Computing Polynomial Solutions and Annihilators of Integro-Differential Operators with Polynomial

25 Pages·2017·0.8 MB·English

Computing Polynomial Solutions and Annihilators of
Integro-Differential Operators with Polynomial
Coefficients
Alban Quadrat, Georg Regensburger
To cite this version:
Alban Quadrat, Georg Regensburger. Computing Polynomial Solutions and Annihilators of Integro-
Differential Operators with Polynomial Coefficients. [Research Report] RR-9002, Inria Lille - Nord
Europe; Institute for Algebra, Johannes Kepler University Linz. 2016, pp.24. ￿hal-01413907￿
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Computing Polynomial
Solutions and
Annihilators of
Integro-Differential
Operators with
Polynomial Coefficients
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Computing Polynomial Solutions and Annihilators of
Integro-Differential Operators with Polynomial
Coefficients
A.Quadrat*,G.Regensburger†
Project-TeamsNon-A
ResearchReport n°9002—December2016—21pages
Abstract: Inthispaper,westudyalgorithmicaspectsofthealgebraoflinearordinaryintegro-differential
operatorswithpolynomialcoefficients. EventhoughthisalgebraisnotNoetherianandhaszerodivisors,
Bavula recently proved that it is coherent, which allows one to develop an algebraic systems theory over
thisalgebra. Foranalgorithmicapproachtolinearsystemsofintegro-differentialequationswithboundary
conditions, computing the kernel of matrices with entries in this algebra is a fundamental task. As a first
step,wehavetofindannihilatorsofintegro-differentialoperators,which,inturn,isrelatedtothecomputa-
tionofpolynomialsolutionsofsuchoperators. Foraclassoflinearoperatorsincludingintegro-differential
operators,wepresentanalgorithmicapproachforcomputingpolynomialsolutionsandtheindex. Agen-
eratingsetforrightannihilatorscanbeconstructedintermsofsuchpolynomialsolutions. Forinitialvalue
problems,aninvolutionofthealgebraofintegro-differentialoperatorsthenallowsustocomputeleftannihi-
lators,whichcanbeinterpretedascompatibilityconditionsofintegro-differentialequationswithboundary
conditions. WeillustrateourapproachusinganimplementationinthecomputeralgebrasystemMaple.
Key-words: Linearsystemstheory,ringsofintegro-differentialoperators,polynomialsolutions,indicial
equation,compatibilityconditions,computeralgebra
G.RegensburgerwassupportedbytheAustrianScienceFund(FWF):P27229.
*InriaLille-NordEurope,Non-Aproject,ParcScientifiquedelaHauteBorne,40AvenueHalley,Bat.A-ParkPlaza,59650Villeneuve
d’Ascq,France.Email:alban.quadrat@ inria.fr.
†InstituteforAlgebra,JohannesKeplerUniversityLinz,Austria.Email:[e-mail]
RESEARCHCENTRE
LILLE–NORDEUROPE
ParcscientifiquedelaHaute-Borne
40avenueHalley-BâtA-ParkPlaza
59650Villeneuved’Ascq
Calcul des solutions polynomiales et des annulateurs d’opérateurs
intégro-différentiels à coefficients polynomiaux
Résumé : Dans ce papier, nous étudions certains aspects algorithmiques de l’algèbre des opérateurs intégro-
différentiels ordinaires linéaires à coefficients polynomiaux. Même si cette algèbre n’est pas noetherienne et
admetdesdiviseursdezéro,Bavulaarécemmentmontréqu’elleétaitcohérente,cequipermetledéveloppement
d’unethéoriealgébriquedessystèmeslinéairessurcettealgèbre. Pouruneapprochealgorithmiquedessystèmes
linéaires d’équations intégro-différentielles ordinaires avec conditions aux bords, le calcul du noyau de matrices
à coefficients dans cette algèbre est un problème fondamental. Pour cela, dans un premier temps, nous sommes
amenés à calculer les annulateurs d’opérateurs intégro-différentiels, problème qui, à son tour, est relié au prob-
lèmeducalculdessolutionspolynomialesdetelsopérateurs. Pouruneclassed’opérateurslinéairesincluantles
opérateursintégro-différentiels,nousprésentonsuneapprochealgorithmiquepourlecalculdessolutionspolyno-
mialesetdel’indice. Unensemblegénérateurdesannulateursàdroited’unopérateurintégro-différentielestalors
construitgrâceaucalculdesolutionspolynomiales. Pourlesproblèmesavecconditionsinitiales, uneinvolution
del’algèbredesopérateursintégro-différentielsnouspermetensuitedecalculerlesannulateursàgauche,quipeu-
ventêtreinterprétéscommedesconditionsdecompatibilitéd’équationsintégro-différentiellesavecconditionsaux
bords. Nousillustronsnotreapprocheàl’aided’uneimplémentationdanslesystèmedecalculformelMaple.
