Finite-time state observer for delayed reaction-diffusion genetic regulatory networks
Introduction
In recent years, genetic regulatory networks (GRNs) has become a hot topic in many disciplines, for example, mathematics, statistics, biology and medicine, and aroused the attention of experts and scholars. As a result, a great deal of very important research results (see [1], [2], [3], [4], [5], [6], [7] and the references therein) have been achieved. GRNs, as highly complex network models, describe genetic expression and regulation behavior. The transcription and translation are the most important and most complex processes in the GRNs.
Currently, mathematical models have been one of the main tools to analyze GRNs. Due to the different forms of the GRNs and the different research purpose and methods, several GRN models have been established. For example, the Bayesian model [8], the Boolean model [7], [9] and the functional differential equations model [5], [10], [11]. A functional differential equation model describes the continuous change of mRNA and protein concentrations which have two merits: (i) the slow processes of the transcription and translation are characterized by time delays; and (ii) the continuous change of mRNA and protein concentrations are expressed as the derivatives of the unknown functions. So, functional differential equation models have been widely applied to understand the nonlinearity and complexity of GRNs. It should be emphasized that time delays are one of main sources for causing instability and/or poor performance [12], [13], [14], [15]. Accordingly, stability analysis of functional differential equation models has aroused increasing research interests, and a great number of outstanding results have been reported (see [6], [7], [16], [17], [18], [19], [20], [21], [22], [23], [24] and the references therein). The stability criteria presented in these literature are divided into two kinds: delay-dependent stability ones and delay-independent stability ones. In general, delay-dependent stability criteria are less conservative than delay-independent ones. Please refer to [25], [26], [27], [28], [29], [30], [31], [32] for effective approaches to establish delay-dependent stability criteria.
In some mathematical modeling, it is assumed that GRNs are spatially homogeneous, namely, the concentrations of mRNAs and proteins are homogenous in space at all times. However, in some cases, it is need to introduce reaction-diffusion terms into models [19], [33], [34], [35], [36], [37], [38], [39]. Especially, it is necessary to consider the diffusion of mRNAs and proteins [19], [37], [38], [39]. Thus, it is imperative to introduce reaction-diffusion terms into the continuous-time GRN models. To the best of authors' knowledge, the stability problem for delayed reaction-diffusion genetic regulatory networks (DRDGRNs) has been only studied in [19], [22], [23], [24]. Ma et al. [24] introduced DRDGRNs for the first time and established delay-dependent asymptotic stability criteria. Ma et al.'s results have been gradually improved in [19], [23] by introducing novel LKF and utilizing Wirtinger-type integral inequality approach. The problem of finite-time robust stochastic stability analysis for uncertain stochastic DRDRNs has been studied in [22].
With the change of environment, the usual feedback loops existing in GRNs may be destroyed. This will make GRNs' performance worse, and eventually lead to some fatal disease like cancer [40]. Therefore, it is necessary to adjust the feedback loops by artificial input control. For this end, the exact concentrations of the mRNAs and proteins (that is, the states of continuous GRN models) are needed. However, due to the complexity of GRNs, it is almost impossible to measure the exact concentrations. Hence, the state estimation for GRNs has been one of available methods to investigate dynamical behaviors; see, eg., [41], [42], [43], [44]. To the best knowledge of authors, the state estimation problem for DRDGRNs is only in [45], although the reaction-diffusion-free case has been researched (see [5], [41], [46], [47], [48], [49] and the references therein). This motivates our research interests.
The aim of this paper is to design a finite-time state observer for estimating concentrations of the mRNAs and proteins of DRDGRNS. A novel LKF is first constructed. Then its derivative is estimated by employing Wirtinger-type integral inequality [50], Gronwall inequality [51], convex technique and reciprocally convex technique [52]. As a result, a reaction-diffusion-dependent and delay-dependent sufficient condition is given to ensure that the error system is finite-time stable. This is different from [45] wherein the asymptotic stability of error systems are involved. The stability criterion is given in the form of linear matrix inequalities (LMIs), which can be solved by applying the Toolbox LMI or YALMIP of MATLAB. Thereby, we design a finite-time state observer whose gain matrices are described based on a feasible solution to these LMIs. In addition, two numerical examples are presented to illustrate the theoretical results obtained in this paper.
