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 ID: 25226
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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.LetusÞrstconsideraverytransparentscalarexample.Consideradifferentialequationwithatimedelay,i.e., zy,yThesolutionclearlyhastheformwhereisarootofthespecialentirefunction)=1+(calledbelowaquasi-polynomial).Oneseesthatthesystem(1.6)isstableiftherootsoftheentirefunctionalllieinthelefthalfplane.Theabovefunctionbelongstoamoregeneralclassofquasi-polynomialsdeÞnednext.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.TheÞrstresultsonstabilityofquasi-polynomialsandentirefunctionswereobtainedinthepioneeringworksofPontryagin[P42]andofChebotarev-Meiman[CM49].Theyweremotivatedbycontrollerdesign[speciÞcally,ofregulatorsandservomechanismsdrivingthetrackingerrortozero];modellingahitinapipeline.Themorerecentrelevantliteratureisnotexhaustive,andreading[BC63],[L80],[HVL93],[BCK95],[L96],[KZ97],[DV98],[OS02][DHB03],[KTMOM03]givesanintroductioninthecurrentstateofartinthisarea[seealsothereferencestherein,nopossibleomissionsareintentional].1.3.Mainresultsandthestructureofthepaper.AverycloseproblemofstudyingthestabilityofcertainÒperturbedÓfamiliesofentirefunctionswasaddressed,e.g.,in[BCK95],[DHB03],[KTMOM03].However,theßavoroftheirresultsseemstobedifferent.Firstly,theyconsideredÒuncertainfamiliesÓthatarenotintervalperturbationsasin(1.2).Secondly,theirsufÞcientconditionsarearenotoftheform(1.3),andmoreovertheirnumbercanbelarge.Themainresultofthispaperisdifferent:itisadirectgeneralizationoftheKharitonovÕstheorem.Inparticular,weconsiderexactlythestabilityofintervalentirefunctionsofthetype(1.2),andweshowthatfulÞllmentofthefourconditionsexactlyofthetype(1.3)sufÞcesforstability.Thepaperisstructuredasfollows.InSec2welaythegroundworkforfurthergener-alizationsbyrecallingthestandardproofoftheKharitonovtheorembasedontheclassicalHermite-Biehlercriterion[B1879].Simpleexamplesareincludedtorecallthatthelattercriterioncannotbecarriedovertoarbitraryentirefunctions.ThesectionidentiÞestwodifÞ-cultiestoovercomeinthenexttwosections. Suchconditionsinvolvecertainhyperplanes[BCK95],ortheyareoftheso-calledsomevertex-type InSec3wediscussthepropertiesoftheclassforwhichweshallprovethegeneralizedKharitonovtheorem.WeÞrstrecallhowtoovercometheÞrstdifÞcultybydeÞningtheHBclassofentirefunctions.Secondly,weshowhowtoovercometheseconddifÞcultyandsuggestananalogofnotchangingdegreeproperty.InSec4weusetheresultsofthetwopreceedingsectionstoformulateageneralizationoftheKharitonovtheoremforaclassofentirefunctions.Sec5containsproofsofthemainresults.InSec6wesuggestatechniquethatisusefulforsolvingsomeKharitonov-likeproblemsformatrixpolynomialsandmatrixentirefunctions.2.TwodifÞculties.2.1.TheHermite-BiehlertheoremandtheproofofthepolynomialKharitonovthe-orem.First,werecalltheclassicalHermire-Biehlertheorem,itplaysanimportantroleinestablishingtheKharitonovÕsresultforthepolynomialsandinextendingittoentirefunc-beapolynomialin(1.1)andde netwopolynomials,andeven,andanoddanddenote Thenthepolynomialisstableifandonlyifthefollowingtwoconditionsholdtrue.Therootsofthepolynomialsareallrealandtheyinterlace.ThereisatleastonepointsuchthatThecondition1isequivalenttothefactthatthattherootsofthepolynomialsareallpurelyimaginaryandtheyinterlace.Ifthe(2.2)isful lledjustforonepointthenitisvalidforallFortherealthecondition(2.2)simplymeansthatwherearethecoef cientsofin(1.1).TheproofoftheKharitonovtheoremisnowimmediate.Figure1(thatillustratestheremark1andtheorem2.1)showsthatanyintervalperturbationoftheoddtermsgivesusastablepolynomial.FixingtheperturbedoddpartandapplyingasimilarargumenttotheevenpartoneÞnallyobtainsthetheorem1.1.Inthenexttwosubsectionsweshowthatthereareatleasttwoproblemsincarryingovertheaboveargumentstoarbitraryentirefunctions.2.2.TheÞrstdifÞculty.TheÞxeddegreepropertyandthenumberofroots.1indicatesthatintervalperturbationsdonotdestroyinterlacingoftherootsof.However,itisimplicitlyassumedthattheallpolynomialsin(1.1)and(1.2)havethesamedegree.Hencethenumberoftherootsofeachofthepolynomialsin(2.1)staysthesame. Fo,max