Mots-clés : Théorie des systèmes linéaires, anneaux d’opérateurs intégro-différentiels, solutions polynomiales,
équationindicielle,conditionsdecompatibilité,calculformel
PolynomialSolutionsandAnnihilatorsofIntegro-DifferentialOperators 3
1 Introduction
Ringsoffunctionaloperators(e.g., ringsofordinarydifferential(OD)operators, partialdifferential(PD)opera-
tors,differentialtime-delayoperators,differentialdifferenceoperators)wererecentlyintroducedinmathematical
systems theory. Since many control linear systems can be defined by means of a matrix with entries in a skew
polynomial ring, in an Ore algebra or in an Ore extension of functional operators (i.e., classes of univariate or
multivariatenoncommutativepolynomialrings)[16,39],theclassicalpolynomialapproachtolinearsystemsthe-
orycanbegeneralizedyieldingamodule-theoreticapproachtolinearfunctionalsystems[19,33,34,41,45,47].
Symboliccomputationtechniques(e.g.,Gröbnerbasistechniques)andcomputeralgebrasystemscanthenbeused
todevelopdedicatedpackagesforalgebraicsystemstheory[17,29].
Algebrasofordinaryintegro-differential(ID)operatorshaverecentlybeenstudiedwithinanalgebraicapproach
in[8,9,10,11]andwithinanalgorithmicapproachin[40,42,43,22]. Thegoalofthelatterworksistoprovide
analgebraicandalgorithmicframeworkforstudyingboundaryvalueproblemsandGreen’soperators.
TheringofIDandtime-delay/dilatationoperatorswasintroducedin[37]todevelopapurelyalgorithmicap-
proach to standard Artstein’s transformation of linear differential systems with delayed inputs. This work also
advocatesfortheeffectivestudyoftheringofIDtime-delay/dilatationoperators. Thenormalformsofelements
ofthisnoncommutativealgebrawillbestudiedinafuturepublicationbasedontheneweffectivetechniquesintro-
ducedin[22,23]. Inthispaper,wefocusonitssubringofIDoperators. Wealsonotethateffectivecomputations
overIDalgebrasplayanimportantroleinparameterestimationproblemsasshownin[15].
EventhoughlinearsystemsofIDequationsplayanimportantroleindifferentdomainsandapplications(e.g.,
PIDcontrollers),itdoesnotseemthattheyhavebeenextensivelystudiedbythemathematicalsystemscommunity.
Forboundaryvaluesystems, wereferto[20,21]andthereferencestherein. Thefirstpurposeofthispaperisto
introduceconcepts,techniques,andresultsdevelopedintheaboverecentsworks. Inparticular,weemphasizethat
thealgebraicstructureoftheringofIDoperatorswithpolynomialcoefficientsismuchmoreinvolved(e.g.,zero
divisors,non-Noetherianity)thantheoneoftheringofODoperatorswithpolynomialcoefficients(theso-called
Weylalgebra). Thefundamentalissueofcomputingleft/rightkernelofamatrixofIDoperatorshastobesolved
towardsdevelopingasystem-theoreticapproachtolinearIDsystems. Formoredetails,see[16,36].
ThesecondgoalofthispaperistostudythisproblemforasingleIDoperator,thatis,computingitsannihilator.
Withinarepresentationapproach,weshowthatthisproblemisrelatedtothecomputationofpolynomialsolutions
of ID operators, a problem that is also studied in detail. To solve this problem, we introduce the concept of a
rational indicial equation for a linear operator acting on the polynomial ring. This approach allows us to find
againandgeneralizestandardresultsontheindicialequationclassicallyusedinthetheoryoflinearODequations
[2,3,5].
