Bahaa∗ Department of Mathematics, Faculty of Science, Taibah University, Al-madinah Al-munawwarah, P.O.. 30002, Saudi Arabia Abstract Keywords:Boundary control;n×nparabolic systems; Mult
Trang 1Journal of Taibah University for Science 7 (2013) 146–161
G.M Bahaa∗
Department of Mathematics, Faculty of Science, Taibah University, Al-madinah Al-munawwarah, P.O 30002, Saudi Arabia
Abstract
Keywords:Boundary control;n×nparabolic systems; Multiple time-varying lags; Distributed control problems; Neumann conditions; Existence
1 Introduction
Distributedparameter systems with delays canbe used todescribe many phenomenain the real world Asis wellknown,heatconduction,propertiesofelastic-plasticmaterial,fluiddynamics,diffusion-reactionprocesses,the transmissionofthesignalsatacertaindistancebyusingelectriclonglines,etc.,allliewithinthisarea.Theobjectthat
wearestudying(temperature,displacement,concentration,velocity,etc.)isusuallyreferredtoasthestate
Duringthelasttwentyyears,equationswithdeviatingargumenthavebeenappliednotonlyinappliedmathematics, physicsandautomaticcontrol,butalsoinsomeproblemsofeconomyandbiology.Currently,thetheoryofequations withdeviatingargumentsconstitutesaveryimportantsubfieldofmathematicalcontroltheory
Consequently,equationswithdeviatingargumentsarewidelyappliedinoptimalcontrolproblemsofdistributed parametersystemwithtimedelays[41]
Theoptimalcontrolproblemsofdistributedparabolicsystemswithtime-delayedboundaryconditionshavebeen widelydiscussedinmanypapersandmonographs.Afundamentalstudyofsuchproblemsisgivenby[48]andwasnext developedby[29,49].Itwasalsointensivelyinvestigatedby[7–9,11–13,25,26,30,34–45],inwhichlinearquadratic
E-mail address:Bahaa GM@hotmail.com
http://dx.doi.org/10.1016/j.jtusci.2013.05.004
Trang 2G.M Bahaa / Journal of Taibah University for Science 7 (2013) 146–161 147
problemforparabolicsystemswithtimedelaysgiveninthedifferentform(constant,timedelays,time-varyingdelays, timedelaysgivenintheintegralform,etc.)werepresented
The necessaryandsufficient conditionsof optimalityfor systems consistsof onlyoneequation andfor (n×n)
systemsgovernedbydifferenttypesofpartialdifferentialequationsdefinedonspacesoffunctionsofinfinitelymany variablesandalsoforinfinite ordersystems arediscussed forexample in[25–28,42,45]inwhichthe argumentof
[46,47]wereused
Making use of theDubovitskii–Milyutin Theorem in[1–6,10,33,34] the necessaryandsufficient conditions of optimalityfor similarsystemsgovernedbysecondorderoperatorwithaninfinitenumberof variablesandalsofor infiniteordersystemswereinvestigated.Theinterestinthestudyofthisclassofoperatorsisstimulatedbyproblems
inquantumfieldtheory
Inparticular,thepapersof[43,44]presentsnecessaryandsufficientoptimalityconditionsfortheNeumannproblem withquadraticperformancefunctionals,appliedtoasingleoneequationofsecondorderparabolicsystemwithboundary conditionsinvolvingconstanttime-varyinglagandmultipletime-varyinglagsrespectively
Alsoin[42,45]presentstime-optimalboundarycontrolforasingleoneequationdistributedinfiniteorderparabolic andhyperbolicsystemsinwhichconstanttimelagsappearintheintegralformbothinthestateequationandinthe Neumannboundarycondition.Somespecificpropertiesoftheoptimalcontrolarediscussed
