# Distributed Control and the Lyapunov Characteristic Exponents in the Model of Infectious Diseases

Authors: M. Bershadsky; M. Chirkov; A. Domoshnitsky; S. Rusakov; I. Volinsky
Journal: Complexity
 (2019)

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## Abstract

The Marchuk model of infectious diseases is considered. Distributed control to make convergence to stationary point faster is proposed. Medically, this means that treatment time can be essentially reduced. Decreasing the concentration of antigen, this control facilitates the patient’s condition and gives a certain new idea of treating the disease. Our approach involves the analysis of integro-differential equations. The idea of reducing the system of integro-differential equations to a system of ordinary differential equations is used. The final results are given in the form of simple inequalities on the parameters. The results of numerical calculations of simulation models and data comparison in the case of using distributive control and in its absence are given.


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## Body

## 1. Introduction

Distributed feedback control system is a challenging problem, but only a few papers were devoted to it (see, for example, [1] and literature therein). A noise in the feedback delay control is one of the main reasons for developing mathematical models with distributed inputs. The second one is resulted from the fact that the dynamics of the processes rather depends on the average value of the process than on the value at a corresponding moment.We propose a method reducing the stability analysis of the system with memory to the analysis of a corresponding finite-dimensional system. This idea was proposed, as well as we know, in [2] and realized, in finite spectrum assignment (see, for example, [3–6]). Our realization of this idea is different. We propose a simple method that allows to reduce the analysis of systems with memory to one of a higher order but finite-dimensional systems. Questions of stability and estimates of solutions can be considered on this base.The model of infectious diseases, constructed by Marchuk in his well-known book [7], reflects the most significant patterns of the immune system acting during these diseases. This model was studied in many works; note, for example, the recent papers [8–12] and the bibliography therein. The adding control was proposed, for example, in [11–14]. In [10, 15], the basic mathematical model that takes into account the concentrated control of the immune response was proposed. It can be noted that the use of information about behavior of a disease and the immune system for a long time (defined by distributed control, for example, in the form of an integral term) looks very natural in choosing strategy of a possible treatment. Optimal control in the basic model of the infectious diseases was considered in the work [15], where the control function characterizing the realization of an immunotherapy which consists in administration of ready immunoglobulins or donor antibodies is proposed. In [16], the model of influence of an immunotherapy on dynamics of an immune response which represents a generalization of basic model was considered. On the basis of the proposed model, the problem of determination of coefficients on the basis of laboratory data was considered and also the management was proposed in [13, 14, 17]. Such task is called control in uncertain conditions [11]. The control algorithm in the case of uncertain conditions was proposed in the work ([15], see pages 71–73).The Lyapunov exponents characterize the rate at which solutions approach each other with increasing time. In our case, one of these solutions is a certain stationary state of the system. The stationary state determines the conditions of recovery of the body after exposure to infection. The speed of approaching this state has a very important role. It allows us to estimate the duration of treatment. In some cases, this can determine the choice of treatment strategy. It is enough to imagine a situation when the patients’ body is weakened by some chronic diseases and is not able to endure too long treatment.Our approach to treatment modeling can be described as follows. We introduce a distributed feedback control in the form of an integral term. This management is based on long-term monitoring of the patient’s condition and comparing the patients’ condition with a certain stationary state over a long period of time. Thus, a closed controlled system, the purpose of which is to improve the patient’s condition, is created. Intuitively, such a treatment strategy is understandable. Further, the scheme of reducing the integro-differential system to a system of ordinary differential equations is proposed. This ordinary differential system is studied. The technique based on properties of the Cauchy matrices is used to estimate possible deviations of solutions in the case of various modifications of this model, such as nonlinear models or delay models, from the solution of this system ordinary differential equation. Note the papers [1, 18–21] where the distributed control was also considered for various problems.In this paper, we consider the Marchuk model of infectious diseases [7]:(1)dVdt=βVt−γFtVt,dCdt=ζmαFtVt−μcCt−C∗,dFdt=ρCt−ηγFtVt−μfFt,dmdt=σVt−μmmt,where Vt is the antigen concentration rate, Ct is the plasma cell concentration rate, Ft is the antibody concentration rate, and mt is the relative features of the body. Denote α,β,γ,ρ,σ,η,μc,μf,andμm as corresponding coefficients described in [7]. ζm takes into account the destruction of the normal functioning immune system, ζ0=1. Denote C∗ as the plasma rate concentration of the healthy body and F∗ the antibody concentration that we wish to achieve after the treatment.Let us discuss every equation in the model (1) in more detail. The first equation dV/dt=βVt−γFtVt presents the block of the virus dynamics. It describes the change in the antigen concentration rate and includes the amount of antigen in the blood. Antigen concentration decreases as a result of interaction with antibodies. The immune process is characterized by the number of antibodies, whose concentration is described by the equation dF/dt=ρCt−ηγFtVt−μfFt. The value of Ft decreases as a result of the interaction and mutual destruction with antigens. The amount of the antibody cells also decreases as a result of natural destruction. However, the plasma restores antibodies and therefore the plasma state plays an important role in the immune process. Thus, the change in the plasma cell concentration rate is included in several equations of the system. Taking into account the healthy body level of plasma cells and their natural aging, the term μcCt−C∗ is included in the second equation of system (1). It is assumed that the plasma is restored as a result of the interaction of the antigen and the antibody cells. The second and third equations present the block of the humoral immune response dynamics.Concerning the last equationdm/dt=σVt−μmmt of system (1), we can note the following. The value of m increases with the antigen’s concentration rate Vt. The maximum value of m is 1 in the case of 100% organ damage and is equal to 0 for a fully healthy organ. The coefficient μm describes the rate of degeneration of the target organ.The main goal of this paper is to demonstrate new ideas in the use of distributed control in the model of infectious diseases [7]. Our goal is to make convergence to a stationary point faster. This allows us to decrease the duration of disease’s treatment. We add a distributed control in the equation describing antibody concentration rate. Numerical simulations demonstrate that this approach could open some new possibilities for a treatment.