Fo Fo,min Fe .2.1.IllustrationfortheProofoftheclassicalKharitonovtheoremforpolynomialsviatheHermite-Biehler.However,ifwereanentirefunctionthenitsdegreeisÒinÞnite,Óandhenceonehastotakecareofpreventingnewrootsfromoccurring.Graphically,onehastopreventthephenomenonshowninFigre2bymeansofimposingcertainadditionalconstrainstotheÞxed-degreeproperty. eFoFo,minFo,max .2.2.AnillustrationfortheÞrstdifÞculty.Onehastoimposeadditionalconditionspreventingarisingofthenewrootsof2.3.TheseconddifÞculty.TheHermite-Biehlertheoremdoesnotcarryovertoentirefunctions.Interlacingoftherootsisnotnecessary.Thefunction doesnothaverootsatallandhencestable.However,usingthede nition(2.3)weseethat)=cos)=sinhavenon-realroots,e.g., InterlacingoftherootsisnotsufÞcient.Again,usingthede nition(2.3)weseethatdonothaverootsatall,andhencetheinterlacingpropertyisful lled.Moreover,forany.Nevertheless Again,theseconddifÞcultyalsoindicatesthatitishopelesstoextendtheKharitonovÕstheoremtotheclassofallentirefunctions.Similarlytotheremarkmadeattheendofthesubsection2.2,thechallengehereistoidentifyaclassofentirefunctionsforwhichthetwodifÞcultiescanberemoved,andthentotrytoprovetheKharitonovÕstheoremforthatclass.Thisispreciselywhatisdoneintherestofthepaper.3.RemovingthetwodifÞculties.3.1.RemovingtheFirstDifÞculty.AnanalogueoftheÞxeddegreeproperty.thatevenforpolynomialstheconditions)=deg)=degareimplicitlyincludedintheclassicalKharitonovtheoremandtheyarecrucialforitsvalidity.(HereweusedthedeÞnitionin(2.3).)InourgeneralizedKharitonovtheoremtheseconditionswillbeincludedinadifferentform Fz)=OFe(z) thatappliesnotonlytopolynomialsbuttoentirefunctionsaswell.Thelatterexpressionsmakesenseforsincealltherootsofarepurelyimaginary,cf.withtheremark2.3.2.RemovingtheSecondDifÞculty.TheclassHP.Theindicatorfunctionproperty.3.2.1.Classicalresults.TheclassP.InthissubsectionwerecallthebasicdeÞnitionsandresultsthatcanbefoundin[CM49]and[L80],[L96].Theywillbeusedinwhatfollows.Wediscussherethesituationinwhichdoesnothaverootsinthelowerhalfplane.Inthenextsectionsthesesettingswillbeadjustedforstabilityof)=(EFINITIONbeanentirefunctionof niteexponentialtype.TocharacterizethegrowthofPhragmenandLindelofintroducedthefunction | whichiscalledanindicatorfunction Thequantity 2)h( iscalledtheofthefunctionAnentirefunctionof niteexponentialtypeissaidtobeintheclassPifithasnozerosintheopenlowerhalf-planeandTheindicatorfunctionplaysacrucialroleontherestofthepaper.Onereasonisthatitsbehaviorisconnectedwithinterlacingoftheroots,asdescribednext.([CM49],[L96],[K])Letuspartitiontheentirefunctionof niteexpo-nentialtypesothatarerealentirefunctions.belongstotheclassPifandonlyif1.therootsofareallrealandinterlacing;2.theindicatorfunctionsof3.atsomerealpointwehave(Ifthelatterconditionisful lledjustforonepointthenitisvalidforall3.2.2.AslightmodiÞcation.TheclassHP.Applicationtostability.ItisconvenienttoexplicitlytranslatetheresultsofSec3.2.1tothelefthalfplanecase(stability)andtousethemthereafter.EFINITIONbeanentirefunctionof niteexponentialtype.WesuggesttorefertothequantityastheofthefunctionWeshallreservethenametheHPclassforentirefunctionsof niteexponentialtypewithnozerosintheopenlefthalf-planeandsatisfying(cf.withtheorem3.2)Letuspartitiontherealentirefunctionof niteexponentialtype evendegreeterms odddegreetermsbelongstotheclassHPifandonlyif1.therootsofareallpurelyimaginaryandinterlacing; 2.theindicatorfunctionsof3.WehaveTheorem3.4yieldsthefollowingresult.