Thischapterisbasedontheconferencepaper[38].Itincludesaself-containedintroductiontoordinaryintegro-
differentialoperatorswithpolynomialcoefficientswithseveralevaluationsincludingnormalforms(Sections2–4).
Allothersectionshavebeenrevisedandextended.
2 The ring of Ordinary Integro-Differential Operators with Polynomial
Coefficients
BeforediscussingtheringofIDoperatorswithpolynomialcoefficients,asanintroducingexample,wefirstrecall
twostandardconstructionsoftheringAofODoperatorswithpolynomialcoefficients(alsocalledtheWeylalgebra
anddenotedbyA (k),wherek isafield). Thefirstconstructionisasthesubalgebrak(cid:104)t,∂(cid:105)ofalllinearmapson
1
thepolynomialringk[t]andthesecondisbymeansofgeneratorsandrelations.
Inwhatfollows,letkdenoteafixedfield,whichcontainsQ. Letend (k[t])denotethek-algebraformedbyall
k
k-linearmapsfromthepolynomialringk[t]toitself. Weconsiderthek-subalgebrak(cid:104)t,∂(cid:105)ofend (k[t])generated
k
bythefollowingtwok-linearmaps
t: tn(cid:55)−→tn+1 and ∂: tn(cid:55)−→ntn−1
definedonthebasis(tn)n∈N ofk[t]. Theyrespectivelycorrespondtothemultiplicationoperatorandthederivation
onthepolynomialringk[t],namely:
t: k[t] −→ k[t] ∂: k[t] −→ k[t]
and (1)
p (cid:55)−→ tp, p (cid:55)−→ dp.
dt
RRn°9002
4 Quadrat&Regensburger
Oneimmediatelyverifiesthatwehave
d(tp) dp
∀p∈k[t], (∂◦t)(p)= =t +p=(t◦∂+id)(p),
dt dt
whereid(alsodenotedby1)istheidentitymaponk[t]. ItshowsthattheLeibnizrule
∂◦t=t◦∂+id
holdsintheoperatoralgebrak(cid:104)t,∂(cid:105).
Using the Leibniz rule, we can define the Weyl algebra also by generators and relations: Let k(cid:104)T,D(cid:105) be the
free associative k-algebra on the set {T,D}, that is, the k-vector space with the basis formed by all words over
{T,D}andthemultiplicationofbasiselementsdefinedbyconcatenation. Letnow
J=(DT−TD−1)⊆k(cid:104)T,D(cid:105)
denotethetwo-sidedidealgeneratedbyDT−TD−1anddefinethek-algebra:
A=k(cid:104)T,D(cid:105)/J.
Bydefinition,theLeibnizrule
DT ≡TD+1 modJ
holdsinA. Usingthisidentity,eachelementofd∈Acanuniquelybewrittenasafinitesum
d≡∑a TiDj modJ
ij
withcoefficientsa ∈k.
ij
Toseethatthetwoconstructionsaboveareequivalent,onecanusethefactthatAisasimplering,thatis,its
onlypropertwo-sidedidealisthezeroideal(seeforexample,[18]). Henceeveryringhomomorphismisinjective
andsothek-algebrahomomorphismA−→k(cid:104)t,∂(cid:105)mapping
T+J(cid:55)−→t and D+J(cid:55)−→∂
isanisomorphism. Inotherwords,eachd∈Acanbeidentifiedwiththefollowingcorrespondingk-linearmap
L : k[t] −→ k[t],
d
p (cid:55)−→ d(p),
whered(p)denotestheactionofd on p.
Inthefollowing,weuseasimilarapproachtointroduceandstudythealgebraofIDoperatorswithpolynomial
coefficients. IDoperatorswithpolynomialcoefficientswerestudiedin[8,10]asageneralizedWeylalgebra[7,6].
See [40]for theconstruction of ordinaryID operatorswith polynomialcoefficients as afactor algebraof a skew
polynomialring(see,e.g.,[16,31]andthereferencestherein). FortheconstructionofthealgebraofIDoperators
F (cid:104)∂, (cid:105) defined over an ordinary ID algebra F and endowed with a set of characters (that is, multiplicative
Φ (cid:114)
linearfunctionals)Φ,wereferto[42,43]. ThisconstructionisbasedonaparametrizednoncommutativeGröbner
basis; seeSection3forthecaseofpolynomialcoefficients. Forabasis-freeconstructionusingafinitereduction
systemintensoralgebras,wereferto[22]. Incontrastto[8,10],thelasttwoapproachesallowsonetohavemore
thanonepointevaluationasdescribedinSection4,whichiscrucialforthestudyofboundaryproblems.
