PreliminaryresultsStabilizationof linear systemsoffinite dimension
1.1Introduction
Wedevelopinthischapterthebasicproblemarisingfromstabilization problemsoffinite-dimension.Sincethecelebratedpoleassignmenttheory[70] (seealso[56,68])forlinearcontrolsystemsoffinitedimensionappeared,the theoryhasbeenappliedtovariousstabilizationproblemsbothoffinite dimensionandinfinitedimensionsuchastheonewithboundary output/boundaryinputscheme(see,e.g.,[12,13,28,37–40,42–45,47–50, 53,58,59]andthereferencestherein).Thesymbol Hn, n = 1, 2,... ,hereafter willdenoteafinite-dimensionalHilbertspacewithdim Hn = n,equippedwith innerproduct ⟨ , ⟩n andnorm ∥ ∥.Thesymbol ∥ ∥ isalsousedforthe L (Hn)-norm.Let L, G,and W beoperatorsin L (Hn), L (CN ; Hn),and L (Hn; CN ),respectively.Hereandhereafter,thesymbol L (R; S), R and S beinglinearspacesoffiniteorinfinitedimension,meansthesetofalllinear boundedoperatorsmapping R into S.Theset L (R; S) formsalinearspace. When R = S, L (R; R) isabbreviatedsimplyas L (R).Given L, W ,andanyset of n complexnumbers, Z = {ζi}1 i n,theproblemistoseekasuitable G such that σ(L GW )= Z,where σ(L GW ) meansthespectrumoftheoperator
L GW .Oralternatively,given L and G,itsalgebraiccounterpartistoseeka W suchthat σ(L GW )= Z.Stimulatedbytheresultof[70],variousapproaches andalgorhismsforcomputationof G or W havebeenproposedsincethen(see, e.g.,[7,10,14]).However,eachapproachneedsmuchpreparationandadeep backgroundinlinearalgebratoachievestabilizationanddeterminethe necessaryparameters.Explicitrealizationsof G or W sometimesseem complicated.Oneforthisisnodoubtthecomplexityoftheprocessin determining G or Wexactly satisfyingtherelation, σ(L GW )= Z
Letusdescribeourcontrolsystem:Oursystem,consistingofastate u(·) ∈ Hn,output y = Wu ∈ CN ,andinput f ∈ CN ,isdescribedbyalinear differentialequationin Hn,
( )T denotingthetransposeofvectorsormatricesthroughoutthemonogtaph.
Thevectors wk ∈ Hn denotegivenweightsoftheobservation(output);and gk ∈ Hn areactuatorstobeconstructed.Bysetting f = y in(1.1),thecontrolsystem yieldsafeedbacksystem,
+(L GW )u = 0, u(0)= u0 ∈ Hn.
Accordingtothechoiceofabasisfor Hn,theoperators L, G,and W are identifiedwithmatricesofrespectivesize.Wehereafteremploytheabove symbolssomewhatdifferentfromthosefamiliarinthecontroltheory communityoffinitedimension,inwhichstateofthesystem,forexample, wouldbeoftenrepresentedas x(·);output Cx;input u;andequation dx dt = Ax + Bu =(A + BC)x, u = Cx.
Thereasonforemployingpresentsymbolsisthattheyareconsistentwiththose insystemsofinfinitedimensiondiscussedinlaterchapters.
Letusassumethat σ(L) ∩ C = ∅,sothatthesystem(1.1)with f = 0is unstable.Givena µ > 0,the stabilizationproblem forthefinitedimensional controlsystem(1.3)istoseeka G or W suchthat e t(L GW ) const e µt , t 0 (1 4)
Thepoleassignmenttheory[70]playsafundamentalroleintheaboveproblem, andhasbeenappliedsofartovariouslinearsystems.Thetheoryisconcretely
statedasfollows: LetZ = {ζi}1 i n beanysetofncomplexnumbers,wheresome ζi maycoincide.Then,thereexistsanoperatorGsuchthat σ(L GW )= Z, ifandonlyifthepair (W, L) isobservable.Thus,iftheset Z ischosensuch thatminζ∈Z Re ζ,say µ (= Re ζ1) ispositive,andifthereis no generalized eigenspaceof L GW correspondingto ζ1,weobtainthedecayestimate(1.4).
