKktdt≥λ1k+μ1ΘtIkt+λ2k+μ2Fkt−εKkt.
Whent⟶∞, from (21), we obtain(25)dKktdt≥λ1k+μ1ΘtIkt−εKkt.
Fort>t4, there exists t4>0 such that(26)Kkt≥λ1k+μ1ε−ε3ΘtIkt,where ε3 is an arbitrary constant.
LetWt=∑kkpkSkt; we have(27)W′t=∑kkpkδλ1k+μ1ε−ε3ΘtIkt−γSkt≥∑kkpkSktδλ1k+μ1ε−ε31k∑kkpk1−σ−γ=γ∑kkpkSktR01−η−1−1−1kδε31−σγ=γ∑kkpkSktR01−σ−1−η=ψWt.
For allt>maxt3,t4, there is(28)ψ=γδεγλ1k2k−1+μ11−1k1−σ−η,where(29)η=1−1kδε31−σγ.
Therefore, whent⟶∞, Wt⟶∞, which contradicts the assumption; hence, the theorem holds.


## 4. Optimal Control Strategy

### 4.1. Analysis of Optimal Control

As we all know, knowledge dissemination has a positive impact on enterprise development and social innovation. Therefore, intervention control strategies can be introduced into the knowledge diffusion system inspired by the infectious disease control model (2). The benefits of control strategies are capable of increasing the number of spreader individuals to speed up knowledge diffusion, but the introduction of the control strategy will also increase a lot of costs inevitably, so the optimal control is considered here; that is, the objective function should not only maximize the effect of knowledge diffusion but also reduce business costs as much as possible.This section mainly modulates the knowledge dissemination process by introducing an intervention controller. The specific details of the IKSF model with two controllers are shown in Figure3. Specific descriptions of the interventions are as follows.Figure 3
Schematic diagram of knowledge dissemination with controllers.


#### 4.1.1. Motivation

By establishing a reward mechanism, the willingness of knower individuals can be increased, which in turn motivates them to continue learning. When they keep learning and receiving material rewards, the probability of knower individuals choosing to become knowledge spreaders isι10≤ι1≤1 under a predefined period of time. Furthermore, continuing learning intensity is denoted by u1. The minimum sleep time for people is 6 h [42]. We assume that the time spent engaging in activities necessary for survival is 1 h and that people are fully engaged in learning. Therefore, the maximum learning intensity u1max is 17 h per day, and u1 can be expressed as(30)u1=u1¯u1max,where u1¯ is the current learning intensity.


#### 4.1.2. Enhancement

We reduce the knowledge degradation of individual spreaders by establishing a supervision mechanism and formulating excellent learning methods. This can be achieved by increasing the frequency of review. Similarly, the probability of a knowledge spreader becoming a knowledge forgetter after completing the above behavior isι20≤ι2≤1 under a predefined period of time. In addition, the review frequency is denoted by u2. The shortest memory cycle is 5 min [43]. Therefore, the maximum revision frequency u2max is 288 per day, and u2 can be expressed as(31)u2=u2¯u2max,where u2¯ is the current review frequency.According to the above description, when we add the two intervention controllers into model (2), the system is written as(32)dIktdt=−λ1k+μ1IktΘt+εKkt,dKktdt=λ1k+μ1IktΘt+λ2k+μ2Fkt−δ+ι1u1Kkt−εKkt,dSktdt=δ+ι1u1Kkt−γ−ι2u2Skt,dFktdt=γ−ι2u2Skt−λ2k+μ2Fkt.According to the above description, the cost function should be set to the number of knowledge disseminators minus the cost of the controller. The objective function of the control model (32) is defined as follows:(33)Ju1,u2=∫0tfSk−c12u12−c22u22dt,where tf represents the final time, c1 and c2 are the weight constants to increase the willingness and reduce the degradation of knowledge, respectively, and c1/2u12 and c2/2u22 are represented the cost of two control strategies. The objective function is as follows:(34)Ju1∗,u2∗=maxu1,u2∈ΨJu1,u2,where u1 and u2 are unknown control rate, and the control range is as follows:(35)Ψ=u1,u20≤ui≤1,uiislebesguemeasurable,i=1,2..The existence of the optimal control pair can be obtained using a result by Fleming and Rishel in [44] and by Lukes in [45]. When ui=0,i=1,2, it means that there is no investment in the control strategy. When ui=1,i=1,2, it means that the investment cost is the largest, and the control strategy has the best effect. For the maximization problem (34) of system (32), it can be transformed into a Hamiltonian problem of maximization by applying Pontryagin’s maximum value theorem. Here, the Hamiltonian of the control system (32) is denoted as H:(36)H=Sk−c12u12−c22u22+η1dIkdt+η2dKkdt+η3dSkdt+η4dFkdt=Sk−c12u12−c22u22+η1−λ1k+μ1IktΘt+εKkt+η2λ1k+μ1IktΘt+λ2k+μ2Fkt−δ+ι1u1Kkt−εKkt+η3δ+ι1u1Kkt−γ−ι2u2Skt+η4γ−ι2u2Skt−λ2k+μ2Fkt.The optimal solution of the control system (32) is discussed next.Theorem 4.
Under the initial conditionIk0,Kk0,Sk0,Fk0, problem (32) has a unique solution Ik∗0,Kk∗0,Sk∗0,Fk∗0 and u1∗,u2∗ on 0,tf, where(37)u1∗=max0,minη3−η2ι1Kkc1,1,u2∗=max0,minη3−η4ι2Skc2,1.
Furthermore, there exists an adjoint function satisfied as(38)dη1dt=−∂Ht,x,u,η∂Ik=η1−η2λ1k+μ1Θt,dη2dt=−∂Ht,x,u,η∂Kk=δ+ι1u1+εη2−η1ε−η3δ+ι1u1,dη3dt=−∂Ht,x,u,η∂Sk=−1+η1−η2λ1k+μ1Ikt∑kkpkk+η3−η4γ−ι2u2,dη4dt=−∂Ht,x,u,η∂Fk=η4−η2λ2k+μ2,where the transversal conditions are η1tf=0,η2tf=0,η3tf=0,andη4tf=0.Proof.
We apply Pontryagin’s maximum value theorem to Hamiltonian to find the optimal solution of the system. Ifx,u is the optimal solution to an optimal control problem, where x=Ik,Kk,Sk,FkT and u=u1,u2T, there is a function η=η1,η2,η3,η4 that satisfies the following equation:(39)dη1dt=−∂Ht,x,u,η∂η1,dη2dt=−∂Ht,x,u,η∂η2,dη3dt=−∂Ht,x,u,η∂η3,dη4dt=−∂Ht,x,u,η∂η4,dη1dt=−∂Ht,x,u,η∂Ik=η1−η2λ1k+μ1Θt,dη2dt=−∂Ht,x,u,η∂Kk=δ+ι1u1+εη2−η1ε−η3δ+ι1u1,dη3dt=−∂Ht,x,u,η∂Sk=−1+η1−η2λ1k+μ1Ikt∑kkpkk+η3−η4γ−ι2u2,dη4dt=−∂Ht,x,u,η∂Fk=η4−η2λ2k+μ2,where the transversal conditions are η1tf=0,η2tf=0,η3tf=0,andη4tf=0.
Next, we will solve the maximization problem of equation (33); let(40)0=∂Ht,x,u,η∂u1,0=∂Ht,x,u,η∂u2.
We obtain(41)u1∗=0,u1≤0,η3−η2ι1Kkc1,0≤u1≤1,1,u1≥1,u2∗=0,u2≤0,η3−η4ι2Skc1,0≤u2≤1,1,u2≥1.
Then, we get that the optimal control problem (32) has a unique optimal solution Ik∗t,Kk∗t,Sk∗t,Fk∗t and u1∗,u2∗ on 0,tf to maximize the objective function (32).
Since the purpose of introducing the control strategy in this paper is to promote faster knowledge diffusion and increase the number of knowledge disseminators, so default hereR0>1; we do not discuss when R0<1.Remark 2.
WhenR0>1, the stability of the knowledge-endemic equilibrium of the control system (32) based on optimal control is globally asymptotically stable.