Definition1. Thek-algebraofordinaryIDoperatorswithpolynomialcoefficientsisdefinedasthek-subalgebra
k(cid:104)t,∂, (cid:105)⊆end (k[t]),
(cid:114) k
withtheoperatorst and∂ definedasin(1)and
: k[t] −→ k[t]
(cid:114)
tn (cid:55)−→ tn+1/(n+1),
definedonthebasis(tn)n∈Nofk[t].
Inria
PolynomialSolutionsandAnnihilatorsofIntegro-DifferentialOperators 5
Theintegraloperator correspondstotheusualintegralstartingat0:
(cid:114)
: k[t] −→ k[t]
(cid:114)
p (cid:55)−→ (cid:82)tp(s)ds.
0
Onecanverifydirectlythatthefundamentaltheoremofcalculus
∂◦ =id
(cid:114)
holds. Moreover,weseethat
E=id− ◦∂
(cid:114)
correspondstotheevaluationat0:
E: k[t] −→ k[t]
p (cid:55)−→ p(0).
Hence,assoonaswehaveanintegral,wealsohaveoneevaluationmaptotheconstantsk“forfree”,whichallows
us to define and study initial value problems in terms of integro-differential operators. Note that the operator E
naturallyinducestheexistenceofzerodivisors. Forinstance,wehave:
E◦t=0.
Basedonthebasicidentitiesabove,wecanconstructthealgebraofintegro-differentialoperatorswithpolyno-
mialcoefficientsalsobygeneratorsandrelations.
Definition2. Wedefinethek-algebra
I=k(cid:104)T,D,I,E(cid:105)/J,
whereJisthetwo-sidedidealofrelationsgeneratedbythefollowingelements:
DT−TD−1, DI−1, ID+E−1, ET. (2)
WenotebyT =T+J(resp.,D=D+J,I=I+J,E=E+J)theresidueclassofT (resp.,D,I,E)inI.
3 Normal Forms
Since we have now four defining identities for I (see (2)) instead of one as for the Weyl algebra A, it is more
involvedtoobtainthenormalformofanelementofI,i.e.,itsuniqueexpressionasanoncommutativepolynomial
in the operators T,D,I and E modulo the relations (2). In this section, we informally discuss the construction
ofanoncommutativeGröbnerbasisforthedefiningidealfollowingBuchberger’salgorithm. Forbackgroundon
noncommutativeGröbnerbases,wereferto[12,32,46,13]. Inthenoncommutativecase,notethatBuchberger’s
algorithmdoesnotterminateingeneralandthepropertyofhavingafiniteGröbnerbasisisundecidable. However,
inourcasewecan“guess”aparametrizedGröbnerbasisfromthecorrespondingS-polynomialcomputations.
See[42,43]forfurtherdetailsonaparametrizedGröbnerbasisforthedefiningrelationsforintegro-differential
operatorsoveranordinaryIDalgebraandthecorrespondingnormalforms. Ananalogousfinitetensorreduction
systemandtherelatedS-polynomialcomputationsusingthepackageTenRescanbefoundin[22,23].
WedenotetheS-polynomialbetweentwopolynomialsoftheform
UV−P and VW−Q,
with“leadingterms”UV andVW by:
S(UV,VW)=PW−UQ.
Inthefollowing,weconsideragradedpartialorderwithD>T andI>T. WefirstcomputetheS-polynomial
betweenthepolynomials
DI−1 and ID+E−1
andobtain:
S(DI,ID)=1D−D(1−E)=DE.
RRn°9002
6 Quadrat&Regensburger
Soweneedtoaddthepolynomial
DE
to the generators of our ideal, which corresponds to the evaluation mapping to k. The S-polynomial between
ID+E−1andthenewpolynomialgives:
S(ID,DE)=(1−E)E.
Soweobtain
E2−E,
whichcorrespondstotheevaluationactingasaprojectorontok. Since
S(ID,DI)=(1−E)I−I1=−EI,
wealsohavetoaddthepolynomial
EI
toourgenerators,whichcorrespondstotheintegral(cid:82)t evaluatedat0being0.