Nowweask:Doweneed all informationon σ(L GW ) forstabilization? Infact,toobtainthedecayestimate(1.4),itisnotnecessarytodesignate all elementsoftheset Z:Whatisreallynecessaryisthenumber, µ = minζi∈Z Re ζi,say = Re ζ1,andthespectralpropertythat ζ1 doesnotallow anygeneralizedeigenspace;thelatteristherequirementthat no factorof algebraicgrowthintimeisaddedtotheright-handsideof(1.4).Infact,when analgebraicgrowthisadded,thedecaypropertybecomesalittleworse,and thegainconstant ( 1) in(1.4)increases.Theaboveoperator L GW also appears,asa pseudo-substructure,inthestabilizationproblemsofinfinite dimensionallinearsystemssuchasparabolicsystemsand/orretardedsystems (see,e.g.,[16]):Thesesystemsaredecomposedintotwo,andunderstoodas compositesystemsconsistingoftwostates;onebelongingtoafinite dimensionalsubspace,andtheothertoaninfinitedimensionalone.Itis impossible,however,tomanagetheinfinitedimensionalsubstructures.Thus,no matterhow precisely thefinitedimensionalspectrum σ(L GW ) couldbe assigned,itdoesnotexactlydominatethewholestructureofinfinite dimension.Inotherwords,theassignedspectrumoffinitedimensionisnot necessarilyasubsetofthespectrumoftheinfinite-dimensionalfeedback controlsystem.
Inviewoftheaboveobservations,ouraiminthischapteristodevelopa newapproachmuchsimplerthanthoseinexistingliterature,whichallowsusto constructadesiredoperator G orasetofactuators gk ensuringthedecay(1.4) inasimplerandmoreexplicitmanner(see(2.10)justbelowLemma2.2).The resultis,however,notassharpasin[70]inthesensethatitdoesnotgenerally providethepreciselocationoftheassignedeigenvalues.Fromtheabove viewpointofinfinite-dimensionalcontroltheory,however,theresultwouldbe meaningfulenough,andsatisfactoryforstabilization.Wenotethatourresult exactlycoincideswiththestandardpoleassignmenttheoryinthecasewherewe canchoose N = 1(seeProposition2.3inSection2).Theresultsofthischapter arebasedonthosediscussedin[48,51,52].
OurapproachisbasedonSylvester’sequationoffinitedimension. Sylvester’sequationininfinite-dimensionalspaceshasalsobeenstudied extensively(see,e.g.,[6]forequationsinvolvingonlyboundedoperators),and eventheunboundednessofthegivenoperatorsareallowed[37,39,40,42–45, 47,49,50,53].Sylvester’sequationinthischapterisoffinitedimension,sothat therearisesnodifficultycausedbythecomplexityofinfinitedimension.Its infinite-dimensionalversionandthepropertiesarediscussedlaterin Chapters4,
6,and 7.Givenapositiveinteger s andvectors ξk ∈ Hs,1 k N,letus considerthefollowingSylvester’sequationin Hn:
XL MX = ΞW, Ξ ∈ L (CN ; Hs), where Ξz = N ∑
1 zN )T
CN (1.5)
Here, M denotesagivenoperatorin L (Hs),and ξk vectorstobedesignedin Hs Apossiblesolution X wouldbelongto L (Hn; Hs).AnapproachviaSylvester’s equationsisfound,e.g.,in[7,10],inwhich,bysetting n = s,aconditionforthe existenceoftheboundedinverse X 1 ∈ L (Hn) issought.Choosingan M such that σ(M) ⊂ C+,itisthenprovedthat
