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A GENERALIZED KHARITONOV THEOREM FOR QUASIPOLYNOMIALS ENTIRE FUNCTIONS AND MATRIX POLYNOMIALS VADIM OLSHEVSKY AND LEV SAKHNOVICH Abstract

The classical Kharitonov theorem on interval stability cannot be carried over from polynomials to arbitrary entire functions In this paper we identify a class of entire functions for which the desired generalization of the Kharitonov theorem can be

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A GENERALIZED KHARITONOV THEOREM FOR QUASIPOLYNOMIALS ENTIRE FUNCTIONS AND MATRIX POLYNOMIALS VADIM OLSHEVSKY AND LEV SAKHNOVICH Abstract






Presentation on theme: "A GENERALIZED KHARITONOV THEOREM FOR QUASIPOLYNOMIALS ENTIRE FUNCTIONS AND MATRIX POLYNOMIALS VADIM OLSHEVSKY AND LEV SAKHNOVICH Abstract"— Presentation transcript:

AGENERALIZEDKHARITONOVTHEOREMFORQUASI-POLYNOMIALS,ENTIREFUNCTIONS,ANDMATRIXPOLYNOMIALSVADIMOLSHEVSKYLEVSAKHNOVICHTheclassicalKharitonovtheoremonintervalstabilitycannotbecarriedoverfrompolynomialstoarbitraryentirefunctions.Inthispaperweidentifyaclassofentirefunctionsforwhichthedesiredgeneralizationof SomeearlyresultsonthestabilityofsuchintervalpolynomialswereobtainedbyFaedoin  ifthefollowingfourpolynomialsarestable:where 0+ f2x2+f 4x4+ f6x6+x 0+f 2x2+ f4x4+f 6x6+x 1x+ f3x3+f 5x5+ f7x7+x 1x+f 3x3+ p5x5+f ThenotationsintheKharitonovstheoremhavethefollowingmeaning.Ifwepartition eventerms oddterms Fo( F Thelattertheoremwasimmediatelyfollowedbyavastliterature;theresulthasbeengen-eralizedinmanyways,anditenjoyedanumberofapplicationsinmechanicalandelectrical1.2.Entirefunctions,quasi-polynomials,andstability.Stabilityproblemsforpoly-nomialswereintensivelystudiedformanydecades,andbynowtheyarequitewellunder-stood.Similarproblemsforentirefunctionsappearinseveralapplications.However,theyaretypicallymuchmoreinvolvedandmuchmorechallenging.Letusrstconsideraverytransparentscalarexample.Consideradifferentialequationwithatimedelay,i.e., zy,yThesolutionclearlyhastheformwhereisarootofthespecialentirefunction)=1+(calledbelowaquasi-polynomial).Oneseesthatthesystem(1.6)isstableiftherootsoftheentirefunctionalllieinthelefthalfplane.Theabovefunctionbelongstoamoregeneralclassofquasi-polynomialsdenednext.EFINITIONbepolynomials.Afunctionoftheform iscalleda.(Pontryagyn[P42]usedthetermforthemorenarrowclassoffunctionsoftheformx,e,wherex,zisapolynomialintwovariables.)Letusnowconsideramorepracticalexamplegivingrisetoquasi-Manyproblemsincontrolengineeringinvolve(multiple)timedelaysmod-elledby Thismodelcanbeconstruedasarepresentativedynamicsoffullstatefeedbacksystemswithmultiplecomputationalandactuationdelays.Thedynamicsin(1.8)isalsocalledretardedtimedelaysystem,becausethehighestorderderivativetermsarenotaffectedbythedelays.AftertheLaplacetransformationonegetsthecharacteristicequation)=det(givingrisetothequasi-polynomialoftheformwherearepolynomials.Again,thestabilityofthefeedbacksystemwithmultipledelays(1.8)isequivalenttothelocationofalltherootsoftheentirefunctioninthelefthalfplane.Therstresultsonstabilityofquasi-polynomialsandentirefunctionswereobtainedinthepioneeringworksofPontryagin[P42]andofChebotarev-Meiman[CM49].Theyweremotivatedbycontrollerdesign[specically,ofregulatorsandservomechanismsdrivingthetrackingerrortozero];modellingahitinapipeline.Themorerecentrelevantliteratureisnotexhaustive,andreading[BC63],[L80],[HVL93],[BCK95],[L96],[KZ97],[DV98],[OS02][DHB03],[KTMOM03]givesanintroductioninthecurrentstateofartinthisarea[seealsothereferencestherein,nopossibleomissionsareintentional].1.3.Mainresultsandthestructureofthepaper.Averycloseproblemofstudyingthestabilityofcertainperturbedfamiliesofentirefunctionswasaddressed,e.g.,in[BCK95],[DHB03],[KTMOM03].However,theavoroftheirresultsseemstobedifferent.Firstly,theyconsidereduncertainfamiliesthatarenotintervalperturbationsasin(1.2).Secondly,theirsufcientconditionsarearenotoftheform(1.3),andmoreovertheirnumbercanbelarge.Themainresultofthispaperisdifferent:itisadirectgeneralizationoftheKharitonovstheorem.Inparticular,weconsiderexactlythestabilityofintervalentirefunctionsofthetype(1.2),andweshowthatfulllmentofthefourconditionsexactlyofthetype(1.3)sufcesforstability.Thepaperisstructuredasfollows.InSec2welaythegroundworkforfurthergener-alizationsbyrecallingthestandardproofoftheKharitonovtheorembasedontheclassicalHermite-Biehlercriterion[B1879].Simpleexamplesareincludedtorecallthatthelattercriterioncannotbecarriedovertoarbitraryentirefunctions.Thesectionidentiestwodif-cultiestoovercomeinthenexttwosections. Suchconditionsinvolvecertainhyperplanes[BCK95],ortheyareoftheso-calledsomevertex-type InSec3wediscussthepropertiesoftheclassforwhichweshallprovethegeneralizedKharitonovtheorem.WerstrecallhowtoovercometherstdifcultybydeningtheHBclassofentirefunctions.Secondly,weshowhowtoovercometheseconddifcultyandsuggestananalogofnotchangingdegreeproperty.InSec4weusetheresultsofthetwopreceedingsectionstoformulateageneralizationoftheKharitonovtheoremforaclassofentirefunctions.Sec5containsproofsofthemainresults.InSec6wesuggestatechniquethatisusefulforsolvingsomeKharitonov-likeproblemsformatrixpolynomialsandmatrixentirefunctions.2.Twodifculties.2.1.TheHermite-BiehlertheoremandtheproofofthepolynomialKharitonovthe-orem.First,werecalltheclassicalHermire-Biehlertheorem,itplaysanimportantroleinestablishingtheKharitonovsresultforthepolynomialsandinextendingittoentirefunc-beapolynomialin(1.1)anddenetwopolynomials,andeven,andanoddanddenote Thenthepolynomialisstableifandonlyifthefollowingtwoconditionsholdtrue.Therootsofthepolynomialsareallrealandtheyinterlace.ThereisatleastonepointsuchthatThecondition1isequivalenttothefactthatthattherootsofthepolynomialsareallpurelyimaginaryandtheyinterlace.Ifthe(2.2)isfullledjustforonepointthenitisvalidforallFortherealthecondition(2.2)simplymeansthatwherearethecoefcientsofin(1.1).TheproofoftheKharitonovtheoremisnowimmediate.Figure1(thatillustratestheremark1andtheorem2.1)showsthatanyintervalperturbationoftheoddtermsgivesusastablepolynomial.Fixingtheperturbedoddpartandapplyingasimilarargumenttotheevenpartonenallyobtainsthetheorem1.1.Inthenexttwosubsectionsweshowthatthereareatleasttwoproblemsincarryingovertheaboveargumentstoarbitraryentirefunctions.2.2.Therstdifculty.Thexeddegreepropertyandthenumberofroots.1indicatesthatintervalperturbationsdonotdestroyinterlacingoftherootsof.However,itisimplicitlyassumedthattheallpolynomialsin(1.1)and(1.2)havethesamedegree.Hencethenumberoftherootsofeachofthepolynomialsin(2.1)staysthesame. Fo,max Fo Fo,min Fe .2.1.IllustrationfortheProofoftheclassicalKharitonovtheoremforpolynomialsviatheHermite-Biehler.However,ifwereanentirefunctionthenitsdegreeisinnite,andhenceonehastotakecareofpreventingnewrootsfromoccurring.Graphically,onehastopreventthephenomenonshowninFigre2bymeansofimposingcertainadditionalconstrainstothexed-degreeproperty. eFoFo,minFo,max .2.2.Anillustrationfortherstdifculty.Onehastoimposeadditionalconditionspreventingarisingofthenewrootsof2.3.Theseconddifculty.TheHermite-Biehlertheoremdoesnotcarryovertoentirefunctions.Interlacingoftherootsisnotnecessary.Thefunction doesnothaverootsatallandhencestable.However,usingthedenition(2.3)weseethat)=cos)=sinhavenon-realroots,e.g., Interlacingoftherootsisnotsufcient.Again,usingthedenition(2.3)weseethatdonothaverootsatall,andhencetheinterlacingpropertyisfullled.Moreover,forany.Nevertheless Again,theseconddifcultyalsoindicatesthatitishopelesstoextendtheKharitonovstheoremtotheclassofallentirefunctions.Similarlytotheremarkmadeattheendofthesubsection2.2,thechallengehereistoidentifyaclassofentirefunctionsforwhichthetwodifcultiescanberemoved,andthentotrytoprovetheKharitonovstheoremforthatclass.Thisispreciselywhatisdoneintherestofthepaper.3.Removingthetwodifculties.3.1.RemovingtheFirstDifculty.Ananalogueofthexeddegreeproperty.thatevenforpolynomialstheconditions)=deg)=degareimplicitlyincludedintheclassicalKharitonovtheoremandtheyarecrucialforitsvalidity.(Hereweusedthedenitionin(2.3).)InourgeneralizedKharitonovtheoremtheseconditionswillbeincludedinadifferentform Fz)=OFe(z) thatappliesnotonlytopolynomialsbuttoentirefunctionsaswell.Thelatterexpressionsmakesenseforsincealltherootsofarepurelyimaginary,cf.withtheremark2.3.2.RemovingtheSecondDifculty.TheclassHP.Theindicatorfunctionproperty.3.2.1.Classicalresults.TheclassP.Inthissubsectionwerecallthebasicdenitionsandresultsthatcanbefoundin[CM49]and[L80],[L96].Theywillbeusedinwhatfollows.Wediscussherethesituationinwhichdoesnothaverootsinthelowerhalfplane.Inthenextsectionsthesesettingswillbeadjustedforstabilityof)=(EFINITIONbeanentirefunctionofniteexponentialtype.TocharacterizethegrowthofPhragmenandLindelofintroducedthefunction | whichiscalledanindicatorfunction Thequantity 2)h( iscalledtheofthefunctionAnentirefunctionofniteexponentialtypeissaidtobeintheclassPifithasnozerosintheopenlowerhalf-planeandTheindicatorfunctionplaysacrucialroleontherestofthepaper.Onereasonisthatitsbehaviorisconnectedwithinterlacingoftheroots,asdescribednext.([CM49],[L96],[K])Letuspartitiontheentirefunctionofniteexpo-nentialtypesothatarerealentirefunctions.belongstotheclassPifandonlyif1.therootsofareallrealandinterlacing;2.theindicatorfunctionsof3.atsomerealpointwehave(Ifthelatterconditionisfullledjustforonepointthenitisvalidforall3.2.2.Aslightmodication.TheclassHP.Applicationtostability.ItisconvenienttoexplicitlytranslatetheresultsofSec3.2.1tothelefthalfplanecase(stability)andtousethemthereafter.EFINITIONbeanentirefunctionofniteexponentialtype.WesuggesttorefertothequantityastheofthefunctionWeshallreservethenametheHPclassforentirefunctionsofniteexponentialtypewithnozerosintheopenlefthalf-planeandsatisfying(cf.withtheorem3.2)Letuspartitiontherealentirefunctionofniteexponentialtype evendegreeterms odddegreetermsbelongstotheclassHPifandonlyif1.therootsofareallpurelyimaginaryandinterlacing; 2.theindicatorfunctionsof3.WehaveTheorem3.4yieldsthefollowingresult.