# The Strong Stability of Optimal Nonlinear Dynamical System in Batch Fermentation

Authors: Hongli Wang; Bing Tan; Xiaohua Gao; Enmin Feng
Journal: Journal of Mathematics
 (2022)

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## Abstract

For the bio-dissimilation of glycerol to 1,3-propanediol byKlebsiella pneumoniae, the nonlinear dynamical system of the complex metabolism in microbial batch fermentation is studied in this study. Since the analytical solution and equilibrium point cannot be found for the nonlinear dynamical system of batch fermentation, the system stability cannot be analyzed using general methods. Therefore, in this study, the stability of the system is analyzed from another angle. We present the corresponding linear variational system for the solution to the nonlinear dynamical system of complex metabolism. In addition, the boundedness of fundamental matrix solutions for the linear variational system is obtained. With this in mind, strong stability with respect to the perturbation of the initial state vector is proved for the nonlinear dynamical system of the complex metabolism.


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## Body

## 1. Introduction

The chemical substance 1,3-propanediol (simply denoted as 1,3-PD) is an important raw material for many products. It is produced by using two methods: chemical synthesis and microbial fermentation. As for the chemical synthesis, the raw material is petroleum, and the catalyst is precious metal, and serious environmental pollution and high costs are generated in the production process. As for microbial fermentation, waste glycerol from biodiesel production is used as raw material. Microbial fermentation has the advantages of environmental friendliness, simple operation, and fewer by-products. Hence, 1,3-PD production through the microbial fermentation of glycerol has been widely studied due to its great research and practice value [1–4].During 1,3-PD production through microbial fermentation, the experimental scheme is firstly designed. 1,3-PD is produced by means of microbial fermentation of glycerol, and the extracellular concentration variations of substances in the fermentation are directly tested. Obviously, a substantial amount of human and material resources and funds should be invested in the experiment, and the experimental data measured might be different even under the same conditions due to the long duration of the experiment and the susceptibility to the environment. Most importantly, the intracellular concentration cannot be measured through this experimental method. Therefore, it is challenging to complete considerable experimental schemes limited by time, funds, and human resources.If a numerical experiment of 1,3-PD production through microbial fermentation is conducted on a computer, the data which cannot be collected in the lab will be obtained, such as the intracellular concentration. However, the numerical experiment should be made on the basis of a mathematical model that reflects the real experimental fermentation. It exhibits the main advantages of completing multiple experimental schemes and saving costs of time, human resources, and funds, as well as the disadvantages of requiring reliability tests, namely, identifying the mathematical model.Batch fermentation is a technique to produce 1,3-PD from glycerol (also named substrate). In 1995, Zeng and Deckwer [5] firstly carried out quantitative research on microbial batch fermentation without providing the specific form of the model. In 2000, Xiu et al. [6] proposed the dynamic model during the microbial transformation based on the Monod-type material balance equation and studied the multistability, but the influence of superfluous substrates was not considered in this model. Yuan et al. [7] considered an optimal minimal variation control and proposed a parallel algorithm based on the genetic algorithm and the gradients of the constraint functions with respect to decision variables. The robust biobjective optimal control of 1, 3-PD microbial batch production process is researched in [8]. Wang et al. [9] formulated the batch process as an optimal control problem subject to continuous state constraints and stochastic disturbances and proposed a modified particle swarm algorithm.In the fermentation process, factors such as time delay need to be taken into account. In addition, studies on the stability of the system are necessary. Modelling and optimal state-delay control in microbial batch process is studied in [10]. An optimal state-delay control strategy is proposed in nonlinear dynamical systems [11]. The optimal control of glycerol producing 1,3-PD in the batch process with uncertain time-delay and kinetic parameters is studied and a novel gradient-based solution method is proposed in [8]. Robust optimal control for a batch nonlinear enzyme-catalytic switched time-delayed process with noisy output measurements is given in [12]. Chio and Koo [13, 14] studied the stability of the linear dynamical system and nonlinear dynamical system in accordance with the u∞-similarity. Zheng and Gao [15, 16] studied multiple Lyapunov functions of the stable positive linear system and the switched linear system.In this study, as for the stability of the nonlinear dynamical system without analytical solution and equilibrium point, properties and relevant theory of the fundamental matrix solution of the linear variational system were utilized to prove the strong stability of the optimal nonlinear dynamical system of complex metabolism of a type of batch fermentation.The remainder of the study is organized as follows. Section2 presents the optimal nonlinear dynamical system of complex metabolism. Section 3 provides the linear variational system and fundamental matrix solution. Section 4 analyzes the strong stability of the optimal nonlinear dynamical system of complex metabolism. Finally, Section 5 concludes the study.


## 2. Optimal Nonlinear Dynamical System of Complex Metabolism

With transmembrane transport of glycerol (also named substrate) and 1,3-PD considered, three means of transport are assumed, including active transport and passive diffusion, as well as active transport and passive diffusion, denoted asA, P, and AP, respectively. GTW stands for the transport pathway of glycerol and PDTW represents the pathway of 1,3-PD. D≔ζ,τ∈0,12 is defined as the set of possible metabolic pathways, where ζ and τ are parameters of pathways. As ζ and τ are continuous variables, there will be numerous metabolic pathways. The transport pathway of glycerol corresponding to ζ is as follows:ζ=0, when GTW is P (passive)ζ=1, when GTW is A (active)ζ∈0,1, when GTW is AP (active and passive)The transport pathway of 1,3-PD corresponding toτ, namely, PDTW, is as follows:τ=0, when PDTW is P (passive)τ=1, when PDTW is A (active)τ∈0,1, when PDTW is AP (active and passive)Letxt=x1t,x2t,…,x8t⊤∈ℝ8, t∈0,tf⊂R+, be the state vector whose components are the concentrations of biomass, extracellular glycerol, extracellular 1,3-PD, acetate, ethanol, intracellular glycerol, 3-HPA, and intracellular 1,3-PD at time t∈0,tf, respectively, and x0∈ℝ8 is an initial state. The complex metabolism system corresponding to ζ,τ is denoted as CMSζ,τ, which is represented as nonlinear dynamical system as follows:(1)x˙t=hx,uζ,τ,ζ,τ,t∈0,tf,x0=x0,where uζ,τ=uζ,τ1,uζ,τ2,…,uζ,τ12∈ℝ12 is the system parameter of CMSζ,τ, and the components of hx,uζ,τ,ζ,τ=h1x,uζ,τ,ζ,τ,h2x,uζ,τ,ζ,τ,…,h8x,uζ,τ,ζ,τ⊤ are shown in [17]. The admissible set of system parameters is defined as Uad≔ul;uu=∏j=112ulj,uju⊂ℝ12, and the admissible domain of state variables is defined as W0≔xl;xu=∏j=18xlj,xju⊂ℝ+8. The values of ul, uu, xl, and xu can be obtained from [17].For the metabolic system, there are properties as follows:Property 1.
ζ,τ∈0,1 is given as follows:(I)
∀x∈W0, hx,uζ,τ,ζ,τ about uζ,τ on Uad is continuous.(II)
∀uζ,τ∈Uad, hx,uζ,τ,ζ,τ about x on W0 is Lipschitz continuous.(III)
∀uζ,τ∈Uad, hx,uζ,τ,ζ,τ satisfies the linear growth conditions. That is, there exist constants α,β>0, making hx,uζ,τ,ζ,τ≤αx+β. It is defined that the norm of vector function x=xt:0,tf⟶ℝn is xt=x≔maxs∈0,tfxs, where ⋅ represents the norm of the vector on Rn, that is, ∀x∈ℝn, x≔∑i=1nxi; and the norm of matrix function At:0,tf⟶ℝn×n is At=A≔maxs∈0,tfAs, where ⋅ represents the norm of the matrix on Rn×n, that is, ∀A=aijn×n∈ℝn×n and A≔maxj∈In∑i=1naij.Property 2.
ζ,τ∈0,1 is given, for ∀x0∈W0 and ∀uζ,τ∈Uad, and there is a unique solution in the system CMSζ,τ, denoted by xt;x0,uζ,τ,ζ,τ. The solution xt;x0,uζ,τ,ζ,τ about parameter uζ,τ∈Uad and initial state x0∈W0 is continuous.
Suppose thatIN≔1,2,…,N presents the experimental sequence set, and it is preset that ζ,τ∈0,1, ∀l∈IN, and ∀x0l∈W0. The solution set of CMSζ,τ is denoted as(2)S0ζ,τ≔xt;x0l,uζ,τ,ζ,τ|xt;x0l,uζ,τ,ζ,τ∈W0,x0l∈W0,uζ,τ∈Uad,l∈IN.
