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Asymptotical stability of a nonlinear non-differentiable dynamic system in microbial continuous cultures
- Jiajia Lv^{1}Email authorView ORCID ID profile,
- Liping Pang^{1} and
- Enmin Feng^{1}
https://doi.org/10.1186/s13662-017-1288-x
© The Author(s) 2017
Received: 2 June 2017
Accepted: 19 July 2017
Published: 25 August 2017
Abstract
In this paper, we consider a nonlinear non-differentiable dynamic system in microbial continuous cultures involving all possible metabolic pathways of the inhibition mechanisms of 3-hydroxypropionaldehyde onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane. First, the existence of the equilibrium point of the system proved. And by numerical calculation, the equilibrium point of the system is obtained. Subsequently, we derive the local bounded properties of the Jacobian, tensor and Hessian matrices of the system. Finally, the local asymptotical stability of the system at equilibrium point is proved.
Keywords
- nonlinear dynamic system
- equilibrium point
- tensor
- asymptotical stability
1 Introduction
The bioconversion of glycerol to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae is particularly attractive to industry because of renewable feedbacks and potential uses of 1,3-PD [1]. Especially continuous culture is of interest because of its high productivity, stable product quality and high automation.
In recent years, many efforts have been made to understand and express, in mathematical terms, the above mentioned bioconversion. The original model was proposed by Zeng et al. [2, 3], in which the concentrations of biomass, glycerol and products (1,3-PD, acetate and ethanol) in reactor were considered. In addition, ignoring the influence of acetate and ethanol on the fermentation process, Xiu et al. [4] discussed the multiplicity of a three-dimensional dynamical system. Based on the model, Xiu et al. [5] proposed a five-dimensional one later. On the basis of models, Sun [6] developed and re-constructed an eight-dimensional dynamic system considering intracellular substances: 3-hydroxy propionaldehyde (3-HPA), 1,3-PD and glycerol, which is more reasonable to describe the consumption of substrates and the formation of products.
With the development of fermentation models, there has been a lot of theoretical and numerical research about these models considering the extracellular substances, such as parameter identification works [7–11]. Since the continuous fermentation proceeds under a steady state (that is, a little change of the initial conditions would not cause a great change of the solutions), the existence of an equilibrium solution and the stability analysis are necessary to evaluate a model describing the continuous fermentation. The stability of equilibrium solutions suggests that proper operating conditions could be chosen to obtain the expected productions’ concentrations. However, the stability results are seldom discussed for the complexity of the mathematical models. Li et al. [12] mainly discussed the stability of the three-dimensional system by introducing its linearization. Ye et al. [13] examined the existence of equilibrium points and proposed an efficient method to calculate the equilibrium points of the five-dimensional model in continuous fermentations. Wang et al. [14] discussed the stability of an eight-dimensional system taking the changes of concentrations of intracellular substances into consideration. Chen and Gao [15, 16] studied positive linear systems with the common linear copositive Lyapunov functions and presented four algorithms to compute common infinity-norm Lyapunov functions. Choi et al. [17, 18] characterized the h-stability for nonlinear difference systems. Wang et al. [19] introduced the h-stability for differential systems with different initial time and formulated the stability criteria. However, especially for the eight-dimensional system being in consideration of both the inhibition mechanisms of 3-HPA onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane, the properties of the system and the stability analysis are scarcely discussed.
In this paper, we consider a nonlinear non-differentiable dynamic system in microbial continuous culture, develop Ye’s [13] method to derive the existence of equilibrium points, and present its local asymptotically stability for the eight-dimensional nonlinear dynamical system presented in [8]. Different from the system of [14], the eight-dimensional dynamic system considers all possible metabolic pathways of the inhibition mechanisms of 3-HPA onto the cell growth and the transport systems of glycerol and 1,3-PD across the cell membrane, and it is a non-differentiable system; especially, we construct differentiable domains to derive the local bounded properties of the Jacobian, tensor and Hessian matrices of the system and the local asymptotically stability on these differentiable domains under some conditions. Numerical experiments are carried out using the parameter values of [8] to evaluate the validity of the theoretical work.
The rest of this paper is organized as follows. Section 2 describes the nonlinear non-differentiable dynamic system and present the existence of the equilibrium point of the system. By numerical calculation, the equilibrium point of the system is obtained. The local asymptotically stability of Jacobian and Hessian matrices of the system on differential domains are derived in Section 3. We present the local asymptotical stability under some conditions of the system at equilibrium point. Final conclusions follow in Section 4.
2 Nonlinear non-differentiable dynamical system and existence of the equilibrium point
Assumption 2.1
Assumption 2.2
Assumption 2.3
Let \(f_{7}(x,u)=0\) and \(f_{8}(x,u)=0\), we can find two expressions of \(x_{7}\) and \(x_{8}\). Take note that \(x_{7}=F_{7}(x_{2},x_{3},x_{4},x_{5},x_{6},u)>0\) and \(x_{8}=F_{8}(x_{2},x_{3},x_{4},x_{5},x_{7},u)>0\).
Assumption 2.4
Therefore, we can obtain the properties of \(F(x,u)\) on the basis of Ref. [8].
