Robust stability of Markovian jumping genetic regulatory networks with disturbance attenuation

Abstract

Because of intracellular and extracellular noise perturbations and environment fluctuations, gene regulation is an intrinsically noisy process. In this paper, we present a hybrid genetic regulatory network (GRN) model which is based on the Markov chain. The GRNs are composed of N modes and the network switches from one mode to another according to a Markov chain with known transition probability. Time‐delays here are mode‐dependent. Based on the Lyapunov stability theory and the linear matrix inequality (LMI) technique, sufficient conditions are given to ensure the stochastic stability of the GRNs with polytopic uncertainties and disturbance attenuation. All the conditions are presented in terms of LMIs which are easily verified via the LMI toolbox. Examples are provided to illustrate the effectiveness of the theoretical results. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

Almost Asymptotic Regulation of Markovian Jumping Linear Systems in Discrete Time

“…This paper, however, just provides a feasible and potential way to study the regulation problems for stochastic MJLSs. Theorem 3 implies that the output e(t) to be regulated almost asymptotically tends to zero; it also implies, by virtue of (21), that ξ(t) = x(t) − Qid(t) almost asymptotically tends to zero. We also see from the simulation results that the regulation error oscillates before eventually converging to zero.…”

“…To obtain the controller design and the observer design via error feedback, we use the separation principle here. By means of LMI (21), we know that x k ( ) 0 2,E → and d k ( ) 0 2,E → as k → ∞. Then, we have ||ξ(k)||2,E → 0 as x(k) does.…”

“…Among the jumps of Markov modes, the system evolves according to a linear description associated with the specific Markov state. The existing results about MJLSs cover a large variety of problems, such as stochastic Lyapunov stability, stochastic controllability, and robust filtering [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36].…”

Global Exponential Stability Analysis of Discrete‐Time Genetic Regulatory Networks with Time Delays

“…A genetic regulatory network (GRN) is a dynamic system used to describe interactions among genes and proteins. Recently, with the rapid development of biology [1], GRNs have attracted growing attention, and there is already a series of achievements in this field [2][3][4][5][6][7][8][9][10][12][13][14][15][16]. Generally speaking, there are two main kinds of GRN models, which are the Boolean model (or discrete model) [7,10,11] and the differential equation model (or continuous model) [12][13][14][15][16].…”

Stability Analysis for Delayed Neural Networks with Discontinuous Neuron Activations

“…In recent years, many results have been obtained on the stability of neural networks with continuous neuron activations. Readers can refer to [1][2][3][4][5][6][7][8][9], and the references therein. Nevertheless, as far as we know, the stability results on neural networks with discontinuous neuron activations, even without delays, are few [10][11][12][13][17][18][19].…”