(cf.with[L80],Ch.VI,sec.4.thm8.)Letbeanentirefunctionofniteexponentialtypewithnorootsontheimaginaryaxissatisfyingisstableifandonlyiftherootsofin(3.7)alllieontheimaginaryaxisandinterlace.Stabilityofquasi-polynomials.wherearerealpolynomials,and... 2 21 23 Ifweassume(0)=i.e.,Inaccordancewiththeorem3.5,inthisexampleinterlacingoftherootsof(thatareallpurelyimaginary)isthenecessaryandsufcientconditionforthestabilityof)=sinh()+cosh(wherearepolynomials.InthiscaseInthelatterexamplethecondition(3.10)isnotfullledwhichshowsthatthetheorem3.5isnotuniversal.HencetoderiveageneralizationoftheKharitonovtheoremwemayneedtousetheorem3.4,e.g.,itscondition(3.8).Thisisasubjectofthenextsubsection.3.2.3.RemovingtheSecondDifculty.Theindicatorfunctionproperty.Theclassoffunctionssatisfying(3.6)isanaturalgeneralizationofpolynomials.Indeed,thedenition(3.10)immediatelyyieldsthatifisapolynomialthenandhenceHence,theclassofstablepolynomialscoincideswithpolynomialsbelongingtoThisobservationisoneindicationastowhytheindicatorfunctionproperty(triviallyfullledforpolynomials)wouldbequitenaturalinthecontextofthegeneralizationoftheKharitonovtheoremtoentirefunctions. Alternatively,alltherootsof arerealandinterlace. 4.ThegeneralizedKharitonovtheorem.4.1.Mainresults.Throughoutthissectionweuseadecomposition evendegreeterms odddegreetermsofanarbitraryentirefunction F( betwoentirefunctionsofniteexponentialtypethatbelongtotheclassHPandsatisfy Fz)oforz= Thenallthefunctionsoftheform Fo( F (4.5)(for= )simultaneouslybelongtotheclassHPprovidedthefollowingconditionsare1.Thefunctionsarealloftheniteexponentialtypeandtheysatisfytheindica-torfunctionproperty:2.Ananalogofthexed-degree-property: Fz)=Oz= Weshallprovetheabovelemmaattheendofthissection.Theresultdualtothelemma4.1isstatednext. Thisisananalogoftheintervaluncertainty(1.4). betwoentirefunctionsofniteexponentialtypethatbelongtotheclassHPandsatisfy Fz)eforz= Thenallthefunctionsoftheform )simultaneouslybelongtotheclassHPprovidedthefollowingconditionsare1.Thefunctionsarealloftheniteexponentialtypeandtheysatisfytheindica-torfunctionproperty:2.Ananalogofthexed-degree-property: Fz)=Oz= Thelemmas4.1and4.2implythefollowingresult.GeneralizedKharitonovtheorem.Lettheconditions(4.4),(4.5),and(4.10),(4.11)andtheconditions1)-2)ofeachofthelemmas4.1and4.2befullled.IfonlyfourfunctionsbelongtotheclassHPthenallthefunctionsbelongtotheclassHPaswell.RecallthatforpolynomialswehavesothattheclassHPissimplytheclassofstablepolynomials.Hencetheorem4.3isadirectgeneralizationoftheKharitonovtheorem1.1.Recallthatforthequasi-polynomialsthevalueisgivenbytheclosedformexpression(3.12).Henceitisoftenpossibletoseethattheperturbations(4.5)and(4.11)yieldafamilysatisfyingthecondition(3.10).Ifitisthecasethenalltheresultsofthissubsectionarevalidinastongerformulation,i.e.,inwhichonereplaceatheclassHPbytheclassofstablequasi-polynomials.4.2.Someexamples.ChebotarevandMeiman.Addresingamechani-calproblemsuggestedbyVoznesenskyChebotarevandMeiman[CM49]consideredthefunc- wherewerepolynomialsofdegree5.LetusconsiderherethemoregeneralcasewithThinkingofasofaxedpolynomial,andofasofanintervalpolynomialweconstructforthefourcorrespondingpolynomialsbytheKharitonovrule.Thenthefollowingassertionisvalid.Ifthefourfunctionsarestablethenallarestable,too.wherearepolynomialssatisfyingAgain,letusthinkingofazofaxedpolynomial,andofasofanintervalpolynomials,andletusconstructthefourpolynomialsandthefourpolynomialsusingtheKharitonovrecipe.ROPOSITIONIfthesixteenfunctionsk,larestablethanallofthequasi-polynomialin(4.14)arestable.5.Proofofthelemma4.1..SincebothbelongtotheclassHP,therootsofeachoftheinterlacewiththeonesofbytheorem3.4.Now,(4.5)impliesthefollowingstatement.Betweenanytwosuccessivezerosoftherecouldbeeitherone(cf.withgure2.1)ormore(cf.withgure2.2)zerosof .Weprovethatthereisalwaysonlyonezero.Denotethepositiverootsofthefollowingfunctionsasfollows: functionsafter their(real)roots Fe( {ak} Fo,min( b1k} Fo( bk} F b2k} 11 sothatwehaveWedonotknowyetthatthereisonlyone.Soletusthinkforamomentthattherearemore,butweworkwithonlyasubsetoftherootsof/jzchoosingonlyonesuch.Sincetherootsofthefunc-areimaginaryandsymmetricwithrespecttotheorigin,theirHadamarddecompositions[L96]havetheform(1+ (1+ Letusconstructthefunction(1+ Itfollowsfrom(4.5)thatSincethearethesubsetofalltherootsofisanentirefunction.Now,therelations(4.4),(4.5),and(5.2)-(5.4)imply Fo(z)oforz= Clearly,(5.7)and(5.8)meanthat isboundedon .Interlacingoftherootsandlemma2ofsec27.3in[L96]implySimilarly,interlacingoftherootsof[L96]impliesThelattertwoequations(5.9),(5.10)andthecondition(4.6)yield Now,alltherootsarepurelyimaginaryand HencethefunctionbytheresultofLecture5in[L96]musthavethecompletelyregulargrowthSinceoneofthefactorsin(5.7)hascompletelyregulargrowth,bytheorem5ofChapterIIIin[L80]wehaveThelatterequationmeanthatthefunctionisoftheminimaltype.BythePhragmofprinciple[L96]isboundedeverywhereentirefunctionandhenceaconstant.Inviewof(5.4)and =lim wehave(0)=1.Hencetherootsofdointerlacewiththeonesofyieldingthecondition1)oftheorem3.4.Secondly,thecondition(3.8)followsfrom(4.6)andthefactthatisintheHPFinally,(3.9)isfullledthanksto(5.5)andexpressions(5.2),(5.3),(5.4).belongtotheclassHPbytheorem3.4Thelemmaisproven.6.Thematrixcase.Inmanyappliedproblemsquasi-polynomialsoccurascharacteris-ticpolynomialsofacertainsystem.However,theintervalfamilytheoriginalsystemsisnottranslatedimmediatelyintotheintervalfamilyofthecorrespondingquasi-polynomials.Herewesuggestacertainalternative,namelytotrytoconstructtheedgepolynomialsfortheintervalfamilyofquasi-polynomialsdirectlyintermsoftheoriginalsystem.Werstformulatethegeneralsuggestion,andthenconsidercertainexamples.6.1.Ageneralsuggestion.isanarbitraryentirefunctionsuchthat Again,werepresent evendegreeterms odddegreetermsNow,supposethatonesucceededtoconstructfourmatricessuchthatforallvectorsz,hh,k Anentirefunctionisreferredtoasafunctionofcompletelyregulargrowthifthefunctionr, uniformlyconvergestoalmosteverywhere. z,hh,kz,hsatisfytheconditionsofthegeneralizedKharitonovtheorem.Thenisastablematrix6.2.Someexamples.Retardeddifferentialequation. whereisanvectorfunctionandisanmatrix,andisanumber.ThecharacteristicmatrixofthisequationhastheformwhereWesupposethatthereexisttwonumberssuchthatROPOSITIONThematrixfunctionisstableifwhere andifistherootoftheequationLetusprovethelatterproposition.Indeed,usingaresultofHyes[H50](seealso[HVL93])wededucethatthefunctionsareallstable.TheprooffollowsimmediatelyfromthematrixversionofthegeneralizedKharitonovtheorem.LetusconsiderthematrixfunctionwherearematricessatisfyingwithcertainnumbersROPOSITIONThematrixfunctionisstableif 2,14 wherearetherootsof ,0 TheprooffollowsimmediatelyfromthematrixversionofthegeneralizedKharitonovtheoremandthefactthatallthefunctionsarestable.Thestabilityofthefourscalarquasi-polynomialsin(6.3)followsfrom13.9of[BC63],andmoreove,theresultoftheabovepropositionmakessenseeveninthesimplestscalarcase.