### 4.2. Guaranteed Cost Control under Uncertain Parameters

When there are multiple time-varying parameters in the system, the stability of the optimal controller of the system may not be guaranteed. As shown in Figure4, when the willingness rate δ changes with time, the number of knowledge disseminators will no longer be stable, and the stability of the system may not be guaranteed. However, the control strategy of guaranteed cost control can effectively solve this problem.Figure 4
Whenδ changes over time, the density changes the curve of Skt.Taking into account the uncertainty of the parameters, we get system (42), and the uncertain parameters of the control system (42) are expressed as follows:(42)dIktdt=−λ1tk+μ1tIktΘt+εtKkt,dKktdt=λ1tk+μ1tIktΘt+λ2tk+μ2tFkt−δt+ι1u1Kkt−εtKkt,dSktdt=δt+ι1u1Kkt−γt−ι2u2Skt,dFktdt=γt−ι2u2Skt−λ2tk+μ2Fkt.Unlike most existing knowledge propagation models with constant parameters, the time-varying uncertainty parametersλ1t,λ2t,μ1t,μ2t,εt,δt,γt in the above control system are expressed as follows:(43)λ1t=λ1+∆λ1,λ2t=λ2+∆λ2,μ1t=μ1+∆μ1,μ2t=μ2+∆μ2,εt=ε+∆ε,δt=δ+∆δ,γt=γ+∆γ,where λ1,λ2,μ1,μ2,ε,δ,γ are constant, representing the normal state of knowledge disseminating individuals without external intervention. And ∆λ1,∆λ2,∆μ1,∆μ2,∆ε,∆δ,∆γ are uncertain parameters, which can be time-varying, representing the fluctuation of knowledge dissemination to individuals around the range of constant parameters under the intervention of external conditions. And satisfying norm-bounded uncertainty, u1 and u2 are the control input. By considering the individual’s time-varying willingness parameter, time-varying self-learning parameter, time-varying communication parameter, and time-varying degradation parameter, the system can be rewritten to obtain(44)dIktdt=−λ1+∆λ1k+μ1+∆μ1IktΘt+ε+∆εKkt,dKktdt=−λ1+∆λ1k+μ1+∆μ1IktΘt+λ2+∆λ2k+μ2+∆μ2Fkt−δ+∆δ+ι1u1Kkt−ε+∆εKkt,dSktdt=δ+∆δ+ι1u1Kkt−γ+∆γ−ι2u2Skt,dFktdt=γ+∆γ−ι2u2Skt−λ2+∆λ2k+μ2+∆μ2Fkt.Letut=u1,u2T and xt=Ik,Kk,Sk,FkT; we rewrite system (44) as the following state feedback control system:(45)x˙t=Axxt+∆Axxt+But,where(46)Ax=−λ1k+μ1Θε00λ1k+μ1Θ−δ+ε0λ2k+μ20δ−γ000γ−λ2k+μ2,∆Ax=−∆λ1k+∆μ1Θ∆ε00∆λ1k+∆μ1Θ−∆δ+∆ε0∆λ2k+∆μ20∆δ−∆γ000∆γ−∆λ2k+∆μ2,B=00−ι1Kk0ι1Kkι2Sk0ι2Sk.Considering an uncertain continuous nonlinear system,x˙t=Axxt+∆Axxt+But, where Ax is a polynomial matrix with respect to x.Because0≤λ1,λ2,μ1,μ2,ε,δ,γ≤1 and 0≤λ1t,λ2t,μ1t,μ2t,εt,δt,γt≤1, then 0≤∆λ1,∆λ2,∆μ1,∆μ2,∆ε,∆δ,∆γ≤1. ∆Ax is the norm-bounded uncertainty caused by the time-varying parameters of the system, satisfying the following formula:(47)∆Ax=DFxtE,where(48)D=−k∑kkpkk0−∑kkpkk0100k∑kkpkkk∑kkpkk1−1−10000001−10−k0−1001,E=1000000110000001010001000010,Fxt=∆λ1Skt0000000∆λ20000000∆μ1Skt0000000∆μ20000000∆ε0000000∆δ0000000∆γ.Because of0≤∆λ1Sktand∆λ2,∆μ1Skt,∆μ2,∆ε,∆δ,∆γ≤1, the elements of the uncertain matrix Fxt are Lebegue measurable and satisfy(49)FTxtFxt≤I,where I is the identity matrix of the appropriate dimension and D and E are the known matrices of the appropriate dimension. For the uncertain item ∆Ax, when it satisfies the above conditions, it is said to be admissible.For the guaranteed cost control of knowledge dissemination, a lower cost should be guaranteed when the level of knowledge dissemination is high. The requirement for knowledge dissemination becomes even more important when investment costs are low. We consider the number of knowledge disseminators in the system as the performance of knowledge dissemination. The cost functions that are defined as related to knowledge propagation control are as follows:(50)J=∫0∞xt−x∗TQxt−x∗+ut−u∗TRut−u∗dt,where Q and R are positive definite weighting matrices and x∗ and u∗ are the optimal state and control action calculated by Theorem 4. The first part of the integral in (50) represents the performance level of knowledge in the process of dissemination and the second part represents the cost input in the process of knowledge dissemination. To solve the optimal control of knowledge dissemination, we design the control law as(51)ut=Kxt,where K is the income matrix, such that the closed-loop system is cost-guaranteed for all admissible uncertainties (47), and the corresponding performance metric does not exceed a certain upper bound:(52)x˙t=Ax+DFE+BKxt.Lemma 1.
(Schur complementary lemma [46]). For a given symmetric matrix,(53)S=S11S12S21S22,with S11T=S11,S22T=S22, and the following three conditions are equivalent:(1)
S<0(2)
S11<0,S22−S12TS11−1S12<0(3)
S22<0,S11−S12S22−1S12T<0Lemma 2.
(see [47]). Given matrices Y, D, and E of appropriate dimensions, where Y is symmetric, then there is the following formula: Y+DFE+ETFTDT<0. For the matrix F, there is a condition for FTtFt≤I to be established; the necessary and sufficient condition for the Y+DFE+ETFTDT<0 condition to be established is that there is a constant ε>0 that satisfies Y+εDDT+ε−1ETE<0.Definition 1.
For the knowledge propagation control model (44) and the cost function (50), if there is a controller ut and a positive value J∗ such that the input cost conforms to the performance level in the knowledge propagation process and the system is asymptotically stable, then J∗ is called the upper bound of the performance of system (45) and ut is the state feedback guaranteed cost controller of the uncertain control system, where the value of its cost function (50) satisfies J≤J∗.Theorem 5.
For uncertain continuous systems (45) and performance indicators (50), if there is a symmetric positive definite matrix L, the positive definite matrix P and matrix K are such that, for all allowable uncertainties, inequality (54) is satisfied:(54)Q+KTRK+PDFE+BK+DFE+BKTP+LTP+PL<0.Then, (51) is called the guaranteed cost control law of system (45); J=xT0Px0 is the corresponding upper bound of system performance at this time.Proof.
Suppose there are matricesP and K that satisfy the conditions; for uncertainties, inequality (54) will hold, and let ut=Kxt.
We change system (52) to(55)x˙t=ft,x+DFE+BKxt.
According to the strategy of [48], where Ax in the function is a piecewise continuous function with respect to t in any finite time interval t0,t1, the elements of Ax are bounded, which can be simplified by the elementary transformation of the matrix as follows:(56)Ax=−λ1k+μ1Θε000−δγ000−γλ2k+μ20000,where(57)Ax=max−λ1k+μ1Θ,ε−δ,0,λ2k+μ2.
Therefore,Ax≤a; for all x,y∈Rn and t∈t0,t1, there is ft,x−ft,y=Axx−Ayy≤ax−y=Lx−y. So, ft,x=Axx is a nonlinear function satisfying the Lipschitz condition and ft,x≤Lx.
We define the Lyapunov functionVxt=xTtPxt.
Taking the derivation of the above formula along system (52), we obtain(58)V˙xt=x˙TtPxt+xTtPx˙t=f+DFE+BKxTPx+xTPf+DFE+BKx=fTPx+xTDFE+BKTPx+xTPf+xTPDFE+BKx≤xTLTPx+xTDFE+BKTPx+xTPLx+xTPDFE+BKx=xTLTP+DFE+BKTP+PL+PDFE+BKx.
From (54), we obtain(59)V˙xt<xTt−KTRK−Qxt.
Furthermore, we integrate the timet from 0 to ∞ on both sides of the above formula, since system (52) is stable, so Vx∞=0; then, we obtain(60)∫0∞xTt−KTRK−Qxtdt≤Vx0=xT0Px0,where Q+KTRK is the income brought by the weight of the performance indicator function to the state quantity plus the weight of the control quantity.
Defined by Definition1, the proof is completed.
The condition in Theorem5 contains the uncertainty matrix F, so all allowable uncertainty matrices F are tested. We give an equivalent characterization of this condition by the following theorem.Theorem 6.
There exist matricesP, L, and K such that, for all admissible uncertainties, matrix inequality (54) holds if and only if there exists a scalar ε>0; symmetric positive definite matrices X and W such that(61)BW+BWT+XLT+LX+εDDT∗∗∗W−R−1∗∗EX0−εI∗X00−Q−1<0.
If matrix inequality (61) has a feasible solution ε,W,X, then(62)ut=WX−1xt,is a guaranteed cost control law of system (45), and the maximum of system performance is(63)J¯=trX−1=J¯∗.Proof.
We defineY=Q+KTRK+PBK+BKTP+LTP+PL.
Then, the matrix inequality (54) can be written as Y+PDFE+ETFTPDT<0. According to Lemma 2, the matrix inequality above holds for all indeterminate matrices F satisfying FTtFt≤I if there is a scalar ε>0, such that Y+εPDDTPT+ε−1ETE<0. Further applying the matrix Schur complement property, we obtain(64)PBK+BKTP+LTP+PL+εPDDTP∗∗∗K−R−1∗∗E0−εI∗I00−Q−1<0.
Multiplying left and right bydiagP−1,I,I,I, respectively, in matrix inequality (64), let X=P−1,W=KP−1, then (61) can be obtained; the matrix inequality (64) is a linear matrix inequality about the variable ε,X,W, so the solver feasp in the LMI toolbox can be used to judge the feasibility of the linear matrix inequality. Furthermore, if the inequality has a feasible solution ε,X,W, then (62) gives a parametric representation of the guaranteed cost control law of system (42).