In thispaperwe considertheprobleminamoregeneralformulation.Adistributedparameterfor infiniteorder parabolic(n×n)systemswithmultipletime-varyinglagsinboundaryconditionsisconsidered.Suchaninfiniteorder parabolicsystemcanbetreatedasageneralizationofthemathematicalmodelforaplasmacontrolprocess.Aninitial statecontainedinaspecifiedsetisassumed(theinitialvaluesarenotknown).Thequadraticperformancefunctionals definedoverafixedtimehorizonaretakenandsomeconstraintsareimposedontheinitialstate andtheboundary control.Suchasystemmaybeviewedasalinearrepresentationofmanydiffusionprocesses,inwhichtime-delayed signalsareintroducedataspatialboundary,andthereisafreedominchoosingthecontrolledprocessinitialstate FollowingalineoftheLionsscheme,necessaryandsufficientoptimalityconditionsfortheNeumannproblemapplied
totheabovesystemwerederived.Theoptimalcontrolischaracterizedbytheadjointequations
Thispaperisorganizedas follows.In Section1,weintroducespacesoffunctionsof infiniteorder.InSection3,
we formulatethemixedNeumannproblemfor infiniteorderparabolicoperatorandmultipletime-varyinglags.In Section4,theboundaryoptimalcontrolproblemforthiscaseisformulated,thenwegivethenecessaryandsufficient conditionsforthecontroltobeanoptimal.InSection5,wegeneralizedthediscussiontotwocases,thefirstcase:The optimalcontrolfor(2×2)coupledinfiniteorderparabolicsystemsisstudied.Thesecondcase:Theoptimalcontrol for(n×n)coupledinfiniteorderparabolicsystemshavebeenformulated
2 Sobolev spaces with infinite order
Theobjectofthissectionistogivethedefinitionofsomefunctionspacesofinfiniteorder,andthechainsofthe constructedspaceswhichwillbeusedlater
LetΩbeaboundedopensetofRnwithasmoothboundary Γ,whichisaC∞-manifoldofdimension(n−1) Locally,ΩistotallyononesideofΓ.WedefinetheinfiniteorderSobolevspaceW∞{a α,2}(Ω)ofinfiniteorderof periodicfunctionsφ (x)definedonΩ[22–24]asfollows:
W∞{a α ,2}(Ω)=
⎧
⎨
⎩φ (x)∈C∞(Ω):
∞
|α|=0
a α ||D α φ||2
⎫
⎬
⎭,
whereC∞(Ω)isthespaceofinfinitedifferentiablefunctions,a
α≥0isanumericalsequenceand||·||2isthecanonical norminthespaceL2(Ω),and
D α= ∂ |α|
(∂x1)α1 (∂x n)αn ,
α=(α1, .,α n)beingamulti-indexfordifferentiation,|α|= n
i=1α i
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ThespaceW−∞{a α,2}(Ω)isdefinedastheformalconjugatespacetothespaceW∞{a α,2}(Ω),namely:
W−∞{a α ,2}(Ω)=
⎧
⎨
⎩ψ (x):ψ (x)=
∞
|α|=0
(−1)|α| a α D α ψ α (x)
⎫
⎬
⎭, whereψ α∈L2(Ω)and ∞
ThedualitypairingofthespacesW∞{a α,2}(Ω)andW−∞{a α,2}(Ω)ispostulatedbytheformula
(φ, ψ)=
∞
|α|=0
a α Ω
ψ α (x)D α φ (x)dx,
where
φ∈W∞{a α ,2}(Ω), ψ∈W−∞{a α ,2}(Ω)
Fromabove,W∞{a α,2}(Ω)iseverywheredenseinL2(Ω)withtopologicalinclusionsandW−∞{a α,2}(Ω)denotes thetopologicaldualspacewithrespecttoL2(Ω),sowehavethefollowingchainofinclusions:
W∞{a α ,2}(Ω)⊆L2(Ω)⊆W−∞{a α ,2}(Ω)
Wenow introduce L2(0, T;L2(Ω)) whichwe shall denoted byL2(Q), whereQ=Ω×]0,T[, denotesthe space of measurablefunctionst→φ (t)suchthat
||φ|| L2(Q)= T
0
||φ(t)||2
1/2
< ∞,
endowedwiththescalarproduct(f, g)= T
0 (f (t), g (t)) L2(Ω) dt,L2(Q)isaHilbertspace
InthesamemannerwedefinethespacesL2(0,T;W∞{a α,2}(Ω)),andL2(0,T;W−∞{a α,2}(Ω)),asitsformal conjugate
Also,wehavethefollowingchainofinclusions:
L2(0, T;W∞{a α ,2}(Ω))⊆L2(Q)⊆L2(0, T;W−∞{a α ,2}(Ω)),
TheconstructionoftheCartesianproductofn-timestotheaboveHilbertspacescanbeconstruct,forexample
(W∞{a α ,2}(Ω)) n=W ∞{a α ,2}(Ω)×W∞{a α ,2}(Ω)×···×W∞{a α ,2}(Ω)
=
n
i=1
(W∞{a α ,2}(Ω)) i
,
withnormdefinedby:
||φ|| (W∞{a α ,2}(Ω)) n =
n
i=1
||φ i||W∞{a α ,2}(Ω) ,
whereφ=(φ1, φ2, , φ n)=(φ i)n i=1isavectorfunctionandφ i∈W∞{a α,2}(Ω).