[B1879]M.Biehler,Suruneclassedequationsalgebriquesdonttouteslesracinessontr,J.ReineAngew.Math.87(1879)350352.[BC63]R.BellmanandK.Cooke,Differential-DifferenceEquations,AcademicPress,1963.[BCK95]S.P.Bhattacharyya,H.Chapellat,L.H.Keel,RobustControl-TheParametricApproach,PrenticeHall,1995.[CM49]N.Chebotarev,N.Meiman,TheRouth-Hurwitzprobelmforpolynomialsandentirefunctions,TrudyMIAN,1949,vol.26.[DHB03]A.Datta,M.-T.HoandS.P.Bhattacharyya,StructureandSynthesisofPIDControllers,SpringerVerlag,2003.[DV98]L.DugardandE.Verriest(eds),Stabilityandcontroloftime-delaysystems,SpringertVerlag1998,[F53]S.Faedo,Unnuovoproblemadistabilitaperleequazionialgebricheecoefcientreali,Ann.Scuolanormalesuperiore,Pisa,Sci.fos.mat.,1953,vol,7,no1-2.p.53-63[H1856]Hermite,C.,ExtraitdunelettredeMr.Ch.HermitedeParisaMr.BorchardtdeBerlin,surlenom-bredesracinesduneequationalgebriquecomprisesentredeslimitsdonJ.ReineAngew.(1856),39-51.[H1895]A.Hurwitz,UberdieBedingungen,unterwelcheneineGleichungnurWurzelnmitnegativenreellenTeilenbesitzt,Math.Ann.46(1895)273284.[H50]N.D.Hayes,Rootsofthetranscedentalequationassociatedwithacertaindifferentialdifference,J.LondonMath.Society,25(1950),226-232.[HVL93]J.K.HaleandS.VerdunLunel,IntroductiontoFunctionalDifferentialEquations,Springer-Verlag,NewYork,AppliedMathematicalSciencesVol.99,1993.[K]M.G.Krein.Asremarkedin[L96]thetheorem3.2wasalsoobtainbyM.G.Krein,butitwasneverpublishedbyhim.[K78]KharitonovV.L.,Asymptoticstabilityofanequilibriumpositionofafamilyofsystemsofdifferen-tialequations,Differenzyalnyeuravneniya,(1978),2086-2088.[KTMOM03]Kharitonov,V.L.;Torres-Munoz,J.A.;Ortiz-Moctezuma,M.B.;Polytopicfamiliesofquasi-polynomials:vertex-typestabilityconditions,CircuitsandSystemsI:FundamentalTheoryandApplications,IEEETransactionson[seealsoCircuitsandSystemsI:RegularPapers,IEEETransactionson],Volume:50,Issue:11,Nov.2003Pages:1413-1420[KZ97]Kharitonov,V.L.andZyabkoA.,RobustStabilityofTimeDelaySystems,IEEETrans.onAuto-maticControl,39(1994),2388-2397.[L80]B.Ya.Levin.Distributionofzerosofentirefunctions.Revisededition.Amer.Math.Soc.Providence,RI,1980.[L96]B.Ya.Levin,LecturesonEntireFunctions,AMSpublications,1996.onlineversion:http://www.ams.org/online [M1868]J.Maxwell.OnGovernors,ProceedingsoftheRoyalSociety,No.100.1868 [OS02]NejatOlgacandRifatSipahi,AnExactMethodfortheStabilityAnalysisofTime-DelayedLinearTime-Invariant(LTI)Systems,IEEETrans.onAutomaticControl,vol.47,no.5,May2002,[OS03]V.OlshevskyandL.Sakhnovich,AnOperatorIdentitiesApproachtoBezoutians.AGeneralSchemeandExamples,2003,preprint.[P42]L.Pontryagin,Onthezerosofsometranscedentalfunctions,IANUSSR,Math.series,vol.6,115-134,1942.[R1977]E.J.Routh,Stabilityofagivenstateofmotion,London,1877.[VL02]S.VerdunLunel,SpectralThgeoryforneutraldelayequationswithapplicationstocontrolandsta-,inMath.SystemsTheoryinBiology,CommunicationsandFinance,(J.Rosenthal,D.Gillivan,eds),p.415-469Springer,2002.