The feasible set of the parameter vectorsu is defined as(3)US0ζ,τ≔uζ,τ∈Uad|xt;x0l,uζ,τ,ζ,τ∈S0ζ,τ,∀t∈0,tf,l∈IN.
According toH3 and Property 3 [18], for ∀ζ,τ∈0,1, S0ζ,τ and Us0ζ,τ are nonempty sets, and for given ζ,τ∈0,1, S0ζ,τ and Us0ζ,τ are compact.
Due to the lack of data on the concentration of intracellular substances, biological robustness is introduced to judge the reliability of the numerical solution of System (1). The smaller the value of biological robustness, the more robust the system is [19]. The identification model (abbreviated as IM) of CMSζ,τ about the pathway variable and the system variable is given as follows:(4)IM:minJuζ,τ,ζ,τ,s.t.uζ,τ∈Uwζ,τ,ζ,τ∈0,1,where Juζ,τ,ζ,τ is the biological robustness for uζ,τ of the solution to system CMSζ,τ.(5)Juζ,τ,ζ,τ:=1N⋅tf∑l=1N∫Uϕuζ,τ′⋅∫0tfxt;x0l,uζ,τ′,ζ,τ−xt;x0l,uζ,τ,ζ,τ/xt;x0l,uζ,τ,ζ,τdtuζ,τ′−uζ,τduζ,τ′,where U⊂Uwζ,τ is the perturbation space of uζ,τ and ϕuζ,τ′ is the probability density function of the uniform distribution in uζ,τ′ on U. Uwζ,τ≔uζ,τ|uζ,τ∈Us0ζ,τ,REuζ,τ,ζ,τ≤ε is the set of uζ,τ, and ε>0 is the allowable error limit of REu,ζ,τ. Assume that there exits ζ,τ such that Uwζ,τ is nonempty. Since Us0ζ,τ is compact, the set Uwζ,τ is compact for some ζ,τ∈0,1. The relative error (abbreviated as RE) between calculated concentration xjt;x0l,uζ,τ,ζ,τ, j∈I3 and measured concentration yjlt, j∈I3 of extracellular components is defined as follows:(6)REuζ,τ,ζ,τ≔13⋅N⋅tf∑l=1N∑j=13∫0tfxjt;x0l,uζ,τ,ζ,τ−yjlt2yjlt2dt.
The existence of the optimal solution ofIM is given in Theorem 1 of [18]; that is, given ζ,τ∈0,1, there exists uζ,τ∗∈Uwζ,τ such that Juζ,τ∗,ζ,τ≤Juζ,τ,ζ,τ.
By constructing the parallel particle swarm optimization algorithm, the optimal solution of the problemIM is calculated as ζ∗=0.9 and τ∗=0.9, and the value of parameter uζ∗,τ∗∗ is given as follows [18]:(7)u0.9,0.9∗=34.0297,107.204,150.417,4.71128,115.59,57.2065,1.98077,17.7808,10.6496,5263.6,44.8808,68.8345.
So, the optimal nonlinear dynamical system of complex metabolism of batch fermentation is obtained as(8)x˙t=hx,uζ,τ∗,ζ∗,τ∗,t∈0,tf,x0=x0,where(9)h1x,uζ,τ∗,ζ∗,τ∗=μx1,h2x,uζ,τ∗,ζ∗,τ∗=−q2ζ∗,τ∗x1,h3x,uζ,τ∗,ζ∗,τ∗=q3ζ∗,τ∗x1,h4x,uζ,τ∗,ζ∗,τ∗=q4ζ∗,τ∗x1,h5x,uζ,τ∗,ζ∗,τ∗=q5ζ∗,τ∗x1,h6x,uζ,τ∗,ζ∗,τ∗=1uζ,τ9∗ζ∗46.1633x2x2+2.7467+1−ζ∗μζ,τ10∗x2−x6⋅NR+x2−x6−q20−μx6,h7x,uζ,τ∗,ζ∗,τ∗=uζ,τ11∗x60.531+x7/185.242+x6−45.9992x70.14+x71+x7/1.3341−μx7,h8x,uζ,τ∗,ζ∗,τ∗=45.9992x70.14+x71+x7/1.3341−τ∗6.9401x8x8+26.6321−1−τ∗⋅uζ,τ12∗x8−x3NR+x8−x3−μx8.
The consumption rate and the specific production rate in the microbial transformation are(10)q20=3.1453+μ0.0066+36.8436x2x2+16.2526,μ=0.67x2x2+0.28∏j=251−xjxj∗,q2ζ∗,τ∗=ζ∗u1∗x2x2+3.87+1−ζ∗u2∗x2−x6NR+x2−x6,q3ζ∗,τ∗=τ∗u3∗x8x8+u4∗+1−τ∗u5∗x8−x3NR+x8−x3,q4=−1.0522+μu6∗,q5=u7∗+μu8∗,where NR+θ is the indicator function; if θ>0, then NR+θ=1; otherwise, NR+θ=0.