## 2. Modeling Distributed Control

Let us consider the following system:(2)dVdt=βVt−γFtVt,dCdt=ζmαFtVt−μcCt−C∗,dFdt=ρCt−ηγFtVt−μfFt−but,dmdt=σVt−μmmt.Comparing this system with the model of Marchuk (1), we see the control ut added in the third equation describing the antibody concentration rate. Note that this control is a reasonable one from the medical point of view [10, 15].We choose the controlut in the following form:(3)ut=∫0tFs−F∗e−kt−sds.It can be noted that influence of a corresponding average value instead ofFt−F∗ at the point t looks reasonable since the control ut is rather dependent on an “average” value of the difference Fs−F∗ on a corresponding time-interval than on this difference at the moment t only. The integral term in (3) increases the influence of the previous moments which are closer to the current moment t.Following [15], we can pass to the dimensionless case, substituting Vt=vtVm,Ct=stC∗,Ft=ftF∗,andut=u˜tF∗ in (2). In the dimensionless case, system (2) is of the following form:(4)dvdt=βvt−γF∗ftvt,dsdt=αVmF∗C∗ζmftvt−μcst−1,dfdt=ρC∗F∗st−ηγVmftvt−μfft−b∫0tfs−1e−kt−sds,dmdt=σVmvt−μmmt.Consider the system(5)dvdt=βvt−γF∗ftvt,dsdt=αVmF∗C∗ζmftvt−μcst−1,dfdt=ρC∗F∗st−ηγVmftvt−μfft−bu˜t,dmdt=σVmvt−μmmt,du˜dt=ft−1−ku˜t.Denoting in (5)(6)α1=β,α2=γF∗,α3=αVmF∗C∗,α4=μf=ρC∗F∗,α5=μc,α6=σVm,α7=μm,α8=ηγVm,we obtain(7)dvdt=α1vt−α2ftvt,dsdt=α3ζmftvt−α5st−1,dfdt=α4st−ft−α8ftvt−bu˜t,dmdt=α6vt−α7mt,du˜dt=ft−1−ku˜t.Let us find possible stationary points of system (7). Their coordinates v,s,f,m,andu˜ satisfy the following algebraic system:(8)α1v−α2fv=0,α3ζmfv−α5s−1=0,α4s−f−α8fv−bu˜=0,α6v−α7m=0,f−1−ku˜=0.Only in two cases, the first equation is satisfied:v=0 or f=α1/α2. Let us start with the first case. If v=0, then s=1 from the second equation and m=0 from the fourth one. We have the system for f and u˜:(9)α4−α4f−bu˜=0,f−ku˜=1.From the last equation, we haveu˜=f−1/k, and substituting this into the first one, we obtain f=1, and then u˜=0. Thus, starting with v=0, we can come only to the stationary point v=m=u˜=0,s=f=1. In the second case of f=α1/α2, we consider only the case of v≠0, since in the case of v=0, we come to the same stationary point. From the fourth equation α6v=α7m, we obtain that m≠0. This means that there is no possibility to completely disinfect the affected organ. That is why we study below the behavior of solutions only in the neighborhood of the noted above stationary point v=m=u˜=0,s=f=1.Remark 1.
It was obtained in [15] on the basis of the laboratory data that α1=0.25;α2=8.5000332;α3=1792175675;α4=1.95992344⋅10−7;α5=0.5;α6=10;α7=0.4;α8=1.7⋅10−3.
It was noted above thatζ0=1. Linearizing systems (7) and (4) in the neighborhood of the stationary point, we obtain the corresponding linear systems:(10)dx1dt=α1−α2x1,dx2dt=α3x1−α5x2,dx3dt=−α8x1+α4x2−α4x3−bx5,dx4dt=α6x1−α7x4,dx5dt=x3−kx5,(11)dx1dt=α1−α2x1,dx2dt=α3x1−α5x2,dx3dt=−α8x1+α4x2−α4x3−b∫0tx3se−kt−sds,dx4dt=α6x1−α7x4,where x1=v,x2=s−1,x3=f−1,x4=m,x5=u˜, respectively. Denote the matrix of the coefficients of system (10):(12)A=α1−α20000α3−α5000−α8α4−α40−bα600−α700010−k.
Note that a corresponding linear system for the Marchuk model of infectious diseases without control (see system (1)) can be written in the form:(13)dx1dt=α1−α2x1,dx2dt=α3x1−α5x2,dx3dt=−α8x1+α4x2−α4x3,dx4dt=α6x1−α7x4.