L +(X 1Ξ)W = X 1MX, σ(X 1MX)= σ(M) ⊂ C+, theleft-handsideofwhichmeansadesiredperturbedoperator.Theprocedureof itsderivationis,however,rathercomplicated,andthechoiceofthe ξk isunclear. Infact, X 1 mightnotexistsometimesforsome ξk
Theapproachinthischapterisnewandratherdifferent.Letuscharacterize theoperator L in(1.5).Thereisasetofgeneralizedeigenpairs {λi, φij} withthe followingproperties:
(i) σ(L)= {λi;1 i ν ( n)}, λi = λj for i = j;and (ii) Lφ
Let Pλi betheprojectorin Hn correspondingtotheeigenvalue λi.Then,wesee that Pλi u = ∑mi j=1 uijφij for u ∈ Hn.Therestrictionof L ontotheinvariant subspace Pλi Hn is,inthebasis {φi1,..., φimi },isrepresentedbythe mi × mi uppertriangularmatrix Λi,where
i|( j, k) =
αi kj, j < k, λi, j = k, 0, j > k. (1.6)
Ifweset Λi = λi + Ni,thematrix Ni isnilpotent,thatis, Nmi i = 0.Theminimum integer n suchthatker Nn i = ker Nn+1 i ,denotedas li,iscalledthe ascent of λi L. Itiswellknownthattheascent li coincideswiththeorderofthepole λi ofthe resolvent (λ L) 1.Laurent’sexpansionof (λ L) 1 inaneighborhoodofthe pole λi ∈ σ(L) isexpressedas (λ L) 1 = li ∑ j=1 K j (λ λi) j + ∞ ∑ j=0 (λ λi) jKj, where li mi, Kj = 1 2πi ∫|ζ λi|=δ (ζ L) 1 (ζ λi) j+1 dζ, j = 0, ±1, ±2,.... (1.7)
Notethat K 1 = Pλi .Theset {φij;1 i ν, 1 j mi} formsabasisfor Hn. Each x ∈ Hn isuniquelyexpressedas x = ∑i, j xijφij.Let T beabijection,defined as Tx = (x11 x12 xνmν )T.Then, L isidentifiedwiththeuppertriangularmatrix Λ;
TLT 1 = Λ = diag (Λ1 Λ2 ... Λν) . (1.8)
Letusturntotheoperator M in(1.5).Let {ηij;1 i n, 1 j ℓi}
beanorthonormalbasisfor Hs.Thennecessarily s = ∑n i=1 ℓi n.Everyvector v ∈ Hs isexpressedas
Let {µi}n 1=1 beasetofpositivenumberssuchthat0 < µ1 < < µn,andset
(1.9) for v = ∑i, j vijηij.Itisapparentthat(i) σ(M)= {µi}n i=1;and(ii) (µi M)ηij = 0, 1 i n,1 j ℓi.Theoperator M isself-adjoint,andpotive-definite,
Let Qµi betheprojectorin Hs correspondingtotheeigenvalue µi ∈ σ(M),say Qµi v = ∑ℓi j=1 vijηij for v = ∑i, j vijηij.Weputanadditionalconditionon M: σ(L) ∩ σ(M)= ∅ (1 10)
Assuming(1.10),wederiveourfirstresultasProposition1.1.Sincetheproofis carriedoutinexactlythesamemannerasin[37,44,45,50],itisomitted.
Proposition1.1. Supposethatthecondition (1.10) issatisfied.Then, Sylvester’sequation (1 5) admitsauniqueoperatorsolutionX ∈ L (Hn; Hs). ThesolutionXisexpressedas
=
TheoryofStabilizationforLinearBoundaryControlSystems
whereCdenotesaJordancontourencircling σ(M) initsinside,with σ(L) outsideC.TheabovefirstexpressionisthesocalledRosenblumformula [6].
The mainresultsarestatedasTheorem2.1andProposition2.2inthenext section,whereamoreexplicitandconcreteexpressionthaneverbeforeofaset ofstabilizingactuators gk in(1.3)isobtained.Asweseeinthenextsection,an advantageofconsideringtheoperator X ∈ L (Hn; Hs) with s n isthatthe boundedinverse (X ∗X) 1 isensuredunderareasonableassumptiononthe operator Ξ.Anumericalexampleisalsogiven.Finally,Proposition2.3is stated,whereourfeedbackschemeexactlycoincideswiththestandardpole assignmenttheory[70]inthecasewherewecanchoose N = 1.