0
TheS-polynomialbetween
ID−1+E and DT−TD−1
isgivenby:
S(ID,DT)=(1−E)T−I(TD+1)=T−ET−ITD−I.
UsingthepolynomialET fromtheoriginalgenerators,weseethatweneedtoaddthepolynomial:
ITD−T+I.
ThisgivesrisetonewS-polynomialswithDT−TD−1andoneseesinductivelythatweneedtoaddthefamily
∀n≥1, ITnD−Tn+nITn−1
toourgenerators,correspondingtointegrationbyparts. ComputingtheS-polynomialswiththisfamilyandDE,
wethenobtain
IE−TE,
and
∀n≥1, ITnE−Tn+1/(n+1)E
whichcorrespondstothek-linearityoftheintegral.
Finally,theS-polynomialbetween
ITD−T+I and DI−1
isgivenby:
S(ITD,DI)=(T−I)I−IT.
Soweobtainthepolynomial
I2−TI+IT,
allowingtoreduceaniteratedintegraltoasumoftwosingleintegrals. Again,thisidentitygivesrisetoaninfinite
family
∀n≥1, ITnI−(Tn+1I+ITn+1)/(n+1)
ofnewgenerators.
Collectingalltheidentitiesabove,onecanverifythatallparametrizedS-polynomialsnowreducetozeroand
wehaveindeedaGröbnerbasisforthedefiningidentities(comparewith[42, Proposition13]and[22, Theorem
5.1]).
Inria
PolynomialSolutionsandAnnihilatorsofIntegro-DifferentialOperators 7
Theorem1. Thegenerators
DT−TD−1, DI−1, ID+E−1, ET,
DE, E2−E, EI, IE−TE, I2−TI+IT,
andtheparametrizedgenerators
 ITnD−Tn+nITn−1,

∀n≥1, ITnE−Tn+1/(n+1)E,
 ITnI−(Tn+1I+ITn+1)/(n+1),
formanoncommutativeGröbnerbasisfortheidealJofI(seeDefinition2)withrespecttoagradedpartialorder
withD>T andI>T.
BythenormalformcorrespondingtotheGröbnerbasisfromTheorem1,usingthenotationsofDefinition2,
eachd∈Icanuniquelybewrittenasasum
d=d +d +d ,
1 2 3
where
d =∑a TiDj, d =∑b TiITj, d =∑f TiEDj (3)
1 ij 2 ij 3 ij
arerespectivelyanODoperator,anintegraloperator,andaboundaryoperator,witha ,b ,and f ∈k,andd ,
ij ij ij 1
d ,andd containonlyfinitelynonzerosummands.
2 3
To see that the definition of integro-differential operators via generators and relations and Definition 1 are
equivalent, we can use the fact that I is “almost” a simple ring. The only nonzero proper two-sided ideal is the
ideal (E) generated by the “evaluation” E. This was first proved by Bavula in [8]. Here we give an alternative
proofbasedonthenormalformsanddirectsumdecompositionabove,whichalsogeneralizestothemoregeneral
settingincludingseveralevaluationsmentionedinthenextsection.
Proposition1. Theonlynonzeropropertwo-sidedidealofIis(E).
Proof. Letd∈I\(E)withd≡d +d +d asin(3)andd +d (cid:54)=0byassumption. Usingtheidentities
1 2 3 1 2
DT =TD+1, DI=1, DE=0,
wecanfindak∈Nsuchthat
k
D d∈A\{0}
isanonzerodifferentialoperatorandthestatementfollowssinceAisasimplering.
Corollary1. Thek-algebrahomomorphismχ: I−→k(cid:104)t,∂, (cid:105)mapping
(cid:114)
T (cid:55)−→t, D(cid:55)−→∂, E(cid:55)−→E, I(cid:55)−→
(cid:114)
isanisomorphism.
Inotherwords,wecanidentifyeachd∈Iwiththecorrespondingk-linearmap
L : k[t] −→ k[t],
d
(4)
p (cid:55)−→ d(p),
whered(p)denotestheactionofd on p.
Finally,using(3),uptoisomorphism,wehavethefollowingdirectsumdecomposition
I=A⊕k[t] k[t]⊕(E)
(cid:114)
withthetwo-sidedideal(E)ofboundaryoperatorsgeneratedbyE.
RRn°9002