## 3. About Comparison of the Lyapunov Exponents

Our approach is based on the following auxiliary assertion.Lemma 1.
The solution-vectorcolvt,st,ft,mt of system (4) and 4 first components of the solution-vector colvt,st,ft,mt,u˜t of system (5) satisfying the condition u˜0=0 coincide.Proof of Lemma 1.
Let us consider the last equation of system (5). Using the representation of solution of the first-order scalar equation, we can write u˜t=∫0te−kt−sfs−1ds+u˜0e−kt. Note that system (5) is considered with the condition u˜0=0. Substituting now this representation of u˜t into the third equation of system (5), we obtain the third equation of system (4). Thus in systems (4) and (5), the first, second, and fourth equations coincide and the third equation of (4) is equivalent to the last equation of system (5) with the initial condition u˜0=0.
This completes the proof of Lemma 1.Remark 2.
It is clear that Lemma 1 can be used also for integro-differential system (11) and system of ordinary differential equations (10). Every solution-vector x1t,x2t,x3t,x4t of integro-differential system (11) coincides with the first 4 components of the solution-vector.
Takex1t,x2t,x3t,x4t,x5t of system (10) with the initial condition x50=0. Thus, if we take a part of the space of solutions of ordinary differential system (10), satisfying the initial condition x50=0 and delete then the fifth component of solution-vectors, we come to the 4-space of solutions of integro-differential equation (11). Now, it is clear that the exponential stability of the 5-dimensional ordinary differential system (10) implies the exponential stability of 4-dimensional integro-differential system (11). The roots of the characteristic equation for ordinary differential system (10) can be used in the representation of vector-solution in the space of solutions of integro-differential system (11). Negativity of real parts of the roots of the characteristic equation of system (10) with constant coefficients allows to make the conclusion about the exponential stability of the integro-differential system (11). The root with maximal real part of the characteristic equation for (11) cannot be greater than the one of the characteristic equations for (10).
The characteristic polynomial of system (10) of ordinary differential equations,(14)P5λ∗=λ∗−α1+α2λ∗+α5λ∗+α7λ∗2+α4+kλ∗+kα4+b,has 5 roots λ1∗,λ2∗,λ3∗,λ4∗,andλ5∗.
Denoteλi,i=1,4¯ as the roots of the characteristic polynomial of systems (13), and λ˜=max1≤i≤4λi,λ∗˜=max1≤j≤5Reλj∗.Theorem 1.
Ifβ<γF∗, b>0 and k>0, then integro-differential system (11) is exponentially stable, and if in addition, the inequality k>α4 is fulfilled then λ˜≥λ∗˜.The proof is based on the following assertion.Lemma 2.
For the roots of the characteristic polynomials of systems (11) and (13), the following facts are true:(15)λ1=λ1∗,λ2=λ2∗,λ3=λ3∗,and if k>α4,b>0, then(16)λ4>Reλ4∗,λ4>Reλ5∗.Proof of Lemma 2.
The characteristic polynomial of systems (13) is(17)P4λ=λ−α1+α2λ+α4λ+α5λ+α7,and the characteristic polynomial of system (10) is (14).
It is clear from the definition of the coefficientsαj (6) that(18)λ2=λ2∗=−α5<0,λ3=λ3∗=−α7<0,λ4=−α4<0.
The inequalityβ<γF∗ in the conditions of Theorem 1 implies that α1−α2<0 and consequently λ1=λ1∗=α1−α2<0. It is clear that(19)λ4∗=−α4+k−α4−k2−4b2,λ5∗=−α4+k+α4−k2−4b2.
In the case of complex rootsλ4∗ and λ5∗, we have Reλ4∗=Reλ5∗<0 if k>0. In the case of real roots λ4∗ and λ5∗, we have λ4∗≤λ5∗<0. Thus we obtain the exponential stability of system (10). According to Remark 2 mentioned above, system (11) is exponentially stable.