Notation: Throughout the paper, for given real symmetric matrices X and Y, means that is positive definite (positive semi-definite). I is the identity matrix of appropriate dimension, represents the transpose of matrix A. Set for any positive integer l. Ω is a compact set in the vector space with smooth boundary . Let be the Banach space of functions which map X into and have the continuous first derivatives. We define a pair of norms on and by as follows:andrespectively. Let and be the (block) diagonal matrix and column matrix formed by the elements in brackets, respectively.
Section snippets
Problem formulation and preliminaries
This paper considers the following DRDGRNs [19]:where, ,
Observer design
In this section, we will design a state observer of the form (5) for DRDGRN (3), that is, find a pair of observer gain matrices Km and Kp so that the trivial solution of (6) is finite-time stable. For convenience, we define
Illustrative examples
In this section, we will give two numerical examples to demonstrate the availability of the state observer proposed previously. Example 1 Consider DRDGRN (3) with measurements (4), where. When , , , T=10 and , by using the Toolbox YALMIP of MATLAB, we find
Conclusions
This paper investigates the finite-time state estimation problem for a class of DRDGRNs under Dirichlet boundary conditions. A finite-time state observer has been designed to estimate the concentrations of mRNAs and proteins based on available network outputs. A LKF including new integral terms is first introduced. Then its derivative is estamated by using the so-called Green's second identity, Wirtinger-type inequality, Gronwall inequality, convex technique and reciprocally convex technique.
Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (11371006), the Natural Science Foundation of Heilongjiang Province (F201326 and A201416), the Fund of Heilongjiang Education Committee (12541603), and the Heilongjiang University Innovation Fund for Graduates (YJSCX2016-027HLJU).
The authors thank the anonymous associate editor and referees for their helpful comments and suggestions.
X.F. Fan was born in Inner Mongolia, China, in 1991. She received the B.S. degree in School of Mathematical Science from Hulunbeir College, Hulunbeir, China, in 2014. She is currently an M.S. student in Heilongjiang University, Harbin, China. Her research interests include genetic regulatory networks and stability analysis of delayed dynamic systems.
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X.F. Fan was born in Inner Mongolia, China, in 1991. She received the B.S. degree in School of Mathematical Science from Hulunbeir College, Hulunbeir, China, in 2014. She is currently an M.S. student in Heilongjiang University, Harbin, China. Her research interests include genetic regulatory networks and stability analysis of delayed dynamic systems.
Y. Xue received the B.S. degree in Automation from Harbin University of Science and Technology, China in 2002; the Ph.D degree in Control Theory and Control Engineering from Harbin Institute of Technology, China in 2007. From April 2007 to August 2009, she is an Engineer in Sichuan Academy of Aerospace Technology. From September 2009 to May 2013, she is a Senior Engineer in Sichuan Academy of Aerospace Technology. In June 2013, she joined Heilongjiang University. Her research interests include stability analysis of delayed dynamic systems, robust control and genetic regulatory networks.
X. Zhang received Ph.D. degree in Control Theory from Queen's University of Belfast in UK in 2004. Since 2004 he has been at Heilongjiang University, where he is currently a Professor in the School of Mathematical Science. His current research interests include neural networks, genetic regulatory networks, mathematical biology and stability analysis of delayed dynamic systems. He has received the Second Class of Science and Technology Awards of Heilongjiang Province. He is a member of the IEEE, and a Vice President of Mathematical Society of Heilongjiang Province. Since 2006, he has been serving as an Editor of the Journal of Natural Science of Heilongjiang University. He has authored more than 100 research papers.
J. Ma was born in Heilongjiang, China, in 1979. She received the M.E. and Ph.D. degrees from Heilongjiang University, Harbin, China, in 2007 and 2012, respectively. Her current research interests include the networked systems filtering and information fusion filtering.