1.2MainResults
Weassumethat σ(L) ∩ C = ∅,sothatthesemigroup e tL , t 0,is unstable.Weconstructsuitableactuators gk ∈ Hn in(1.3)suchthat e t(L GW ) hasapreassigneddecayrate,say µ1 (see(1.9)).Theoperator (WWL WLn 1)T belongsto L (Hn; CnN ).Theobservabilitycondition onthepair (W, L) meansthattheaboveoperatorisinjective,inotherwords, ker (WWL WLn 1)T = {0}.Throughoutthesection,theseparation condition(1.10)isassumedinSylvester’sequation(1.5).Then,weobtainone ofthemainresults:
Theorem2.1. Assumethattheconditions
ker (WWL ... WLn 1)T = {0}, and ker Qµi Ξ = {0}, 1 i n (2.1)
aresatisfied.Then, kerX = {0}.
Proof. Let Xu = 0.InviewofProposition1.1,weseethat Qµi ΞW (µi L) 1 u = 0, 1 i n.
Since ker Qµi Ξ = {0},1 i n,by(2.1),weobtain W (µi L) 1 u = 0, 1 i n, or ⟨(µi L) 1 u, wk⟩n = 0, 1 k N, 1 i n. (2.2)
Set fk(λ; u)= ⟨(λ L) 1u, wk⟩n.Byrecallingthat T (λ L) 1T 1 = (λ Λ) 1 (see(1.8)), fk(λ; u) isrewrittenas ⟨(λ Λ) 1Tu, (T 1)∗ wk⟩Cn . Eachelementofthe n × n matrix (λ Λ) 1 isarationalfunctionof λ;its denominatorconsistsofapolynomialoforder n;andthenumeratoratmostof order n 1.Thismeansthateach fk(λ; u) isarationalfunctionof λ,the
denominatorofwhichisapolynomialoforder n,andthenumeratoroforder n 1.Sincethenumeratorof fk has atleastn distinctzeros µi,1 i n,by (2.2),weconcludethat fk(λ; u)= ⟨(λ L) 1 u,
Let c beanumbersuchthat c ∈ ρ(L),andset Lc = L + c.Inviewoftheidentity (λ L) 1 = Lc(λ L) 1Lc 1 = Lc 1 +(λ + c)(λ L) 1Lc 1 , letusintroduceaseriesofrationalfunctions f l k(λ; u), l = 0, 1,..., as f 0 k (λ; u)= fk(λ; u), f l+1 k (λ; u)= f l
Itiseasilyseenthat f l k(λ; u)= ⟨(λ L) 1L l c u, wk⟩n l ∑ i=1 1 (λ + c)i ⟨L (l+1 i) c u, wk⟩n (2.5) and f l k(λ; u)= 0, λ ∈ ρ(L) \{−c}, 1 k N, l 0.
InviewofLaurent’sexpansion(1.7)of (λ L) 1 inaneighborhoodof λi,we obtaintherelation
0 = fk(λ; u) = li ∑ j=1 ⟨K ju, wk⟩n (λ λi) j + ∞ ∑ j=0 (λ λi) j ⟨Kju, wk⟩n , 1 k N, inaneighborhoodof λi.Calculationoftheresidueof fk(λ; u) at λi impliesthat
⟨K 1u, wk⟩n = ⟨Pλi u, wk⟩n = 0, 1 i ν, 1 k N, or WPλi u = 0, 1 i ν (2.6)
Asfor f l k(λ; u), ℓ 1,wehaveasimilarexpressioninaneighborhoodof λi, f l k(λ; u)= li ∑ j=1 ⟨A jL l c u, wk⟩n (λ λi) j + ∞ ∑ j=0 (λ λi) j ⟨A jLc l u, wk⟩n l ∑ i=1 1 (λ + c)i ⟨L (l+1 i) c u, wk⟩n = 0
TheoryofStabilizationforLinearBoundaryControlSystems
by(2.5).Notethat K 1L l c u = Pλi L l c u = L l c Pλi u.Calculationoftheresidueof f l k(λ; u) at λi similarlyimpliesthat
K 1L l c u, wk⟩n = ⟨L l c Pλi u, wk⟩n = 0, 1 i ν, 1 k N, or WL l c Pλi u = 0, 1 i ν, l 1.
