In this paper, the robust exponential stability problem is investigated for a class of Markovian jumping genetic networks which involve both uncertain parameters and stochastic disturbances. Under the assumption that the jumping parameters are generated from a continuous-time discrete-state homogeneous Markov process, the stability problem is first studied for a deterministic genetic model. By constructing suitable Lyapunov functionals and conducting some stochastic analysis, the stability criteria are derived in the form of linear matrix inequalities (LMIs), which can be easily checked in practice. Then, based on the derived results, sufficient LMI conditions are obtained explicitly for an indeterministic genetic system where the parameter uncertainties are norm-bounded. An illustrative example is presented to demonstrate the effectiveness and usefulness of the proposed stability criteria.
ChenT.HeH. L.ChurchG. M.‘Modeling gene expression with differential equations.’ In Proceedings of the 4th Pacific Symposium onBiocomputing, Hawaii, USA, 1999, pp. 29–40 (World Scientific, Hawaii).
2.
de JongH.‘Modelling and simulation of genetic regulatory systems: A literature review.’J. Comput. Biol.9 (2002): 67–103
3.
ElwitzM. B.LeiberS.‘A synthetic oscillatory network of transcriptional regulator.’Nature403 (2000): 335–338
ElowitzM. B.LevineA. J.SiggiaE. D.SwainP. S.‘Stochastic gene expression in a single cell.’Science297 (2002): 1183–1186
6.
LiC.ChenL.AiharaK.‘Stochastic stability of genetic networks with disturbance attenuation.’IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process.54 (2007): 892–896
7.
PaulssonJ.‘Summing up the noise in gene networks.’Nature427 (2004): 415–418
8.
TomiokaR.KimurabH.KobayashibT. J.AiharaK.‘Multivariate analysis of noise in genetic regulatory networks.’J. Theor. Biol.229 (2004): 501–521
9.
ChenL.AiharaK.‘Stability of genetic regulatory networks with time delay.’IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.49 (2002): 602–608
10.
LiC.ChenL.AiharaK.‘Stability of genetic networks with SUM regulatory logic: Lure system and LMI approach.’IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.53 (2006): 2451–2458
11.
CaoJ.RenF.‘Exponential stability of discrete-time genetic regulatory networks with delays.’IEEE Trans. Neural Net.19 (2008): 520–523
12.
CaoJ.WangJ.‘Global asymptotic and robust stability of recurrent neural networks with time delays.’IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.52 (2005): 417–426
13.
MaoX.KorolevaN.RodkinaA.‘Robust stability of uncertain stochastic differential delay equations.’Syst. Control Lett35 (1998): 325–336
14.
WangZ.LauriaS.FangJ.LiuX.‘Exponential stability of uncertain stochastic neural networks with mixed time-delays.’Chaos Solitons Fractals32 (2007): 62–72
15.
ChesiG.HungY. S.‘Stability analysis of uncertain genetic sum regulatory networks.’Automatica44 (2008): 2298–2305
16.
HeW.CaoJ.‘Robust stability of genetic regulatory networks with distributed delay.’Cog. Neurodyn.2 (4) (2008): 355–361
17.
RenF.CaoJ.‘Robust stability of genetic regulatory networks with interval time-varying delays.’Neurocomputing71 (2008): 834–842
18.
WangG.CaoJ.‘Robust exponential stability analysis for stochastic genetic networks with uncertain parameters.’Commun. Nonlinear Sci. Numer. Simul.14 (2009): 3369–3378
TianT.BurrageK.‘Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage λ.’J. Theor. Biol.227 (2004): 229–237
21.
LoingerA.LipshtatA.BalabanN.BihamO.‘Stochastic simulations of genetic switch systems.’Phys. Rev. E75 (2007): 021904
22.
AtkinsonM. R.SavageauM. A.MyersJ. T.NinfaA. J.‘Development of genetic circuitry exhibiting toggle switch or oscillatory behavior in escherichia coli.’Cell113 (2003): 597–607
23.
LewisM.‘The lac repressor.’C. R. Biol.328 (2005): 521–548
24.
MahmoudM. ‘Stochastic stability analysis of gene regulatory networks using hybrid systems theory.’ Intelligent Systems and Control, Cambridge, Massachusetts, USA, 2005.
25.
ZeiserS.FranzU.WittichO.LiebscherV.‘Simulation of genetic networks modelled by piecewise deterministic Markov processes.’IET Syst. Biol.2 (3) (2008): 113–135
26.
MaoX.‘Exponential stability of stochastic delay interval systems with Markovian switching.’IEEE Trans. Autom. Control47 (2002): 1604–1612
27.
MaritonM.Jump linear systems in automatic control, (Marcel Dekker New York1990)
28.
WangZ.LiuY.YuL.LiuX.‘Exponential stability of delayed recurrent neural networks with Markovian jumping parameters.’Phys. Lett. A356 (2006): 346–352
29.
XieL.FuM.de SouzaC. E.‘H∞ control and quadratic stabilization of systems with uncertainty via output feedback.’IEEE Trans. Autom. Control37 (1992): 1253–1256
30.
BoydS.GhaouiL. E.FeronE.BalakrishnanV.Linear matrix inequalities in system and control theory, (SIAM, Philadelphia, Pennsylvania1994)