Comparing the values of the roots of the characteristic polynomials (17) and (14) we have to verify only the inequality(20)Reλ5∗<λ4,in (16). Substituting the values of λ5∗ and λ4 to (20), we have to obtain the inequalities:(21)−α4+k+α4−k2−4b2<−α4,for the case of real roots λ4∗andλ5∗, and(22)−α4+k2<−α4,for the case of the complex roots λ4∗andλ5∗.
The inequalities (21) and (22) are fulfilled if k>α4 and b>0.
Substituting the values ofαi in Remark 1, we see that λ4>max1≤i≤3λi. If the inequalities k>α4andb>0 are fulfilled, then (16) is true.
Note the assertion following from Theorem 1.Corollary 1.
Let the coefficientsαi be defined by the formulas in Remark 1, and k>1.95992344⋅10−7,b>0, then the Lyapunov exponents of system (11) are less than the ones of system (13).


## 4. Numerical Simulations and Comments

The mathematical model can be represented in the following form:(23)dxdt=Φxt,xt=colvt,st,ft,mt,u˜t,where Φxt is the right-hand side of (7). Values of the parameters are defined in Remark 1, k=4andb=1 and the initial conditions are v0=10−6,f0=1,s0=1,m0=0andu0=0. We use the second-order Runge–Kutta’s scheme with a constant step h.In Figures1–4, the solution of model with the natural flow of data without the control of disease is presented by curves of red color and disease in the case of considered distributed control by curves of green color.Figure 1
Dynamics of the immune response: antigen.Figure 2
Dynamics of the immune response: plasma.Figure 3
Dynamics of the immune response: antibodies.Figure 4
Dynamics of the immune response: rate of the destroyed cells.Figure1 demonstrates the dynamics in antigen concentration during the course of the disease. The insert detailing the process in the first two days was performed on a different scale and demonstrates the fact that the management transfers the disease from the acute form to the subclinical one (the antigen concentration only decreases after injection). Figure 2 demonstrates the dynamics in plasma cell concentration during the disease process. It can be seen from the figure that control leads to a faster increase in the concentration of plasma cells, which in this case ensures a transition to the subclinical form of the disease. In addition, it is necessary to note a fourfold increase in the maximum concentration of plasma cells in the case of control, compared with the option without control. Figure 3 demonstrates the dynamics in antibody concentration during the disease process. The graph shows that the concentration of antibodies in the solution with control practically does not change, because in this case, they are replaced by donor antibodies, which is what the control actually consists of. The dynamics in the proportion of target organ cells destroyed by antigen during the disease process is presented in Figure 4. The values for the variant with control are given with an increase of 104 times. Thus, control allows to reduce the maximum proportion of affected cells of the target organ by more than 2.5×104 times. The dynamics of the control (concentration of donor antibodies) during the disease process is presented in Figure 5.Figure 5
Control function.Remark 3.
The integral term can accumulate small mistakes made in the process of numerical integration. This explains difficulties in the use of numerical methods for solving integro-differential equations. The idea to reduce a system of integro-differential equations to a corresponding system of ordinary differential ones and the use of the well-developed technique for their solution could be one of the possible ways to develop numerical methods for integro-differential equations.


