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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(6),2663-2670
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.126268
êŒÅƒ†VÇÙ››ä
k•žm½
¸¸¸™™™
úô“‰ŒÆêƆOŽÅ‰ÆÆ§úô7u
ÂvFϵ2023c59F¶¹^Fϵ2023c62F¶uÙFϵ2023c69F
Á‡
©Ì‡ïÄêŒÅƒ†VÇÙ››äk•žm-½¯K§·‚‰Ñ˜‡yêŒ
цVÇÙ››äk•žm-½¿‡^‡§¿…‰ÑäNy²5Øã·‚(Ø"
'…c
ŒÜþȧêŒÅƒ†§Ùä§k•žm½
Finite-TimeStabilizationofMarkovian
SwitchingProbabiliticBoolean
ControlNetworks
ShinanLin
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:May9
th
,2023;accepted:Jun.2
nd
,2023;published:Jun.9
th
,2023
Abstract
In this paper, we mainly studiesthe finite timestability problem ofMarkovian switch-
©ÙÚ^:¸™.êŒÅƒ†VÇÙ››äk•žm½[J].A^êÆ?Ð,2023,12(6):2663-2670.
DOI:10.12677/aam.2023.126268
¸™
ingprobabilityBooleancontrolnetworks.Weprovideanecessaryandsufficientcon-
ditiontoverifythefinitetimestabilityofMarkovianswitchingprobabiliticBoolean
controlnetworksandprovidespecificprooftoverifyourconclusion.
Keywords
Semi-TensorProduct,MarkovianSwitch,BooleanNetworks,Finite-Time
Stabilization
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.1262682665A^êÆ?Ð
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DOI:10.12677/aam.2023.1262682666A^êÆ?Ð
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k
(I
ω
(χ
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1
(I
ω
(χ
e
)) ={a∈∆
N
|•3˜‡››u¦
Pr{z(t+1) ∈I
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(χ
e
) |z(t) = a,u(t) = u}= 1}
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k
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DOI:10.12677/aam.2023.1262682668A^êÆ?Ð
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ë•©z
[1]Davidich,M.andBornholdt, S.(2008) TheTransitionfrom DifferentialEquationsto Boolean
Networks:ACaseStudyinSimplifyingaRegulatoryNetworkModel.JournalofTheoretical
Biology,255,269-277.https://doi.org/10.1016/j.jtbi.2008.07.020
DOI:10.12677/aam.2023.1262682669A^êÆ?Ð
¸™
[2]Meng,M.,Xiao,G.,Zhai,C.andLi,C.(2019)ControllabilityofMarkovianJumpBoolean
ControlNetworks.Automatica,106,70-76.https://doi.org/10.1016/j.automatica.2019.04.028
[3]Kauffman,S.(1969)MetabolicStabilityandEpigenesisinRandomlyConstructedGenetic
Nets.JournalofTheoreticalBiology,22,437-467.
https://doi.org/10.1016/0022-5193(69)90015-0
[4]Cheng, D., Qi, H.and Li, Z.(2011) AnalysisandControlof Boolean Networks—ASemi-Tensor
ProductApproach.Springer,London.
[5]R05Li,F.andSun,J.(2011)ControllabilityofProbabilisticBooleanControlNetworks.
Automatica,47,2765-2771.https://doi.org/10.1016/j.automatica.2011.09.016
[6]Cheng,D., Li,C. andHe, F.(2018)Observability ofBooleanNetworksvia SetControllability
Approach.SystemsControlLetters, 115,22-25.https://doi.org/10.1016/j.sysconle.2018.03.004
[7]Cheng,D.,Qi,H.,Li,Z.andLiu,J.(2011)StabilityandStabilizationofBooleanNetworks.
InternationalJournalofRobustandNonlinearControl,21,134-156.
https://doi.org/10.1002/rnc.1581
[8]Alberti, W.,Anderson,G., Bartolucci,A.,Bell,D. and Toni,V. (1995) Chemotherapy in Non-
SmallCellLungCancer:AMeta-AnalysisUsingUpdatedDataonIndividualPatientsfrom
52 RandomisedClinicalTrials.BMJ,311,899-909. https://doi.org/10.1136/bmj.311.7010.899
[9]Li,F.andSun,J.(2012)StabilityandStabilizationofBooleanNetworkswithImpulsive
Effects.SystemsandControlLetters,61,1-5.https://doi.org/10.1016/j.sysconle.2011.09.019
[10]Liu,Y.,Wang,L.,Lu,J.andCao,J.(2019)Sampled-DataStabilizationofProbabilistic
BooleanControlNetworks.SystemsandControlLetters,124,106-111.
https://doi.org/10.1016/j.sysconle.2018.12.012
DOI:10.12677/aam.2023.1262682670A^êÆ?Ð

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