## 4.1. Analysis of Optimal Control

As we all know, knowledge dissemination has a positive impact on enterprise development and social innovation. Therefore, intervention control strategies can be introduced into the knowledge diffusion system inspired by the infectious disease control model (2). The benefits of control strategies are capable of increasing the number of spreader individuals to speed up knowledge diffusion, but the introduction of the control strategy will also increase a lot of costs inevitably, so the optimal control is considered here; that is, the objective function should not only maximize the effect of knowledge diffusion but also reduce business costs as much as possible.This section mainly modulates the knowledge dissemination process by introducing an intervention controller. The specific details of the IKSF model with two controllers are shown in Figure3. Specific descriptions of the interventions are as follows.Figure 3
Schematic diagram of knowledge dissemination with controllers.


### 4.1.1. Motivation

By establishing a reward mechanism, the willingness of knower individuals can be increased, which in turn motivates them to continue learning. When they keep learning and receiving material rewards, the probability of knower individuals choosing to become knowledge spreaders isι10≤ι1≤1 under a predefined period of time. Furthermore, continuing learning intensity is denoted by u1. The minimum sleep time for people is 6 h [42]. We assume that the time spent engaging in activities necessary for survival is 1 h and that people are fully engaged in learning. Therefore, the maximum learning intensity u1max is 17 h per day, and u1 can be expressed as(30)u1=u1¯u1max,where u1¯ is the current learning intensity.


### 4.1.2. Enhancement

We reduce the knowledge degradation of individual spreaders by establishing a supervision mechanism and formulating excellent learning methods. This can be achieved by increasing the frequency of review. Similarly, the probability of a knowledge spreader becoming a knowledge forgetter after completing the above behavior isι20≤ι2≤1 under a predefined period of time. In addition, the review frequency is denoted by u2. The shortest memory cycle is 5 min [43]. Therefore, the maximum revision frequency u2max is 288 per day, and u2 can be expressed as(31)u2=u2¯u2max,where u2¯ is the current review frequency.According to the above description, when we add the two intervention controllers into model (2), the system is written as(32)dIktdt=−λ1k+μ1IktΘt+εKkt,dKktdt=λ1k+μ1IktΘt+λ2k+μ2Fkt−δ+ι1u1Kkt−εKkt,dSktdt=δ+ι1u1Kkt−γ−ι2u2Skt,dFktdt=γ−ι2u2Skt−λ2k+μ2Fkt.According to the above description, the cost function should be set to the number of knowledge disseminators minus the cost of the controller. The objective function of the control model (32) is defined as follows:(33)Ju1,u2=∫0tfSk−c12u12−c22u22dt,where tf represents the final time, c1 and c2 are the weight constants to increase the willingness and reduce the degradation of knowledge, respectively, and c1/2u12 and c2/2u22 are represented the cost of two control strategies. The objective function is as follows:(34)Ju1∗,u2∗=maxu1,u2∈ΨJu1,u2,where u1 and u2 are unknown control rate, and the control range is as follows:(35)Ψ=u1,u20≤ui≤1,uiislebesguemeasurable,i=1,2..The existence of the optimal control pair can be obtained using a result by Fleming and Rishel in [44] and by Lukes in [45]. When ui=0,i=1,2, it means that there is no investment in the control strategy. When ui=1,i=1,2, it means that the investment cost is the largest, and the control strategy has the best effect. For the maximization problem (34) of system (32), it can be transformed into a Hamiltonian problem of maximization by applying Pontryagin’s maximum value theorem. Here, the Hamiltonian of the control system (32) is denoted as H:(36)H=Sk−c12u12−c22u22+η1dIkdt+η2dKkdt+η3dSkdt+η4dFkdt=Sk−c12u12−c22u22+η1−λ1k+μ1IktΘt+εKkt+η2λ1k+μ1IktΘt+λ2k+μ2Fkt−δ+ι1u1Kkt−εKkt+η3δ+ι1u1Kkt−γ−ι2u2Skt+η4γ−ι2u2Skt−λ2k+μ2Fkt.The optimal solution of the control system (32) is discussed next.Theorem 4.
Under the initial conditionIk0,Kk0,Sk0,Fk0, problem (32) has a unique solution Ik∗0,Kk∗0,Sk∗0,Fk∗0 and u1∗,u2∗ on 0,tf, where(37)u1∗=max0,minη3−η2ι1Kkc1,1,u2∗=max0,minη3−η4ι2Skc2,1.
Furthermore, there exists an adjoint function satisfied as(38)dη1dt=−∂Ht,x,u,η∂Ik=η1−η2λ1k+μ1Θt,dη2dt=−∂Ht,x,u,η∂Kk=δ+ι1u1+εη2−η1ε−η3δ+ι1u1,dη3dt=−∂Ht,x,u,η∂Sk=−1+η1−η2λ1k+μ1Ikt∑kkpkk+η3−η4γ−ι2u2,dη4dt=−∂Ht,x,u,η∂Fk=η4−η2λ2k+μ2,where the transversal conditions are η1tf=0,η2tf=0,η3tf=0,andη4tf=0.Proof.
We apply Pontryagin’s maximum value theorem to Hamiltonian to find the optimal solution of the system. Ifx,u is the optimal solution to an optimal control problem, where x=Ik,Kk,Sk,FkT and u=u1,u2T, there is a function η=η1,η2,η3,η4 that satisfies the following equation:(39)dη1dt=−∂Ht,x,u,η∂η1,dη2dt=−∂Ht,x,u,η∂η2,dη3dt=−∂Ht,x,u,η∂η3,dη4dt=−∂Ht,x,u,η∂η4,dη1dt=−∂Ht,x,u,η∂Ik=η1−η2λ1k+μ1Θt,dη2dt=−∂Ht,x,u,η∂Kk=δ+ι1u1+εη2−η1ε−η3δ+ι1u1,dη3dt=−∂Ht,x,u,η∂Sk=−1+η1−η2λ1k+μ1Ikt∑kkpkk+η3−η4γ−ι2u2,dη4dt=−∂Ht,x,u,η∂Fk=η4−η2λ2k+μ2,where the transversal conditions are η1tf=0,η2tf=0,η3tf=0,andη4tf=0.
Next, we will solve the maximization problem of equation (33); let(40)0=∂Ht,x,u,η∂u1,0=∂Ht,x,u,η∂u2.
We obtain(41)u1∗=0,u1≤0,η3−η2ι1Kkc1,0≤u1≤1,1,u1≥1,u2∗=0,u2≤0,η3−η4ι2Skc1,0≤u2≤1,1,u2≥1.
Then, we get that the optimal control problem (32) has a unique optimal solution Ik∗t,Kk∗t,Sk∗t,Fk∗t and u1∗,u2∗ on 0,tf to maximize the objective function (32).
Since the purpose of introducing the control strategy in this paper is to promote faster knowledge diffusion and increase the number of knowledge disseminators, so default hereR0>1; we do not discuss when R0<1.Remark 2.
WhenR0>1, the stability of the knowledge-endemic equilibrium of the control system (32) based on optimal control is globally asymptotically stable.