## 5. System with Delay in Equation of Plasma Concentration

It was explained in [7] why the delay τt can appear in the second equation of system (1) in the model of infectious diseases:(24)dVdt=βVt−γFtVt,dCdt=ζmαFt−τtVt−τt−μcCt−C∗,dFdt=ρCt−ηγFtVt−μfFt,dmdt=σVt−μmmt.We have to define what should be set instead ofVt−τt and Ft−τt when t−τt<0. We consider all delay systems with the initial function Vζ=0 for ζ<0. Let us consider the following system with the control in antibody concentration:(25)dVdt=βVt−γFtVt,dCdt=ςmαFt−τtVt−τt−μcCt−C∗,dFdt=ρCt−ηγFtVt−μfFt−but,dmdt=σVt−μmmt,where ut=∫0tFs−F∗e−kt−sds.We can pass to dimensionless case(26)dvdt=βvt−γF∗ftvt,dsdt=αVmF∗C∗ςmft−τtvt−τt−μcst−1,dfdt=ρC∗F∗st−ηγVmftvt−μfft−bu˜t,dmdt=σVmVt−μmmt,du˜dt=ft−1−ku˜t.Denotingαii=1,…,8, according to (6), we obtain(27)dvdt=α1vt−α2ftvt,dsdt=α3ςmft−τtvt−τt−α5st−1,dfdt=α4st−ft−α8ftvt−μfft−bu˜t,dmdt=α6Vt−α7mt,du˜dt=ft−1−ku˜t.For stability studies, only behavior of solutions for sufficiently larget is considered. Below, we assume that t−τt≥0. We can write the second equation in the following form:(28)dsdt=α3ςmftvt−∫t−τttfφvφ′dφ−α5st−1,dsdt=α3ςmftvt−∫t−τttf′φvφ+fφv′φdφ−α5st−1.We can write the expression under the integral in the following form:(29)f′φvφ+fφv′φ=vφα4sφ−fφ−α8fφvφ−μffφ−bu˜φ+fφα1vφ−α2fφvφ=α4sφ−1+1−fφ−1+1vφ−α8fφv2φ−μffφvφ−bu˜φvφ+α1vφfφ−1+1−α2f2φvφα1vφ.Linearizing system (27) in the neighborhood of the stationary point(30)v=m=u˜=0,s=f=1,we obtain(31)dx1dt=α1−α2x1,dx2dt=α3x1−α5x2−α3∫t−τttα1x1φdφ,dx3dt=−α8x1+α4x2−α4x3−bx5,dx4dt=α6x1−α7x4,dx5dt=x3−kx5.Consider the system(32)X′t=PtXt+Gt,where Pt is a n×n-matrix and Gt is n-vector. The general solution Xt=colx1t,…,xnt can be represented in the following form:(33)Xt=∫0tCt,sGsds+Ct,0X0,where n×n-matrix Ct,s=Cijt,si,j=1n is called the Cauchy matrix. Its j-th column (j=1,…,n) for every fixed s as a function of t is a solution of the corresponding homogeneous system:(34)X′t=PtXt,satisfying the initial conditions xis=δij, where(35)δij=1,i=j,0,i≠j,i=1,…,n.Construction of the Cauchy matrix of system with ordinary differential equations can be found, for example, in [22].Let us denoteτ∗=esssupt≥0τt.Theorem 2.
Ifβ<γF∗,k>0,b>0 and(36)τ∗max1≤j≤5supt≥0∫0tC2jt,sds<1α3α1,where Ct,s=Cijt,si,j=15 is the Cauchy matrix of system (10), then system (31) is exponentially stable.Proof of Theorem 2.