## 3. Linear Variational System and Fundamental Matrix Solution

To simplify notation, we denoteuζ,τ∗ as u∗. In accordance with Property 1, the function hx,u∗,ζ∗,τ∗∈ℝ8 exhibited its continuous partial derivative with respect to x0∈W0 in System (1), so the linear variational system corresponding to System (1) could be built as(11)y˙=∂hx,u∗,ζ∗,τ∗∂x⋅y,t∈I0,where x=xt;x0,u∗,ζ∗,τ∗ is the solution to System (1) with x0;x0,u∗,ζ∗,τ∗=x0 as an initial state. It is assumed that(12)xt=zt+xt;x0,u∗,ζ∗,τ∗,was also the solution to System (1) with the following x(0) as the initial state,(13)x0=z0+x0;x0,u∗,ζ∗,τ∗=x0.Next, the derivative oft on both sides of (12) is calculated as(14)x˙t=z˙t+∂xt;x0,u∗,ζ∗,τ∗∂t=z˙t+hxt;x0,u∗,ζ∗,τ∗,u∗,ζ∗,τ∗.Since (12) is the solution to System (1), we have(15)z˙t+∂xt;x0,u∗,ζ∗,τ∗∂t=hzt+xt;x0,u∗,ζ∗,τ∗,u∗,ζ∗,τ∗.From (14) and (15), it could be calculated that(16)z˙t=hzt+xt;x0,u∗,ζ∗,τ∗,u∗,ζ∗,τ∗−hxt;x0,u∗,ζ∗,τ∗,u∗,ζ∗,τ∗=∂hxt;x0,u∗,ζ∗,τ∗,u∗,ζ∗,τ∗∂x⋅zt+ozt.Whenzt is sufficiently small, we have(17)z˙t=∂hxt;x0,u∗,ζ∗,τ∗,u∗,ζ∗,τ∗∂x⋅zt.Therefore, the variational system (11) is obtained by ignoring ozt in the differential equation of zt.According to Theorem 3.3 in [20], if x=xt;x0,u∗,ζ∗,τ∗ is the solution to System (1) with x0;x0,u∗,ζ∗,τ∗=x0 as an initial state matrix,(18)∂xt;x0,u∗,ζ∗,τ∗∂x0∈ℝ8×8,which is the fundamental matrix solution to the linear variational system (11) with the unit matrix as(19)∂x0;x0,u∗,ζ∗,τ∗∂x0=II∈ℝ8×8unit matrix,as an initial state.According to Theorem 2.6.4 in [20], we have those as follows.Lemma 1.
Suppose thatxt;x0,u∗,ζ∗,τ∗ and xt;y0,u∗,ζ∗,τ∗ are the solutions to System (1) with x0;x0,u∗,ζ∗,τ∗=x0∈W0 and x0;y0,u∗,ζ∗,τ∗=y0∈W0 as an initial state, then(20)xt;y0,u∗,ζ∗,τ∗−xt;x0,u∗,ζ∗,τ∗=∫01Φt,x0+sy0−x0ds⋅y0−x0,t∈I0,where Φ is the fundamental matrix solution to the linear variational system (11), corresponding to the solution xt;x0+sy0−x0,u∗,ζ∗,τ∗ of System (1) with x0+sy0−x0 as an initial state. The term s is used to portray the perturbation process of the initial values.