## 4.1.1. Motivation

By establishing a reward mechanism, the willingness of knower individuals can be increased, which in turn motivates them to continue learning. When they keep learning and receiving material rewards, the probability of knower individuals choosing to become knowledge spreaders isι10≤ι1≤1 under a predefined period of time. Furthermore, continuing learning intensity is denoted by u1. The minimum sleep time for people is 6 h [42]. We assume that the time spent engaging in activities necessary for survival is 1 h and that people are fully engaged in learning. Therefore, the maximum learning intensity u1max is 17 h per day, and u1 can be expressed as(30)u1=u1¯u1max,where u1¯ is the current learning intensity.


## 4.1.2. Enhancement

We reduce the knowledge degradation of individual spreaders by establishing a supervision mechanism and formulating excellent learning methods. This can be achieved by increasing the frequency of review. Similarly, the probability of a knowledge spreader becoming a knowledge forgetter after completing the above behavior isι20≤ι2≤1 under a predefined period of time. In addition, the review frequency is denoted by u2. The shortest memory cycle is 5 min [43]. Therefore, the maximum revision frequency u2max is 288 per day, and u2 can be expressed as(31)u2=u2¯u2max,where u2¯ is the current review frequency.According to the above description, when we add the two intervention controllers into model (2), the system is written as(32)dIktdt=−λ1k+μ1IktΘt+εKkt,dKktdt=λ1k+μ1IktΘt+λ2k+μ2Fkt−δ+ι1u1Kkt−εKkt,dSktdt=δ+ι1u1Kkt−γ−ι2u2Skt,dFktdt=γ−ι2u2Skt−λ2k+μ2Fkt.According to the above description, the cost function should be set to the number of knowledge disseminators minus the cost of the controller. The objective function of the control model (32) is defined as follows:(33)Ju1,u2=∫0tfSk−c12u12−c22u22dt,where tf represents the final time, c1 and c2 are the weight constants to increase the willingness and reduce the degradation of knowledge, respectively, and c1/2u12 and c2/2u22 are represented the cost of two control strategies. The objective function is as follows:(34)Ju1∗,u2∗=maxu1,u2∈ΨJu1,u2,where u1 and u2 are unknown control rate, and the control range is as follows:(35)Ψ=u1,u20≤ui≤1,uiislebesguemeasurable,i=1,2..The existence of the optimal control pair can be obtained using a result by Fleming and Rishel in [44] and by Lukes in [45]. When ui=0,i=1,2, it means that there is no investment in the control strategy. When ui=1,i=1,2, it means that the investment cost is the largest, and the control strategy has the best effect. For the maximization problem (34) of system (32), it can be transformed into a Hamiltonian problem of maximization by applying Pontryagin’s maximum value theorem. Here, the Hamiltonian of the control system (32) is denoted as H:(36)H=Sk−c12u12−c22u22+η1dIkdt+η2dKkdt+η3dSkdt+η4dFkdt=Sk−c12u12−c22u22+η1−λ1k+μ1IktΘt+εKkt+η2λ1k+μ1IktΘt+λ2k+μ2Fkt−δ+ι1u1Kkt−εKkt+η3δ+ι1u1Kkt−γ−ι2u2Skt+η4γ−ι2u2Skt−λ2k+μ2Fkt.The optimal solution of the control system (32) is discussed next.Theorem 4.
Under the initial conditionIk0,Kk0,Sk0,Fk0, problem (32) has a unique solution Ik∗0,Kk∗0,Sk∗0,Fk∗0 and u1∗,u2∗ on 0,tf, where(37)u1∗=max0,minη3−η2ι1Kkc1,1,u2∗=max0,minη3−η4ι2Skc2,1.
Furthermore, there exists an adjoint function satisfied as(38)dη1dt=−∂Ht,x,u,η∂Ik=η1−η2λ1k+μ1Θt,dη2dt=−∂Ht,x,u,η∂Kk=δ+ι1u1+εη2−η1ε−η3δ+ι1u1,dη3dt=−∂Ht,x,u,η∂Sk=−1+η1−η2λ1k+μ1Ikt∑kkpkk+η3−η4γ−ι2u2,dη4dt=−∂Ht,x,u,η∂Fk=η4−η2λ2k+μ2,where the transversal conditions are η1tf=0,η2tf=0,η3tf=0,andη4tf=0.Proof.
We apply Pontryagin’s maximum value theorem to Hamiltonian to find the optimal solution of the system. Ifx,u is the optimal solution to an optimal control problem, where x=Ik,Kk,Sk,FkT and u=u1,u2T, there is a function η=η1,η2,η3,η4 that satisfies the following equation:(39)dη1dt=−∂Ht,x,u,η∂η1,dη2dt=−∂Ht,x,u,η∂η2,dη3dt=−∂Ht,x,u,η∂η3,dη4dt=−∂Ht,x,u,η∂η4,dη1dt=−∂Ht,x,u,η∂Ik=η1−η2λ1k+μ1Θt,dη2dt=−∂Ht,x,u,η∂Kk=δ+ι1u1+εη2−η1ε−η3δ+ι1u1,dη3dt=−∂Ht,x,u,η∂Sk=−1+η1−η2λ1k+μ1Ikt∑kkpkk+η3−η4γ−ι2u2,dη4dt=−∂Ht,x,u,η∂Fk=η4−η2λ2k+μ2,where the transversal conditions are η1tf=0,η2tf=0,η3tf=0,andη4tf=0.
Next, we will solve the maximization problem of equation (33); let(40)0=∂Ht,x,u,η∂u1,0=∂Ht,x,u,η∂u2.
We obtain(41)u1∗=0,u1≤0,η3−η2ι1Kkc1,0≤u1≤1,1,u1≥1,u2∗=0,u2≤0,η3−η4ι2Skc1,0≤u2≤1,1,u2≥1.
Then, we get that the optimal control problem (32) has a unique optimal solution Ik∗t,Kk∗t,Sk∗t,Fk∗t and u1∗,u2∗ on 0,tf to maximize the objective function (32).
Since the purpose of introducing the control strategy in this paper is to promote faster knowledge diffusion and increase the number of knowledge disseminators, so default hereR0>1; we do not discuss when R0<1.Remark 2.
WhenR0>1, the stability of the knowledge-endemic equilibrium of the control system (32) based on optimal control is globally asymptotically stable.