Property 2.1
Suppose that Assumptions 2.1-2.4 hold and \(F(x,u)\in\mathbb{R}^{8}\) is defined by equations (2)-(14), then \(F(x,u)\) is Lipschitz continuous for any \((x,u)\in\mathcal{W}_{a}\times\mathcal{U}_{a}\), and satisfies the linear growth condition.
Property 2.2
According to Property 2.1, we can see that, for any \(u\in\mathcal {U}_{a}\), there exists one unique solution satisfying system (1) and denote it as \(x(t)=x(t,u)\).
On the basis of the above four assumptions and the two properties, the next theorem is concerned with the existence of equilibrium point of system (1).
Theorem 2.1
Suppose that Assumptions 2.1-2.4 hold and \(F(x,u)\) is defined by equations (2)-(14), then, for any \(u\in\mathcal{U}_{a}\), there exists at least one equilibrium point \(\bar{x}\in\mathcal{W}_{a}\) of system (1).
Proof
3 Asymptotical stability of nonlinear dynamical system
Theorem 3.1
Suppose that Assumptions 2.1-2.4 hold and \(F(x,u)\in\mathbb{R}^{8}\) is defined by equations (2)-(14), then \(F(x,u)\) is twice continuously differentiable on \((x,u)\in \mathbb{\bar{B}}(\bar{x},\bar{\delta})\times\mathcal{U}_{a}\).
Proof
From equations (2)-(14) and the definition of \(\mathbb{\bar{B}}(\bar {x},\bar{\delta})\), we can see that there is a two-order partial derivative of \(F(x,u)\) on \((x,u)\in\mathbb{\bar{B}}(\bar{x},\bar {\delta})\times\mathcal{U}_{a}\), and all two-order partial derivatives are continuous on \((x,u)\in\mathbb{\bar{B}}(\bar {x},\bar{\delta})\times\mathcal{U}_{a}\). Thus the proof is completed. □
Next we will discuss the local asymptotical stability in \(\mathbb{\bar {B}}(\bar{x},\bar{\delta})\).
3.1 Local bounded properties of \(F'(x)\), \(F''(x)\) and \(f''_{k}(x)\)
Denote \(F'(x)\) be the Jacobian matrix of \(F(x)\), \(f''_{k}(x), k\in \mathbb{I}_{8}\) be the Hessian matrix of \(f_{k}(x), k\in\mathbb {I}_{8}\), \(F''(x)\) be the Hessian matrix of \(F(x)\). From Theorem 3.1, we can see that \(f_{k}(x),\frac{\partial f_{k}}{\partial x_{i}},\frac{\partial^{2} f_{k}}{\partial x_{i}\,\partial x_{j}}\in\mathbb{C}^{2}(\mathbb{\bar{B}}(\bar {x},\bar{\delta}))\) for any \(i,j,k\in\mathbb{I}_{8}\), so the Jacobian matrix \(F'(x)\), the tensor \(F''(x)\) and the Hessian matrix \(f''_{k}(x)\) exist. To get the local asymptotical stability of the equilibrium point x̄ of system (1), in this section, we will derive the bounded properties of \(F'(x)\), \(F''(x)\) and \(f''_{k}(x),k\in\mathbb{I}_{8}\). For this purpose, the definitions of tensor and norm are shown first.
Definition 3.1
Definition 3.2
Remark 3.1
Theorem 3.2
Suppose that the conditions of Theorem 3.1 hold, then \(F'(x)\), \(F''(x)\) and \(f''_{k}(x), k\in\mathbb{I}_{8}\), are bounded on \(x\in\mathbb {\bar{B}}(\bar{x},\bar{\delta})\), where δ̄ is defined by equation (23).
Proof
3.2 Local asymptotical stability of the equilibrium point
Lemma 3.1
[20]
Suppose that \(\lambda_{i}\) is an eigenvalue of Jacobian matrix \(F'(\bar{x})=(\triangledown f_{1}(\bar{x}),\ldots, \triangledown f_{8}(\bar{x}))^{T}\), and each \(\lambda_{i}\) satisfies the condition \(\operatorname{Re}(\lambda_{i})<0\), then the equilibrium point \(x=\bar{x}\) of system (27) is asymptotically stable.
Lemma 3.2
[21]
From Lemmas 3.1 and 3.2, it can be seen that we can use a linear approximate system to get the local asymptotically stable property of x̄.
Theorem 3.3
Suppose that the conditions of Theorem 2.1 hold, x̄ is an equilibrium point of system (1) corresponding \(u^{*}\in\mathcal {U}_{a}\); for any \(x\in\mathbb{\bar{B}}(\bar{x},\bar{\delta})\), then the equilibrium point x̄ of system (1) is local asymptotically stable on \(\mathbb{\bar{B}}(\bar{x},\bar{\delta})\).
Proof
Remark 3.2
4 Conclusions
In this work, we have considered the stability of a nonlinear non-differentiable dynamic system in a microbial continuous culture. The existence of the equilibrium point has been derived. Because of its non-differentiable property, we have constructed a differentiable domain, in which the stability of the system has been derived. The local bounded properties of the Jacobian, tensor and Hessian matrices of the system have been proved. A numerical experiment has been performed to evaluate its effectiveness.
Declarations
Acknowledgements
The authors would like to thank the reviewers for their constructive and valuable suggestions. This work was supported by the National Natural Science Foundation of China under Grant No. 11371164 and No. 11171050.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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