An integral invariance principle for differential

ANINTEGRALINVARIANCEPRINCIPLEFORDIFFERENTIALINCLUSIONSWITHAPPLICATIONSINADAPTIVECONTROLE.P.RYANAbstract.TheByrnes-MartinintegralinvarianceprincipleforordinarydierentialequationsisextendedtodierentialinclusionsonRN.Theextendedresultisappliedindemonstratingtheexistenceofadaptivestabilizersandservomechanismsforavarietyofnonlinearsystemclasses.Keywords.adaptivecontrol;dierentialinclusions;invarianceprinciples;nonlinearsystems;universalservomechanisms.AMSsubjectclassications.93D05,93D09,93D15,93D21,34D051.Introduction.Supposethat_x=f(x)generatesasemidynamicalsystemonRNwithsemiow’andso,foreachx02RN,x()=’(;x0)istheuniquemaximalforwards-timesolutionoftheinitial-valueproblem_x=f(x),x(0)=x0.In[2],Byrnes&Martinprovethefollowingintegralinvarianceprinciple:if’(;x0)isboundedandR10l(’(t;x0))dt1forsomecontinuousfunctionl:RN!R+:=[0;1),then’(t;x0)tends,ast!1,tothelargestinvariant(withrespecttothedierentialequation)setinl1(0),thezerolevelsetofl.Thisresulthasramicationsinadap-tivecontrol,someofwhicharehighlightedinthepresentpaper.However,wewishtoconsiderthe(adaptive)controlprobleminafairlygeneralsettingthatallowstime-variationintheunderlyingdierentialequations,possiblenon-uniquenessofsolutions,anddiscontinuousfeedbackstrategies:eachofthesefeaturesplacestheproblemout-sidethescopeof[2].Forthisreasonwedevelop,inTheorem2.10,anintegralin-varianceprincipleforinitial-valueproblemsoftheform_x2X(x),x(0)=x0,wheretheset-valuedmapXisdenedonsomeopendomainGRNandisassumedtobeuppersemicontinuouswithnon-empty,convexandcompactvalues.InthecaseG=RN,Theorem2.10containsthefollowinggeneralizationoftheByrnes-Martinresult:ifx():R+!RNisaboundedsolutionandR10l(x(s))ds1forsomelowersemicontinuousl:RN!R+,thenx(t)tends,ast!1,tothelargestweakly-invariant(withrespecttothedierentialinclusion)setinl1(0).OneparticularconsequenceofTheorem2.10istofacilitatethederivationofanonsmoothextension,todierentialinclusions,ofLaSalle’sinvarianceprinciplefordierentialequations:thisextensionmaybeofindependentinterestandispresentedinTheorem2.11.Theremainderofthepaperisdevotedtotheapplication(inacollectionofvelemmas)ofthegeneralizedintegralinvarianceprincipletodemonstrate,byconstructionandforavarietyofnonlinearsystemclasses,theexistenceofasingleadaptivecontrollerthatachieves(withoutsystemidentication,parameterestimationorinjectionofprobingsignals)someprescribedobjectiveforeverysystemintheunderlyingclass.2.Dierentialinclusions.Someknownfacts(tailored1toourimmediatepur-pose)pertainingtodierentialinclusionsarerstassembled.DepartmentofMathematicalSciences,UniversityofBath,BathBA27AY,UnitedKingdom.E-mail:[e-mail]1VariantsofPropositions2.2,2.4and2.8canbefoundin,forexample,[8],[19]:forgeneraltreatmentsofdierentialinclusionsandrelatedtopicsinset-valuedanalysis,nonsmoothcontrolandoptimizationsee[1],[4],[6],[7],[8]and[12].12E.P.RYAN2.1.Maximalsolutions.Considerthenon-autonomousinitial-valueproblem_x(t)2X(t;x(t));x(t)2G;x(t0)=x0;(1)whereG6=;isanopensubsetofRN.Theset-valuedmap(t;x)7!X(t;x)RNin(1)isassumedtobeuppersemicontinuous2onRG,withnon-empty,convexandcompactvalues.Thisissucient(see,forexample,[1,Chapter2,Theorem3])toensurethat,foreach(t0;x0)2RG,(1)admitsasolution:anX-arc3x2AC([t0;!);G),withx(t0)=x0.Definition2.1.Asolutionxof(1)issaidtobemaximal,ifitdoesnothaveaproperrightextensionwhichisalsoasolutionof(1).Proposition2.2.Everysolutionof(1)canbeextendedtoamaximalsolution.Definition2.3.Asolutionx2AC([t0;!);G)of(1)isprecompactifitismaximalandtheclosurecl(x([t0;!)))ofitstrajectoryisacompactsubsetofG.Proposition2.4.Ifx2AC([t0;!);G)isaprecompactsolutionof(1),then!=1.2.2.Limitsets.Here,wespecializetotheautonomouscaseof(1),rewrittenas_x(t)2X(x(t));x(t)2G;x(0)=x0(2)where,withoutlossofgenerality,t0=0isassumed.Themapx7!X(x)RN(withdomainG)isuppersemicontinuouswithnon-empty,convexandcompactvalues.Definition2.5.Letx2AC([0;!);G)beamaximalsolutionof(2).Apointx2RNisan!-limitpointofxifthereexistsanincreasingsequence(tn)[0;!)suchthattn!!andx(tn)!xasn!1.Theset(x)ofall!-limitpointsofxisthe!-limitsetofx.Definition2.6.LetCRNbenon-empty.Afunctionx2AC([0;!);G)issaidtoapproachCifdC(x(t))!0ast!!,wheredCisthe(Euclidean)distancefunctionforCdened(onRN)bydC(v):=inffkvckjc2Cg.Definition2.7.Relativeto(2),SRNissaidtobeaweakly-invariantsetif,foreachx02S\G,thereexistsatleastonemaximalsolutionx2AC([0;!);G)of(2)with!=1andwithtrajectoryx([0;!))inS.Proposition2.8.Ifxisaprecompactsolutionof(2),then(x)isanon-empty,compact,connectedsubsetofG.Moreover,(x)isthesmallestclosedsetapproachedbyxandisweaklyinvariant.2.3.Invarianceprinciples.Forlateruse,thefollowingfact(aspecializationofamoregeneralresult[4,Theorem3.1.7])isrstrecorded.Proposition2.9.LetI=[a;b],letnon-emptyKGbecompact.If(xn)AC(I;K)isasequenceofX-arcsandthereexistsascalarsuchthat,foralln,k_xn(t)kforalmostallt2I,then(xn)hasasubsequenceconverginguniformlytoanX-arcx2AC(I;K).Wenowarriveatthemainresult,whichgeneralizes[2,Theorem1.2].2Theset-valuedmapXisuppersemicontinuousifitisuppersemicontinuousateverypointofitsdo