Combiningthesewiththeaboverelation(2.6),weseethat (WWL 1 c ... WL (n 1) c )T Pλi u = 0, 1 i ν (2 7)
Itisclearthat
ker (WWL ... WLn 1)T = ker (WWLc ... WLn 1 c )T ,
where Lc = L + c.Thus,bythefirstconditionof(2.1),itiseasilyseenthat
ker(WWL 1 c ... WL (n 1) c )T = ker(WWL ... WLn 1)T = {0}.
Thus,(2.7)immediatelyimpliesthat Pλi u = 0for1 i ν,andfinallythat u = 0.
ByTheorem2.1,thereisapositiveconstantsuchthat ∥Xu∥s const ∥u∥ , ∀u ∈ Hn
Thederivationoftheabovepositivelowerboundof ∥Xu∥s isduetoaspecific natureoffinite-dimensionalspaces.Theoperator X ∗X ∈ L (Hn) isself-adjoint, andpositive-definite.Infact,bytherelation const ∥u∥2 ∥Xu∥2 s = ⟨Xu, Xu⟩s = ⟨X ∗Xu, u⟩n ∥X ∗Xu∥∥u∥ , weseethat ∥X ∗Xu∥ const ∥u∥.Thustheboundedinverse (X ∗X) 1 ∈ L (Hn) exists.WegobacktoSylvester’sequation(1.5).Setting X ∗X = X ∈ L (Hn) and X ∗MX = M ∈ L (Hn),weobtaintherelation, L (X ∗X) 1X
Bothoperators X and M areself-adjoint,but X 1M isnot.Thefollowing assertionisthesecondofourmainresults,andleadstoastabilizationresult:
Proposition2.2. Assumethat (2.1) issatisfied.Then, σ(X 1M ) is containedin R1 +.Actually,
Inaddition,thereisnogeneralizedeigenspaceforany λ ∈ σ(X 1M ).
Remark: ByProposition2.2,weobtainadecayestimate exp ( t (L +(X ∗X) 1X ∗ΞW )) = e t(X 1M ) const e µ1t , t 0. (2.10)
Infact,thelastassertionofthepropositionensuresthatnoalgebraicgrowthin timearisesinthesemigroup,regardingthesmallesteigenvalue.Thus,asetof actuators gk = (X ∗X) 1X ∗ξk,1 k N,inotherwords, G = (X ∗X) 1X ∗Ξ explicitlygivesadesiredsetofactuatorsin(1.3).
ProofofProposition2.2. Since X ispositive-definite,wecanfindanonuniquebijection U ∈ L (Hn) suchthat X = X ∗X = U
) thesocalledCholeskyfactorization.Letusdefine
Then, M ′ ∈ L (Hn) isaself-adjointoperator,enjoyingsomepropertiessimilar tothoseof X 1M .Infact,let λ ∈ σ(X 1M ),or (λX M ) u = 0forsome u = 0.Then,since
0 =(λU ∗U M ) u = U ∗ (λ (U ∗) 1MU 1)U u = U ∗ (λ M ′)U u = 0,
weseethat λ belongsto σ(M ′).Theconverserelationisalsocorrect,which meansthat σ(X 1M )= σ(M ′) ⊂ R1 . (2.12)
Inequality(2.9)isachievedbyapplyingthewellknownmin-maxprinciple[11] to M ′,ormoredirectlybythefollowingobservation:Let λ ∈ σ(X 1M ),and (λX M ) u = 0forsome u = 0.Then λ∥Xu∥2 s = λ⟨X u, u⟩n = ⟨M u, u⟩n = ⟨MXu, Xu⟩s µ1∥Xu∥2 s ,
fromwhich(2.9)immediatelyfollows,since Xu = 0.