Consider the nonhomogeneous system corresponding to homogeneous system (31):(37)dx1dt=α1−α2x1+g1t,dx2dt=α3x1−α5x2−α3∫t−τttα1x1φdφ+g2t,dx3dt=−α8x1+α4x2−α4x3−bx5+g3t,dx4dt=α6x1−α7x4+g4t,dx5dt=x3−kx5+g5t,where g1t,…,g5t are measurable essentially bounded functions on the semiaxis (i.e., they are from the space L∞).
We can write system (37) in the following form:(38)x′t=Axt+Fxt+Gt,where(39)xt=colx1t,x2t,x3t,x4t,x5t,Fxt=col0,−α3∫t−τttα1x1φdφ,0,0,0,Gt=colg1t,g2t,g3t,g4t,g5t,and A is defined by (12). It is clear that the use of the standard formula of solutions’ representation leads to the following system:(40)xt=∫0tCt,sFxsds+∫0tCt,sGsds+Ct,0x0.
It follows from Theorem 1 that the Cauchy matrixCt,s satisfies the exponential estimate, i.e., there exist such positive γ and N, then all elements Cijt,s of the Cauchy matrix Ct,s satisfy the following estimate:(41)Cijt,s≤Ne−γt−s,0≤s≤t<∞,i,j=1,…,5.
Now it is clear that the vector-functionG˜t≡∫0tCt,sGsds+Ct,0X0 is bounded for t∈0,∞.
Define the operatorK:C5⟶C5 (C5 is the space of 5-dimensional vector-functions x:0,∞ with continuous components) by the following formula:(42)Kxt=∫0tCt,sFxsds.
System (40) can be rewritten in the form(43)xt=Kxt+G˜t,where(44)G˜t=∫0tCt,sGsds+Ct,0x0.
The inequality (36) implies that the norm of the operator K:C5⟶C5 is less than one, then there exists the operator I−K−1:C5⟶C5 and it is bounded. Thus, the solution-vector xt is bounded for t∈0,∞, for every bounded on the semiaxis right-hand side Gt. Now the exponential estimate of the solution xt of the homogeneous system (31) follows the Bohl–Perron theorem [23, 24]. We have proven the exponential stability of system (31).Remark 4.
It is clear that system (31) is exponentially stable if β<γF∗,k>0,b>0 for sufficiently small delay τt.Remark 5.
Let us compare the difference of solutionxt=colx1t,…,x5t of system (10) and solution yt=coly1t,…,y5t of system (31) with the same initial conditions x0=y0. It is clear that(45)y′t−x′t=Ayt−xt+Fyt.
Denotezt=yt−xt. zt satisfies the following equation:(46)z′t=Azt+Fyt.
The solutionzt can be represented in the following form:(47)zt=∫0tCt,sFysds.
It follows from the exponential estimates ofCt,s and yt that zt tends to zero exponentially, when t⟶∞.
In order to estimatezt, we can use the results of the paper [25], where the elements of the Cauchy matrix Ct,s were constructed.