## 4.2. Guaranteed Cost Control under Uncertain Parameters

When there are multiple time-varying parameters in the system, the stability of the optimal controller of the system may not be guaranteed. As shown in Figure4, when the willingness rate δ changes with time, the number of knowledge disseminators will no longer be stable, and the stability of the system may not be guaranteed. However, the control strategy of guaranteed cost control can effectively solve this problem.Figure 4
Whenδ changes over time, the density changes the curve of Skt.Taking into account the uncertainty of the parameters, we get system (42), and the uncertain parameters of the control system (42) are expressed as follows:(42)dIktdt=−λ1tk+μ1tIktΘt+εtKkt,dKktdt=λ1tk+μ1tIktΘt+λ2tk+μ2tFkt−δt+ι1u1Kkt−εtKkt,dSktdt=δt+ι1u1Kkt−γt−ι2u2Skt,dFktdt=γt−ι2u2Skt−λ2tk+μ2Fkt.Unlike most existing knowledge propagation models with constant parameters, the time-varying uncertainty parametersλ1t,λ2t,μ1t,μ2t,εt,δt,γt in the above control system are expressed as follows:(43)λ1t=λ1+∆λ1,λ2t=λ2+∆λ2,μ1t=μ1+∆μ1,μ2t=μ2+∆μ2,εt=ε+∆ε,δt=δ+∆δ,γt=γ+∆γ,where λ1,λ2,μ1,μ2,ε,δ,γ are constant, representing the normal state of knowledge disseminating individuals without external intervention. And ∆λ1,∆λ2,∆μ1,∆μ2,∆ε,∆δ,∆γ are uncertain parameters, which can be time-varying, representing the fluctuation of knowledge dissemination to individuals around the range of constant parameters under the intervention of external conditions. And satisfying norm-bounded uncertainty, u1 and u2 are the control input. By considering the individual’s time-varying willingness parameter, time-varying self-learning parameter, time-varying communication parameter, and time-varying degradation parameter, the system can be rewritten to obtain(44)dIktdt=−λ1+∆λ1k+μ1+∆μ1IktΘt+ε+∆εKkt,dKktdt=−λ1+∆λ1k+μ1+∆μ1IktΘt+λ2+∆λ2k+μ2+∆μ2Fkt−δ+∆δ+ι1u1Kkt−ε+∆εKkt,dSktdt=δ+∆δ+ι1u1Kkt−γ+∆γ−ι2u2Skt,dFktdt=γ+∆γ−ι2u2Skt−λ2+∆λ2k+μ2+∆μ2Fkt.Letut=u1,u2T and xt=Ik,Kk,Sk,FkT; we rewrite system (44) as the following state feedback control system:(45)x˙t=Axxt+∆Axxt+But,where(46)Ax=−λ1k+μ1Θε00λ1k+μ1Θ−δ+ε0λ2k+μ20δ−γ000γ−λ2k+μ2,∆Ax=−∆λ1k+∆μ1Θ∆ε00∆λ1k+∆μ1Θ−∆δ+∆ε0∆λ2k+∆μ20∆δ−∆γ000∆γ−∆λ2k+∆μ2,B=00−ι1Kk0ι1Kkι2Sk0ι2Sk.Considering an uncertain continuous nonlinear system,x˙t=Axxt+∆Axxt+But, where Ax is a polynomial matrix with respect to x.Because0≤λ1,λ2,μ1,μ2,ε,δ,γ≤1 and 0≤λ1t,λ2t,μ1t,μ2t,εt,δt,γt≤1, then 0≤∆λ1,∆λ2,∆μ1,∆μ2,∆ε,∆δ,∆γ≤1. ∆Ax is the norm-bounded uncertainty caused by the time-varying parameters of the system, satisfying the following formula:(47)∆Ax=DFxtE,where(48)D=−k∑kkpkk0−∑kkpkk0100k∑kkpkkk∑kkpkk1−1−10000001−10−k0−1001,E=1000000110000001010001000010,Fxt=∆λ1Skt0000000∆λ20000000∆μ1Skt0000000∆μ20000000∆ε0000000∆δ0000000∆γ.Because of0≤∆λ1Sktand∆λ2,∆μ1Skt,∆μ2,∆ε,∆δ,∆γ≤1, the elements of the uncertain matrix Fxt are Lebegue measurable and satisfy(49)FTxtFxt≤I,where I is the identity matrix of the appropriate dimension and D and E are the known matrices of the appropriate dimension. For the uncertain item ∆Ax, when it satisfies the above conditions, it is said to be admissible.For the guaranteed cost control of knowledge dissemination, a lower cost should be guaranteed when the level of knowledge dissemination is high. The requirement for knowledge dissemination becomes even more important when investment costs are low. We consider the number of knowledge disseminators in the system as the performance of knowledge dissemination. The cost functions that are defined as related to knowledge propagation control are as follows:(50)J=∫0∞xt−x∗TQxt−x∗+ut−u∗TRut−u∗dt,where Q and R are positive definite weighting matrices and x∗ and u∗ are the optimal state and control action calculated by Theorem 4. The first part of the integral in (50) represents the performance level of knowledge in the process of dissemination and the second part represents the cost input in the process of knowledge dissemination. To solve the optimal control of knowledge dissemination, we design the control law as(51)ut=Kxt,where K is the income matrix, such that the closed-loop system is cost-guaranteed for all admissible uncertainties (47), and the corresponding performance metric does not exceed a certain upper bound:(52)x˙t=Ax+DFE+BKxt.Lemma 1.
(Schur complementary lemma [46]). For a given symmetric matrix,(53)S=S11S12S21S22,with S11T=S11,S22T=S22, and the following three conditions are equivalent:(1)
S<0(2)
S11<0,S22−S12TS11−1S12<0(3)
S22<0,S11−S12S22−1S12T<0Lemma 2.
(see [47]). Given matrices Y, D, and E of appropriate dimensions, where Y is symmetric, then there is the following formula: Y+DFE+ETFTDT<0. For the matrix F, there is a condition for FTtFt≤I to be established; the necessary and sufficient condition for the Y+DFE+ETFTDT<0 condition to be established is that there is a constant ε>0 that satisfies Y+εDDT+ε−1ETE<0.Definition 1.
For the knowledge propagation control model (44) and the cost function (50), if there is a controller ut and a positive value J∗ such that the input cost conforms to the performance level in the knowledge propagation process and the system is asymptotically stable, then J∗ is called the upper bound of the performance of system (45) and ut is the state feedback guaranteed cost controller of the uncertain control system, where the value of its cost function (50) satisfies J≤J∗.Theorem 5.
For uncertain continuous systems (45) and performance indicators (50), if there is a symmetric positive definite matrix L, the positive definite matrix P and matrix K are such that, for all allowable uncertainties, inequality (54) is satisfied:(54)Q+KTRK+PDFE+BK+DFE+BKTP+LTP+PL<0.Then, (51) is called the guaranteed cost control law of system (45); J=xT0Px0 is the corresponding upper bound of system performance at this time.Proof.
Suppose there are matricesP and K that satisfy the conditions; for uncertainties, inequality (54) will hold, and let ut=Kxt.
We change system (52) to(55)x˙t=ft,x+DFE+BKxt.
According to the strategy of [48], where Ax in the function is a piecewise continuous function with respect to t in any finite time interval t0,t1, the elements of Ax are bounded, which can be simplified by the elementary transformation of the matrix as follows:(56)Ax=−λ1k+μ1Θε000−δγ000−γλ2k+μ20000,where(57)Ax=max−λ1k+μ1Θ,ε−δ,0,λ2k+μ2.
Therefore,Ax≤a; for all x,y∈Rn and t∈t0,t1, there is ft,x−ft,y=Axx−Ayy≤ax−y=Lx−y. So, ft,x=Axx is a nonlinear function satisfying the Lipschitz condition and ft,x≤Lx.
We define the Lyapunov functionVxt=xTtPxt.
Taking the derivation of the above formula along system (52), we obtain(58)V˙xt=x˙TtPxt+xTtPx˙t=f+DFE+BKxTPx+xTPf+DFE+BKx=fTPx+xTDFE+BKTPx+xTPf+xTPDFE+BKx≤xTLTPx+xTDFE+BKTPx+xTPLx+xTPDFE+BKx=xTLTP+DFE+BKTP+PL+PDFE+BKx.
From (54), we obtain(59)V˙xt<xTt−KTRK−Qxt.
Furthermore, we integrate the timet from 0 to ∞ on both sides of the above formula, since system (52) is stable, so Vx∞=0; then, we obtain(60)∫0∞xTt−KTRK−Qxtdt≤Vx0=xT0Px0,where Q+KTRK is the income brought by the weight of the performance indicator function to the state quantity plus the weight of the control quantity.
Defined by Definition1, the proof is completed.
The condition in Theorem5 contains the uncertainty matrix F, so all allowable uncertainty matrices F are tested. We give an equivalent characterization of this condition by the following theorem.Theorem 6.
There exist matricesP, L, and K such that, for all admissible uncertainties, matrix inequality (54) holds if and only if there exists a scalar ε>0; symmetric positive definite matrices X and W such that(61)BW+BWT+XLT+LX+εDDT∗∗∗W−R−1∗∗EX0−εI∗X00−Q−1<0.
If matrix inequality (61) has a feasible solution ε,W,X, then(62)ut=WX−1xt,is a guaranteed cost control law of system (45), and the maximum of system performance is(63)J¯=trX−1=J¯∗.Proof.
We defineY=Q+KTRK+PBK+BKTP+LTP+PL.
Then, the matrix inequality (54) can be written as Y+PDFE+ETFTPDT<0. According to Lemma 2, the matrix inequality above holds for all indeterminate matrices F satisfying FTtFt≤I if there is a scalar ε>0, such that Y+εPDDTPT+ε−1ETE<0. Further applying the matrix Schur complement property, we obtain(64)PBK+BKTP+LTP+PL+εPDDTP∗∗∗K−R−1∗∗E0−εI∗I00−Q−1<0.
Multiplying left and right bydiagP−1,I,I,I, respectively, in matrix inequality (64), let X=P−1,W=KP−1, then (61) can be obtained; the matrix inequality (64) is a linear matrix inequality about the variable ε,X,W, so the solver feasp in the LMI toolbox can be used to judge the feasibility of the linear matrix inequality. Furthermore, if the inequality has a feasible solution ε,X,W, then (62) gives a parametric representation of the guaranteed cost control law of system (42).