Nextletusshowthatthereis no generalizedeigenspacefor λ ∈ σ(X 1M ).Let (λ X 1M )2u = 0forsome u = 0.Setting v =(λ X 1M )u,wecalculate
0 = X (λ X 1M )2 u =(λX M )v =(λU ∗U M ) v = U ∗ (λ (U ∗) 1MU 1)U v = U ∗ (λ M ′)w = 0, w = U v,
TheoryofStabilizationforLinearBoundaryControlSystems
or (λ M ′) w = 0.Ontheotherhand,since w = U v = U (λ X 1M )u = U (λ U 1(U ∗) 1M )u =(λ (U ∗) 1MU 1)U u =(λ M ′)U u,
weseethat
0 = (λ M ′)w = (λ M ′)2 U u, U u = 0
But, M ′ isself-adjoint,sothatthereisnogeneralizedeigenspacefor λ ∈ σ(M ′).Thus, U u turnsouttobeaneigenvectorof M ′ for λ,and
0 = U ∗(λ M ′)U u = U ∗(λ (U ∗) 1MU 1)U u =(λU ∗U M )u =(λX M )u.
Thismeansthat u isaneigenvectorof X 1M for λ.
Thefollowingexampleshowsthat λ∗ = min σ(X 1M ) doesnotgenerally coincidewiththeprescribed µ1.
Example: Let n = 3,andset H3 = C3,sothat L isa3 × 3matrix.Let
L = diag (aab) , where a, b 0and a = b.Since n = 3, ν = 2, m1 = 2,and m2 = 1,wechoose N = 2, s = 6, H6 = C6,and ℓ1 = ℓ2 = ℓ3 = 2.Asfortheoperator W ∈ L (C3; C2), letusconsiderthecase,forexample,where w1 =(101)T and w2 =(010)T .
Theoperator W isa2 × 3matrixgivenby (101 010).Thepair (W, L) isthen observable,andthefirstconditionof(2.1)issatisfied.
ToconsiderSylvester’sequation(1.5),let {ηij;1 i 3, j = 1, 2} bea standardbasisfor C6 suchthat η11 =(100 ... 0)T , η12 =(010 ... 0)T , η21 = (001 0)T , ,and η32 =(0 01)T.Set M = diag (µ1 µ1 µ2 µ2 µ3 µ3)
for0 < µ1 < µ2 < µ3.Intheoperator Ξ givenby Ξu = u1ξ1 + u2ξ2 for u =(u1 u2)T ∈ C2 , set ξ1 =(101010)T and ξ2 =(010101)T.Then,weseethatker Qµi Ξ = {0}, 1 i 3,andthesecondconditionof(2.1)issatisfied.Theuniquesolution X ∈ L (C3; C6) toSylvester’sequation(1.5)isa6 × 3matrixdescribedas(u = (u11 u12 u21)T ∈ C3)
Xu =
⟨(µ1 L) 1u, w1⟩3
⟨(µ1 L) 1u, w2⟩3
⟨(µ2 L) 1u, w1⟩3
⟨(µ2 L) 1u, w2⟩3
⟨(µ3 L) 1u, w1⟩3
⟨(µ3 L) 1u, w2⟩3
where ⟨·, ·⟩3 denotestheinnerproductin C3.Setting,forcomputational convenience,
weseethat
Itisapparentthatoneoftheeigenvaluesofthismatrixisthe (
)-element:
andiscertainlygreaterthan µ1.Notethat
Theothereigenvaluesarethoseofthematrix,
Toseethattheseeigenvaluesaregenerallygreaterthan µ1,letusconsidera numericalexample:Let (µ1 µ2 µ3)=(234), a = 0,and b = 1.Then,
oftheeigenvalues a+⟨
> 2 (= µ1).Thematrix(2.14) isthen 1 253 ( 1860 1860 3540 3287),theeigenvaluesofwhicharedenotedas ζ1 and ζ2.Then, µ1 = 2 < ζ1 < 156/61 < ζ2,andthus λ∗ = ζ1 > 2.