(cf.with[L80],Ch.VI,sec.4.thm8.)Letbeanentirefunctionof niteexponentialtypewithnorootsontheimaginaryaxissatisfyingisstableifandonlyiftherootsofin(3.7)alllieontheimaginaryaxisandinterlace.Stabilityofquasi-polynomials.wherearerealpolynomials,and... 2 21 23 Ifweassume(0)=i.e.,Inaccordancewiththeorem3.5,inthisexampleinterlacingoftherootsof(thatareallpurelyimaginary)isthenecessaryandsuf cientconditionforthestabilityof)=sinh()+cosh(wherearepolynomials.InthiscaseInthelatterexamplethecondition(3.10)isnotfulÞlledwhichshowsthatthetheorem3.5isnotuniversal.HencetoderiveageneralizationoftheKharitonovtheoremwemayneedtousetheorem3.4,e.g.,itscondition(3.8).Thisisasubjectofthenextsubsection.3.2.3.RemovingtheSecondDifÞculty.Theindicatorfunctionproperty.Theclassoffunctionssatisfying(3.6)isanaturalgeneralizationofpolynomials.Indeed,thedeÞnition(3.10)immediatelyyieldsthatifisapolynomialthenandhenceHence,theclassofstablepolynomialscoincideswithpolynomialsbelongingtoThisobservationisoneindicationastowhytheindicatorfunctionproperty(triviallyfulÞlledforpolynomials)wouldbequitenaturalinthecontextofthegeneralizationoftheKharitonovtheoremtoentirefunctions. Alternatively,alltherootsof arerealandinterlace. 4.ThegeneralizedKharitonovtheorem.4.1.Mainresults.Throughoutthissectionweuseadecomposition evendegreeterms odddegreetermsofanarbitraryentirefunction F( betwoentirefunctionsof niteexponentialtypethatbelongtotheclassHPandsatisfy Fz)oforz= Thenallthefunctionsoftheform Fo( F (4.5)(for= )simultaneouslybelongtotheclassHPprovidedthefollowingconditionsare1.Thefunctionsareallofthe niteexponentialtypeandtheysatisfytheindica-torfunctionproperty:2.Ananalogofthe xed-degree-property: Fz)=Oz= Weshallprovetheabovelemmaattheendofthissection.Theresultdualtothelemma4.1isstatednext. Thisisananalogoftheintervaluncertainty(1.4). betwoentirefunctionsof niteexponentialtypethatbelongtotheclassHPandsatisfy Fz)eforz= Thenallthefunctionsoftheform )simultaneouslybelongtotheclassHPprovidedthefollowingconditionsare1.Thefunctionsareallofthe niteexponentialtypeandtheysatisfytheindica-torfunctionproperty:2.Ananalogofthe xed-degree-property: Fz)=Oz= Thelemmas4.1and4.2implythefollowingresult.GeneralizedKharitonovtheorem.Lettheconditions(4.4),(4.5),and(4.10),(4.11)andtheconditions1)-2)ofeachofthelemmas4.1and4.2beful lled.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.LetusconsiderherethemoregeneralcasewithThinkingofasofa xedpolynomial,andofasofanintervalpolynomialweconstructforthefourcorrespondingpolynomialsbytheKharitonovrule.Thenthefollowingassertionisvalid.Ifthefourfunctionsarestablethenallarestable,too.wherearepolynomialssatisfyingAgain,letusthinkingofazofa xedpolynomial,andofasofanintervalpolynomials,andletusconstructthefourpolynomialsandthefourpolynomialsusingtheKharitonovrecipe.ROPOSITIONIfthesixteenfunctionsk,larestablethanallofthequasi-polynomialin(4.14)arestable.5.Proofofthelemma4.1..SincebothbelongtotheclassHP,therootsofeachoftheinterlacewiththeonesofbytheorem3.4.Now,(4.5)impliesthefollowingstatement.Betweenanytwosuccessivezerosoftherecouldbeeitherone(cf.withÞgure2.1)ormore(cf.withÞgure2.2)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)isfulÞlledthanksto(5.5)andexpressions(5.2),(5.3),(5.4).belongtotheclassHPbytheorem3.4Thelemmaisproven.6.Thematrixcase.Inmanyappliedproblemsquasi-polynomialsoccurascharacteris-ticpolynomialsofacertainsystem.However,theintervalfamilytheoriginalsystemsisnottranslatedimmediatelyintotheintervalfamilyofthecorrespondingquasi-polynomials.Herewesuggestacertainalternative,namelytotrytoconstructtheÒedgeÓpolynomialsfortheintervalfamilyofquasi-polynomialsdirectlyintermsoftheoriginalsystem.WeÞrstformulatethegeneralsuggestion,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,Unnuovoproblemadistabilitaperleequazionialgebricheecoef 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