## 5. Numerical Simulation

The proposed theoretical results are verified by numerical simulations, where the effects of model parameters are investigated. Supposing there are 2000 nodes in the network, the degree distribution of the vertices is a power-law distribution, the initial number of vertices in the network is 10, the vertices are randomly connected, and we add new nodes with 5 new edges in turn. At the same time, the effectiveness of optimal control and guaranteed cost control of the system is also verified.


### 5.1. Theoretical Verification

Figures5 and 6 show different degrees of density evolution when R0<1 and R0>1. We show that the evolution under R0<1 and R0>1 is different, although the two given R0 are nearly identical, which verifies the setting in Theorem 1. When R0<1, the parameter selects λ1=0.01,λ2=0.5,μ1=0.01,μ2=0.6,ε=0.2,δ=0.01,andγ=0.05; then, there is R0=9.9760e−04<1; when R0>1, we choose parameter λ1=0.4,λ2=0.5,μ1=0.3,μ2=0.6,ε=0.2,δ=0.6,andγ=0.05; then, there is R0=2.2446>1. It can be seen from Figure 5 that when R0<1, the density of Spreader will drop to 0; when R0>1 in Figure 6, the knowledge will exist forever, which can verify Theorems 2 and 3.Figure 5
Diagram of density change of four types of population whenR0<1.Figure 6
Diagram of density change of four types of population whenR0>1.


### 5.2. Parameter Influence

Since we take many factors into account in this model, firstly, the willingness rateδ is studied. Figure 7 shows that as the value of δ increases, so does the number of knowledge disseminators, indicating that the willingness rate δ has a positive impact on knowledge dissemination.Figure 7
The density of knowledge disseminators varies with the value ofδ.Secondly, the initial communication rateλ1 is studied. Figure 8 shows that the larger the value of λ1 in the early stage of knowledge dissemination, the greater the density of knower individuals, indicating that the value of the communication rate λ1 has a good promotion for the early knowledge dissemination process.Figure 8
The density of knower individuals changes with the value ofλ1.Finally, the degradation rate is studied. Figure9 shows that, as the value of γ increases, the density of knowledge disseminators is decreasing, indicating that the degradation rate γ hurts knowledge dissemination.Figure 9
The density of knowledge disseminators varies with the value ofγ.


### 5.3. Optimal Control

WhenR0>1, we conduct knowledge dissemination simulations under various control measures. The performance of the knowledge diffusion model is then validated with three different control strategies. For simplicity, let ι1=0.5 and ι2=0.5.


#### 5.3.1.u1≠0andu2=0

We used controlu1, which increases the rate of willingness by increasing learning intensity and material rewards u1. The optimal control curve u1∗ is shown in Figure 10. u1∗ rapidly decreases to near 0 until 0. Under this control measure, knower individuals become knowledge disseminators by sharing knowledge stimulated by a reward mechanism. Figure 11 mainly describes the changes in the four groups of people under the control strategy u1, in which the spreader individuals increased slightly and the other three groups decreased slightly. Furthermore, the maximum rate of change in individual spreaders compared to the noncontrol system is about 15%, which is consistent with our expected results. When increasing the willingness of knower individuals, the knowledge dissemination model performs better.Figure 10
Diagram of the optimal solutionu1∗ of knowledge propagation.Figure 11
Comparison of population density withu1.
(a)(b)(c)(d)