Weclosethissectionwiththefollowingremark:Thereisacasewhere λ∗ coincideswith µ1.Following[52],letusconsider(1.3)inthespace Hn = Cn (see(1.8)).Alloperators L, G,and W arethenmatricesofrespectivesize.Let σ(L) consistonlyofsimpleeigenvalues,sothat mi = 1,1 i n,and n = ν. Thuswecanchoose N = 1, ℓi = 1,1 i n,andthus s = n.Theoperatorin (2.10)iswrittenas L +(X ∗X) 1X ∗ΞW ,where Ξu = uξ for u ∈ C1,and W =
⟨ , w⟩n, w = (w1 w2 wn)T ∈ Cn.Theobservabilityconditionthenturnsoutto be wi = 0,1 i n.LetusconsiderSylvester’sequation(1.5)in Hs = Cn.By setting ξ = (11 ... 1)T ∈ Cn,thesolution X to(1.5)isan n × n matrix,andhas aboundedinverse: X = Φ ˜ W , (2 15) where
Φ = ( 1 µi λj ; i ↓ 1,..., n j → 1,..., n ) , and ˜ W = diag (w1 w2 wn)
Thus, L + (X ∗X) 1X ∗ΞW = L + X 1ξwT.Itisshown[52]that,givenaset {µi}1 i n,thereisaunique g ∈ Cn suchthat σ(L gwT) = {µi}1 i n,andthat g isconcretelyexpressedas
TheproofwillbegivenlaterinSection4.
Proposition2.3. SupposeinProposition2.2that σ(L) consistsonlyofsimple eigenvalues.Set ξ =(11 1)T asabove.ThenX 1ξ = g,andthus λ∗ = µ1. Infact,wehave
Proof. Therelation, X 1ξ = g isrewrittenas
Inotherwords,weshowthat
Theleft-handsideof(2.17),apolynomialof λi,1 i n,isinparticulara polynomialof λ1 oforder n 1,andthecoefficientof λn 1 1 is ∆1 = ∏2 i< j n(λi λj).For j < k,letuscomparethe jthandthe kthterms. Thefollowinglemmaiselementary:
Lemma2.4. Let 1 j < k n. Intheproduct ∆k,apolynomialof {λi}i=k, set λj = λk.Then ∆k =( 1)k 1+ j∆j
In theleft-handsideof(2.17),set λj = λk.Sincethetermsotherthanthe jth andthe kthtermscontainthefactor (λj λk),theybecometobe0.The kthterm isthen ( 1)k 1∆k ∏ 1 ℓ n, ℓ=i (λk µℓ)=( 1)k 1( 1)k 1 j∆j
)
the jthterm).
Thus theleft-handsideof(2.17)hasfactors λj λk, j < k,andiswrittenas c∆. But, c∆ isapolynomialof λ1 oforder n 1,andthecoefficientof λn 1 1 is c∆1. Thismeansthat c = 1,andtheproofofrelation(2.17)isnowcomplete.
1.3Observability:ReductiontoSubstructures
The firstconditionof(2.1)istheobservabilityconditiononthepair (W, L).The operator (WWL ... WLn 1)T isrewrittenforconvenienceas (WLk; k ↓ 0,..., n 1)
Similarexpressionsforotheroperatorsandmatriceswillbeemployedhereafter withoutanyconfusion.Following[48],weshowinthissectionthatthe observabilityconditionisreducedtoasetofobservabillityconditionson subsystems.Let Li = L|PλiHn betherestrictionof L ontotheinvariantsubspace Pλi Hn,andlet Wi = W |PλiHn .Toobtainthereduction,itisconvenienttoemploy matrixrepresentations.Theset {φij;1 i ν, 1 j mi} introducedin Section1formsabasisfor Hn.Recallthat T isabijection, Tu = ˆ u = (u11 u12 uνmν )T for u = ∑i, j uijφij ∈ Hn.Accordingtothebasis,theoperator W isrewrittenas Wu = ˆ W ˆ u
Then theoperator Wi ∈ L (Pλi Hn; CN ) isclearly Wiu = ˆ Wi ˆ u = (w k ij; j → 1,..., mi k ↓ 1,..., N )
u, u ∈ Pλi Hn.