## 4. Strong Stability of the Optimal Nonlinear Dynamical System of Complex Metabolism

According to Properties1 and 2, there is a unique solution xt;x0,u∗,ζ∗,τ∗∈W0 to ∀x0∈W0 in the optimal nonlinear dynamical system (1) of complex metabolism. However, the analytical solution cannot be calculated at present, and only the numerical solution can be obtained. From the numerical solution, there is no stationary point in System (1), so the strong stability is defined as follows.Definition 1.
Suppose thatxt=xt;x0,u∗,ζ∗,τ∗ is the solution to System (1) with x0=x0∈W0 as an initial state, if ∀ε>0, then there exists δ=δε>0 making the solution yt=yt;y0,u∗,ζ∗,τ∗ of system (1) satisfy(21)y0∈W0,x0−y0<δ,xt−yt<ε,∀t∈I0.
Then, the solutionxt;x0,u∗,ζ∗,τ∗ in System (1) has strong stability for the initial state perturbation.
To prove the strong stability of the solution to System (1), the boundedness of the fundamental matrix solution of the linear variational system (11) corresponding to solution xt;x0,u∗,ζ∗,τ∗ to System (1) must be firstly discussed.Theorem 1.
Ifxt=xt;x0,u∗,ζ∗,τ∗ is the solution to System (1) with x0=x0∈W0 as an initial state and Φt,0,I, t∈I0, is the fundamental matrix solution of the linear variational system (11) corresponding to solution xt;x0,u∗,ζ∗,τ∗ to System (1), then Φt,0,I is bounded on I0⊂ℝ+.Proof.
Φt,0,I≔x1t,0,e1,x2t,0,e2…,x8t,0,e8∈ℝ8×8 is the fundamental matrix solution of the linear variational system (11), namely,(22)x˙it=∂hxt,x0,u∗,ζ∗,τ∗∂x⋅xit,xi0=ei,i∈I8,t∈I0,where ei∈ℝ8 is the ith column of the unit matrix I∈ℝ8×8.
Ashx,u∗,ζ∗,τ∗ about x∈W0 shows continuous partial derivative and W0⊂ℝ8 is also a nonempty bounded closed set, so the given x0∈W0 could make ∂hx,u∗,ζ∗,τ∗/∂x bounded on W0⊂ℝ8, that is, ∃M1>0, such that(23)∂hx,u∗,ζ∗,τ∗∂x≤M1,∀t∈I0.
Let(24)uit=argminuit|uit∈C1I0,ℝ+,uit≥maxj∈I8xjit,t∈I0,(25)Vit,uit≔8⋅M1⋅uit,i∈I8.
Obviously,Vit,uit is continuous on I0×ℝ+. Hence, the system is(26)u˙it=Vit,uit,ui0=1,t∈I0,i∈I8,which has a unique solution uit, t∈I0, and it is also obtained that(27)uit≥1,t∈I0.
Therefore, the right-hand side of the state equation in system (22) is(28)∂hxt,x0,u∗,ζ∗,τ∗∂x⋅xit≤∂hxt,x0,u∗,ζ∗,τ∗∂x⋅xit,(29)≤M1⋅xit,(30)≤M1⋅8⋅maxj∈I8xjit,(31)≤8⋅M1⋅uit,∀t∈I0,(32)≤Vit,uit,∀t∈I0,where (29), (30), and (32) are obtained according to (23), (24), and (25), respectively. It is acquired from the equation above that(33)∂hxt,x0,u∗,ζ∗,τ∗∂x⋅xit≤Vit,uit,∀t∈I0,i∈I8.