#### 5.3.2.u1=0andu2≠0

We used controlu2, which reduces the probability of degradation by increasing the frequency of review. The optimal control curve u2∗ is shown in Figure 12. u2∗ rapidly decreases to near 0 until 0. Under this control, the individual spreader continues to maintain the identity of the knowledge disseminator through regular review of knowledge under a supervision mechanism and excellent learning methods. Figure 13 mainly describes the changes in the four types of populations under the control strategy u2, in which the spreader individuals and knower individuals slightly increased, while the other two populations slightly decreased. Furthermore, the maximum rate of change in individual spreaders compared to the noncontrol system was about 16%, which is consistent with our expected results. When reducing the knowledge degradation of the individual spreader, the knowledge dissemination model performs better.Figure 12
Diagram of the optimal solutionu2∗ of knowledge propagation.Figure 13
Comparison of population density withu2.
(a)(b)(c)(d)


#### 5.3.3.u1≠0andu2≠0

We improve the willingness of knower individualsu1 while reducing the degradation of spreader individuals u2. The optimal control curve u1∗,u2∗ is shown in Figure 14. u1∗,u2∗ rapidly decreases to near 0 until 0. Under this control, the individual spreader continues to maintain the identity of the knowledge disseminator through regular review of knowledge under a supervision mechanism and excellent learning methods. Figure 15 mainly describes the changes in the four groups of people under the two control strategies u1,u2, in which the spreader and knower individuals increased slightly and the other two groups decreased slightly. Furthermore, the maximum rate of change of individual spreaders compared to the noncontrol system is about 25%, which is consistent with our expected results. When increasing the willingness of knower individuals and reducing the degradation of spreader individuals, the knowledge dissemination model performs best.Figure 14
Diagram of the optimal solutionu1∗ and u2∗ of knowledge propagation.Figure 15
Comparison of population density withu1,u2.
(a)(b)(c)(d)


### 5.4. Robustness of Uncertain Parameter Control System

According to the characteristics of knowledge dissemination, the parameters of system (45) are as follows:(65)L=20000003000000−5000000−200,B=00−101101.Uncertain structure selects(66)D=−29.9280−9.9760010029.92839.97601−1−10000001−10−30−1001,E=1000000110000001010001000010,given(67)Q=1000010000100001,R=1001.Applying Theorem6, the corresponding linear matrix inequality (61) is available from the LMI toolbox, and a feasible solution is obtained:(68)X=−1.92921.5244−1.2840e−040.02091.5244−1.2884−0.00370.0764−1.2840e−04−0.00370.0072−0.00110.02090.0764−0.00110.0348,W=−0.02640.0198−5.5194e−05−0.0063−0.04660.0171−1.3533e−054.2526e−04,ε=0.7486.Then, we get the guaranteed cost controller and guaranteed cost index of the system:(69)u∗t=−0.0263−0.0497−0.0423−0.05800.06920.05910.0062−0.1589xt,J¯∗=142.3200.We choose the willingness rate as an uncertain parameter and the time-varying part∆δt=sint. Figure 16 is the simulation diagram of the four types of people in the IKSF dissemination system under the unguaranteed cost control. Figure 17 is the guaranteed cost control effect comparison of the IKSF knowledge dissemination system picture.Figure 16
Diagram of knowledge transmission without guaranteed cost control.Figure 17
Diagram of knowledge transmission with guaranteed cost control.Figure16 shows that when the system parameters fluctuate within the range, the density fluctuations of the four groups of people will be caused. In Figure 17, compared with no controller, the guaranteed cost controller can well control the density fluctuations of the four groups of people brought by the uncertainty parameters of the system so that it can be maintained at the desired given value.Figure18 shows the change of control laws over time when the system is affected by time-varying uncertainty. As can be seen from the figure, the control law initially declines, then continues to fluctuate due to uncertainty, and stabilizes at a specific value, which is in line with our expectations.Figure 18
The temporal variation of the control law.Remark 3.
The aforementioned guaranteed cost control example of uncertain parameters is simulated under the premise ofR0>1, and the simulation is meaningless when R0<1.


## 5.1. Theoretical Verification

Figures5 and 6 show different degrees of density evolution when R0<1 and R0>1. We show that the evolution under R0<1 and R0>1 is different, although the two given R0 are nearly identical, which verifies the setting in Theorem 1. When R0<1, the parameter selects λ1=0.01,λ2=0.5,μ1=0.01,μ2=0.6,ε=0.2,δ=0.01,andγ=0.05; then, there is R0=9.9760e−04<1; when R0>1, we choose parameter λ1=0.4,λ2=0.5,μ1=0.3,μ2=0.6,ε=0.2,δ=0.6,andγ=0.05; then, there is R0=2.2446>1. It can be seen from Figure 5 that when R0<1, the density of Spreader will drop to 0; when R0>1 in Figure 6, the knowledge will exist forever, which can verify Theorems 2 and 3.Figure 5
Diagram of density change of four types of population whenR0<1.Figure 6
Diagram of density change of four types of population whenR0>1.


## 5.2. Parameter Influence

Since we take many factors into account in this model, firstly, the willingness rateδ is studied. Figure 7 shows that as the value of δ increases, so does the number of knowledge disseminators, indicating that the willingness rate δ has a positive impact on knowledge dissemination.Figure 7
The density of knowledge disseminators varies with the value ofδ.Secondly, the initial communication rateλ1 is studied. Figure 8 shows that the larger the value of λ1 in the early stage of knowledge dissemination, the greater the density of knower individuals, indicating that the value of the communication rate λ1 has a good promotion for the early knowledge dissemination process.Figure 8
The density of knower individuals changes with the value ofλ1.Finally, the degradation rate is studied. Figure9 shows that, as the value of γ increases, the density of knowledge disseminators is decreasing, indicating that the degradation rate γ hurts knowledge dissemination.Figure 9
The density of knowledge disseminators varies with the value ofγ.


## 5.3. Optimal Control

WhenR0>1, we conduct knowledge dissemination simulations under various control measures. The performance of the knowledge diffusion model is then validated with three different control strategies. For simplicity, let ι1=0.5 and ι2=0.5.


### 5.3.1.u1≠0andu2=0

We used controlu1, which increases the rate of willingness by increasing learning intensity and material rewards u1. The optimal control curve u1∗ is shown in Figure 10. u1∗ rapidly decreases to near 0 until 0. Under this control measure, knower individuals become knowledge disseminators by sharing knowledge stimulated by a reward mechanism. Figure 11 mainly describes the changes in the four groups of people under the control strategy u1, in which the spreader individuals increased slightly and the other three groups decreased slightly. Furthermore, the maximum rate of change in individual spreaders compared to the noncontrol system is about 15%, which is consistent with our expected results. When increasing the willingness of knower individuals, the knowledge dissemination model performs better.Figure 10
Diagram of the optimal solutionu1∗ of knowledge propagation.Figure 11
Comparison of population density withu1.
(a)(b)(c)(d)


### 5.3.2.u1=0andu2≠0

We used controlu2, which reduces the probability of degradation by increasing the frequency of review. The optimal control curve u2∗ is shown in Figure 12. u2∗ rapidly decreases to near 0 until 0. Under this control, the individual spreader continues to maintain the identity of the knowledge disseminator through regular review of knowledge under a supervision mechanism and excellent learning methods. Figure 13 mainly describes the changes in the four types of populations under the control strategy u2, in which the spreader individuals and knower individuals slightly increased, while the other two populations slightly decreased. Furthermore, the maximum rate of change in individual spreaders compared to the noncontrol system was about 16%, which is consistent with our expected results. When reducing the knowledge degradation of the individual spreader, the knowledge dissemination model performs better.Figure 12
Diagram of the optimal solutionu2∗ of knowledge propagation.Figure 13
Comparison of population density withu2.
(a)(b)(c)(d)