Theobservabilityconditionon (W, L) is,intermsofthesesymbols,equivalent to
ker( ˆ WΛk; k ↓ 0,..., n 1) = {0}, or
rank( ˆ WΛk; k ↓ 0,..., n 1) = n, (3 2)
where Λ = TLT 1 (see(1.8)).Theresultinthissectionisstatedas
Proposition3.1. Inorderthatthepair (W, L) isobservable,itisnecessary andsufficientthatthepairs (Wi, Li), 1 i ν,areobservable,inotherwords ker(WiLk i ; k ↓ 0,..., mi 1) = {0}, 1 i ν. (3.3)
Proof.Theproofiselementary.Supposefirstthat (W, L) isobservable.Then itisclearthat(3.3)holds.Conversely,suppose(3.3),orequivalently
rank( ˆ WiΛk i ; k ↓ 0,..., mi 1) = m
Weapplyelementaryrowoperationstothematrix ( ˆ WΛ
,..., n 1).In ordertoshow(3.2),itisenoughtoprovethat
( ˆ W (
Thematrixjustaboveiswrittenas
Thesubmatrix (
ontherightsideof(3.4)has thefullrank (= m
) bytheassumption(3.3).Thus(3.2)willbeproven,if
TheoryofStabilizationforLinearBoundaryControlSystems
Bysetting Λ′ = diag(Λ1 ... Λν 1) and ˆ W ′ = ( ˆ Wi; i → 1,..., ν 1),theabove relationisequivalentto
rank( ˆ W ′(Λ′ λν)k; k ↓ 0,..., n ′ 1)(Λ′ λν)mν = rank( ˆ W ′(Λ′ λν)k; k ↓ 0,..., n ′ 1) = rank( ˆ W ′Λ′k ; k ↓ 0,..., n ′ 1) = n ′ .
Theproblemisthusreducedtotheproblemofprovingtheobservabilityofthe pair ( ˆ W ′ , Λ′).Bycontinueingthereductionprocedure-viatheassumption(3.3) ateachstage,itfinallyleadstotheproblemofproving
rank( ˆ W1Λk 1 ; k ↓ 0,..., m1 1) = m1, or ker(W1Lk 1; k ↓
However,thisisnothingbutourassumption(3.3)when i = 1.
Remark:Letusconsiderthecasewhere Ni = 0,1 i ν.Thisoccurs,for example,whenthe L isaself-adjointoperator.Inthiscase,therelation(3.3) meansthat rank( ˆ WiΛk i ; k ↓ 0,..., mi 1) = rank(λk i ˆ Wi; k ↓ 0,..., mi 1) = rank ˆ Wi = mi, 1 i ν. (3.5)
Thusweneedtochoosethe N greaterthanorequaltomax1 i ν mi inthiscase. Thiscase: Ni = 0,1 i ν isalreadydiscussedin[58],wheretheresultcan beviewedasaspecialcaseofourProposition3.1.Following[58],letusbriefly giveanalternativeproof.Thematrix ( ˆ WΛk; k ↓ 0,..., n 1) isdecomposed intotheproductoftwomatrices Φ and Ψ: ( ˆ WΛk; k ↓ 0,..., n 1) = (λk i ˆ Wi; i → 1,..., ν k ↓ 0,..., n 1 ) = (λk i IN ; i → 1,..., ν k ↓ 0,..., n 1 ) diag( ˆ W1 ˆ Wν) = ΦΨ, where Φ and Ψ denotethe nN × νN andthe νN × n matrices,respectively. Supposefirstthatrank ˆ Wi = mi,1 i ν.Thentherankof Ψ isclearlyequal to m1 + + mν = n.Itiseasilyseenthattherankof Φ isequalto νN ( n). Thus,weseethat n = rank Φ + rank Ψ νN rank ΦΨ min (rank Φ, rank Ψ)= n.
Conversely,supposethatrank ΦΨ = n.Then,weseethat n = rank Φ Ψ rank Ψ,sothat n columnvectorsofthe νN × n matrix Ψ arelinearly independent.But,thismeansthatrank ˆ Wi = mi,1 i ν.
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