By Theorem 6.1 and Inference 6.3 in [20], the solution uit to System (26) satisfies(34)xit,0,ei≤uit≤uit=maxt∈I0uit,i∈I8.
Because ofuit∈C1I0,ℝ+, let mi denote the supremum of uit; then,(35)uit≤mi<∞,∀t∈I0,and it is obtained from (34) that(36)xit,0,ei≤mi,t∈I0,i∈I8.
According to the definition of norm, it is acquired that(37)xit,0,ei=∑j∈I8xjit,0,ei≤mi,∀t∈I0.
LetM≔∑j∈I8mi; then,(38)Φt,0,I=∑i∈I8∑j∈I8xjit,0,ei≤∑i∈I8mi≤M,t∈I0.
So, it is verified that the fundamental matrix solutionΦt,0,I of the linear variational system (11) is bounded on I0⊂ℝ+.
Then, the strong stability of the optimal nonlinear dynamical system (1) of complex metabolism is considered.Theorem 2.
Ifxt=t;x0,u∗,ζ∗,τ∗ is the solution to System (1) with x0=x0∈W0 as an initial state, the solution xt;x0,u∗,ζ∗,τ∗ is of strong stability about the initial state perturbation.Proof.
Suppose that System (1) with y0=y0∈W0 is an initial state. As for ∀ε>0, the initial state y0∈W0 satisfies(39)x0−y0≤εM=δε,where M>0, and according to Lemma 1, M is constant in Theorem 1.(40)yt;y0,u∗,ζ∗,τ∗−xt;x0,u∗,ζ∗,τ∗=∫01Φt,0,x0+Sy0−x0dS⋅y0−x0.
Hence,(41)yt;y0,u∗,ζ∗,τ∗−xt;x0,u∗,ζ∗,τ∗≤∫01Φt,0,x0+Sy0−x0dS⋅y0−x0≤M⋅y0−x0≤M⋅εM=ε,∀t∈I0.
In accordance to Definition1, the solution xt;x0,u∗,ζ∗,τ∗ to System (1) has strong stability about the initial state perturbation.


## 5. Conclusions

The optimal nonlinear dynamical system of the complex metabolism of batch fermentation and its properties are taken into consideration in this study. It is important to analyze the stability of the system. However, the stability of the dynamical system of the complex metabolism cannot be analyzed by the general methods because the analytical solution and equilibrium point of the system cannot be obtained. To address this problem, we propose the linear variational system and the fundamental matrix solution corresponding to the system. Based on the properties of the system and the proposed content, the strong stability of the system solution about the initial state perturbation is proved.



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