### 5.3.3.u1≠0andu2≠0

We improve the willingness of knower individualsu1 while reducing the degradation of spreader individuals u2. The optimal control curve u1∗,u2∗ is shown in Figure 14. u1∗,u2∗ rapidly decreases to near 0 until 0. Under this control, the individual spreader continues to maintain the identity of the knowledge disseminator through regular review of knowledge under a supervision mechanism and excellent learning methods. Figure 15 mainly describes the changes in the four groups of people under the two control strategies u1,u2, in which the spreader and knower individuals increased slightly and the other two groups decreased slightly. Furthermore, the maximum rate of change of individual spreaders compared to the noncontrol system is about 25%, which is consistent with our expected results. When increasing the willingness of knower individuals and reducing the degradation of spreader individuals, the knowledge dissemination model performs best.Figure 14
Diagram of the optimal solutionu1∗ and u2∗ of knowledge propagation.Figure 15
Comparison of population density withu1,u2.
(a)(b)(c)(d)


## 5.3.1.u1≠0andu2=0

We used controlu1, which increases the rate of willingness by increasing learning intensity and material rewards u1. The optimal control curve u1∗ is shown in Figure 10. u1∗ rapidly decreases to near 0 until 0. Under this control measure, knower individuals become knowledge disseminators by sharing knowledge stimulated by a reward mechanism. Figure 11 mainly describes the changes in the four groups of people under the control strategy u1, in which the spreader individuals increased slightly and the other three groups decreased slightly. Furthermore, the maximum rate of change in individual spreaders compared to the noncontrol system is about 15%, which is consistent with our expected results. When increasing the willingness of knower individuals, the knowledge dissemination model performs better.Figure 10
Diagram of the optimal solutionu1∗ of knowledge propagation.Figure 11
Comparison of population density withu1.
(a)(b)(c)(d)


## 5.3.2.u1=0andu2≠0

We used controlu2, which reduces the probability of degradation by increasing the frequency of review. The optimal control curve u2∗ is shown in Figure 12. u2∗ rapidly decreases to near 0 until 0. Under this control, the individual spreader continues to maintain the identity of the knowledge disseminator through regular review of knowledge under a supervision mechanism and excellent learning methods. Figure 13 mainly describes the changes in the four types of populations under the control strategy u2, in which the spreader individuals and knower individuals slightly increased, while the other two populations slightly decreased. Furthermore, the maximum rate of change in individual spreaders compared to the noncontrol system was about 16%, which is consistent with our expected results. When reducing the knowledge degradation of the individual spreader, the knowledge dissemination model performs better.Figure 12
Diagram of the optimal solutionu2∗ of knowledge propagation.Figure 13
Comparison of population density withu2.
(a)(b)(c)(d)


## 5.3.3.u1≠0andu2≠0

We improve the willingness of knower individualsu1 while reducing the degradation of spreader individuals u2. The optimal control curve u1∗,u2∗ is shown in Figure 14. u1∗,u2∗ rapidly decreases to near 0 until 0. Under this control, the individual spreader continues to maintain the identity of the knowledge disseminator through regular review of knowledge under a supervision mechanism and excellent learning methods. Figure 15 mainly describes the changes in the four groups of people under the two control strategies u1,u2, in which the spreader and knower individuals increased slightly and the other two groups decreased slightly. Furthermore, the maximum rate of change of individual spreaders compared to the noncontrol system is about 25%, which is consistent with our expected results. When increasing the willingness of knower individuals and reducing the degradation of spreader individuals, the knowledge dissemination model performs best.Figure 14
Diagram of the optimal solutionu1∗ and u2∗ of knowledge propagation.Figure 15
Comparison of population density withu1,u2.
(a)(b)(c)(d)


## 5.4. Robustness of Uncertain Parameter Control System

According to the characteristics of knowledge dissemination, the parameters of system (45) are as follows:(65)L=20000003000000−5000000−200,B=00−101101.Uncertain structure selects(66)D=−29.9280−9.9760010029.92839.97601−1−10000001−10−30−1001,E=1000000110000001010001000010,given(67)Q=1000010000100001,R=1001.Applying Theorem6, the corresponding linear matrix inequality (61) is available from the LMI toolbox, and a feasible solution is obtained:(68)X=−1.92921.5244−1.2840e−040.02091.5244−1.2884−0.00370.0764−1.2840e−04−0.00370.0072−0.00110.02090.0764−0.00110.0348,W=−0.02640.0198−5.5194e−05−0.0063−0.04660.0171−1.3533e−054.2526e−04,ε=0.7486.Then, we get the guaranteed cost controller and guaranteed cost index of the system:(69)u∗t=−0.0263−0.0497−0.0423−0.05800.06920.05910.0062−0.1589xt,J¯∗=142.3200.We choose the willingness rate as an uncertain parameter and the time-varying part∆δt=sint. Figure 16 is the simulation diagram of the four types of people in the IKSF dissemination system under the unguaranteed cost control. Figure 17 is the guaranteed cost control effect comparison of the IKSF knowledge dissemination system picture.Figure 16
Diagram of knowledge transmission without guaranteed cost control.Figure 17
Diagram of knowledge transmission with guaranteed cost control.Figure16 shows that when the system parameters fluctuate within the range, the density fluctuations of the four groups of people will be caused. In Figure 17, compared with no controller, the guaranteed cost controller can well control the density fluctuations of the four groups of people brought by the uncertainty parameters of the system so that it can be maintained at the desired given value.Figure18 shows the change of control laws over time when the system is affected by time-varying uncertainty. As can be seen from the figure, the control law initially declines, then continues to fluctuate due to uncertainty, and stabilizes at a specific value, which is in line with our expectations.Figure 18
The temporal variation of the control law.Remark 3.
The aforementioned guaranteed cost control example of uncertain parameters is simulated under the premise ofR0>1, and the simulation is meaningless when R0<1.


## 6. Conclusion

In this paper, the knowledge transfer process in complex networks is modeled and dynamically analyzed by considering the interaction effects of multiple mechanisms of knowledge transfer. Specifically, we established a new IKSF knowledge dissemination model, and the model simultaneously considers multiple factors such as internalization mechanism, degradation mechanism, communication, and willingness. We obtain the basic reproduction number of the scale-free network that depends on the multimechanism and determine the knowledge loss equilibriumE0 and the unique endemic equilibrium E∗. Furthermore, we studied the stability of the equilibrium point. Through numerical simulation, enhancing the willingness of knower individuals and reducing the degradation rate of spreader individuals will help to promote the dissemination of knowledge.In terms of optimal control, to increase the number of knowledge disseminators, we establish incentive mechanisms to stimulate willingness and monitoring mechanisms to reduce knowledge degradation. Experimental results show that the effect is the most noticeable when multiple controls are applied.To study the process of knowledge propagation more precisely, we study the optimal guaranteed cost control problem of knowledge propagation with uncertain parameters and control constraints. Using the Lyapunov stability theory and guarantee cost control technology, the control law of optimal guarantee cost is designed in the form of linear matrix inequality so that the four groups of people in the IKSF model are stabilized in an ideal state, and the optimal guarantee cost of knowledge dissemination is obtained. Experimental results show that the proposed method has a significant effect on dealing with parameter uncertainty problems.However, there are still many shortcomings that need further study.(1)
There are many ways to access knowledge, but it is not clear exactly how knowledge diffusion can be carried out in reality because people always have different ways of acquiring knowledge. This prevents us from thinking about access to knowledge in a particular way.(2)
This paper only conducts numerical simulations on scale-free networks and does not compare experiments with other heterogeneous networks.(3)
System parameters may be affected by various factors. We find that when system parameters change over time, the system may be in a chaotic or bifurcated state.(4)
The robust control of uncertain nonlinear systems based on the LMI method is studied, and the influence of adding different control constraints to the system needs to be further studied.These problems will be considered in future work.



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*Source: 8862323-2023-03-18.xml*