## 6. Construction of the Cauchy Matrix and Stability of Model with Delay

In order to use Theorem 2, we have to obtain the estimates ofmax1≤j≤5supt≥0∫0tC2jt,sds. In this section, we present these estimates.Let us denotec=a12−2a1a2+a1a4+a1k+a22−a2a4−a2k+a4k+b/a1a8−a2a8−a3a4+a5a8,(48)α31=−2ba4−k+a4−k2−4b,α32=−2ba4−k−a4−k2−4b,α24=−a4a5−a4k−a52+a5k−ba4,α34=−a5+k,α15=−ca5+a1−a2,α25=−ca3,α35=a1−a2+k,α45=−a5+a1−a2a6ca1−a2+a7.Lemma 3 (see [25]).
Ifa4−k2>4b, then the second column of the Cauchy matrix of (10) is as follows:(49)C→2t,s=α32−α34α24α31−α3200−2ba4−k+a4−k2−4b01e−a4−k+a4−k2−4b/2t−s−α31−α34α24α31−α3200−2ba4−k−a4−k2−4b01e−a4−k−a4−k2−4b/2t−s+1α240−a4a5−a4k−a52+a5k−ba4−a5+k01e−a5t−s.
Let us denoteβ31=−2b/a4−k,β52=2/a4−k,β24=−a4a5−a4k−a52+a5k−b/a4,β34=−a5+k,β15=−ca5+a1−a2,β25=−ca3,β35=a1−a2+k,β45=−a5+a1−a2a6c/a1−a2+a7.Lemma 4 (see [25]).
fa4−k2=4b, then the second column of the Cauchy matrix of (10) is as follows:(50)C→2t,s=−β34β24β3100−2ba4−k01−β31−β34β31β24β5200002a4−k+t−s00−2ba4−k01e−a4+k/2t−s+1β240−a4a5−a4k−a52+a5k−ba4−a5+k01e−a5t−s.
Let us denoteγ32=4b−a4−k2/2,γ24=−a4a5−a4k−a52+a5k−b/a4,γ15=−ca5+a1−a2,γ25=−ca3,γ35=a1−a2+k,γ45=−a5+a1−a2a6c/a1−a2+a7.Lemma 5 (see [25]).
Ifa4−k2<4b, then the second column of the Cauchy matrix of (10) is as follows:(51)C→2t,s=−1γ24⋅00k−a4201⋅e−a4+k/2t−s⋅cos4b−a4−k22t−s+00−4b−a4−k2200⋅e−a4+k/2t−s⋅sin4b−a4−k22t−s12a4−3k+2a5γ24γ32⋅004b−a4−k2200⋅e−a4+k/2t−s⋅cos4b−a4−k22t−s+00a4−k201⋅e−a4+k/2t−s⋅sin4b−a4−k2/2t−s1γ24⋅0−a4a5−a4k−a52+a5k−ba4a5−k01⋅e−a5t−s.
The following assertions are results of substitutions ofC2jt,s presented in Lemmas 3–5 into inequality (36) in Theorem 2.Theorem 3.
Ifα4−k2>4b, β<γF∗,k>0,b>0, and(52)τ∗maxα32−α34α24α31−α322ba4−k+a4−k2−4b2−a4−k+a4−k2−4b+α31−α34α24α31−α322ba4−k−a4−k2−4b2−a4−k−a4−k2−4b+1α24a4a5−a4k−a52+a5k−ba41a5α32−α34α24α31−α322−a4−k+a4−k2−4b+α31−α34α24α31−α322−a4−k+a4−k2−4b+1α241a51α24a5+ka5<1α3α1,then system (31) is exponentially stable.Theorem 4.
Ifα4−k2=4b, β<γF∗,k>0,b>0, and(53)τ∗max1β24a4a5−a4k−a52+a5k−ba41a5β34β24β314ba42−k2+β31−β34β31β24β522ba4−k4a4+k2+1β24a5+ka5β34β24β312a4+k+β31−β34β31β24β524a42−k2+β31−β34β31β24β524a4+k2+1β241a5<1α3α1,then system (31) is exponentially stable.Theorem 5.
Ifα4−k2<4b, β<γF∗,k>0,b>0, and(54)τ∗max1γ24a5a4a5−a4k−a52+a5k−ba41γ24a4−ka4+k+1γ244b−a4−k2a4+k+12a4−3k+2a5γ24γ324b−a4−k2a4+k+12a4−3k+2a5γ24γ32a4−ka4+k+1γ24a5−ka51γ242a4+k+a4−3k+2a5γ24γ321a4+k+1γ241a5<1α3α1,then system (31) is exponentially stable.



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*Source: 5234854-2019-11-13.xml*