Finally,wehavethefollowingchainofinclusions:
(L2(0, T;W∞{a α ,2}(Ω)))n⊆(L2(Q)) n⊆(L2(0, T;W−∞{a α ,2}(Ω)))n ,
where(L2(0, T;W−∞{a α ,2}(Ω)))n
arethedualspacesof(L2(0, T;W∞{a α ,2}(Ω)))n
.Thespacesconsideredinthis paperareassumedtobereal
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3 Mixed Neumann problem for infinite order parabolic system with multiple time-varying lags
TheobjectofthissectionistoformulatethefollowingmixedinitialboundaryvalueNeumannproblemforinfinite orderparabolicsystemwithmultipletime-varyinglagswhichdefinesthestateofthesystemmodel[3,5,6,27,35–45]
∂y
∂t +A(t)y(x, t)=u, x∈Ω, t∈(0, T ), (1)
∂y
∂ν A =
m
s=1
whereΩhasthesamepropertiesasinSection1.Wehave
y≡y (x, t;u ), y(0)≡y (x,0;u ), y (T)≡y (x, T;u ), u≡u (x, t ), v≡v (x, t ),
Q=Ω×(0, T ), Q=Ω×[0, T ], Q0=Ω×[−Δ(0),0), Σ=Γ ×(0, T ), Σ0=Γ ×[−Δ(0),0),
• Tisaspecifiedpositivenumberrepresentingafinitetimehorizon,
• k s (t), s=1,2, .,standforfunctionsrepresentingmultipletime-varyinglags,
• c s (t), s=1,2, .arerealC∞functiondefinedonΣ,
• Kisaclosed,convexsubsetinthespaceW1/2(Ω, R∞),
• Δ(0)=max{k1(0),k2(0), .,k m(0)},
• yisafunctiondefinedonQsuchthatΩ×(0,T) (x, t)→y(x, t)∈R,
• u, varefunctionsdefinedonQandΣsuchthatΩ×(0,T) (x, t)→u(x, t)∈RandΓ ×(0, T) (x, t)→v (x, t)∈R,
• Φ0isaninitialfunctiondefinedonΣ0suchthatΓ×[−Δ(0),0) (x, t)→Φ0(x, t)∈R.
Theparabolicoperator∂/∂t+A(t)inthestateequation(1)isaninfiniteorderparabolicoperatorandA(t)[24]and (GaliandEl-Saify,1982,1983)and[34]isgivenby:
Ay= ∞
|α|=0
(−1)|α| a
α D2|α| y (x, t ),
and
A= ∞
|α|=0
(−1)|α| a
α D2|α|
isaninfiniteorderself-adjointellipticpartialdifferentialoperatorthatmapsW∞{a α,2}(Ω)ontoW−∞{a α,2}(Ω).
Forthisoperatorwedefinethebilinearformasfollows:
Definition 3.1. Foreacht∈(0,T),wedefineafamilyofbilinearformsonW∞{a α,2}(Ω)by:
π (t; y, φ)=(A(t)y, φ)L2(Ω) , y, φ∈W∞{a α ,2}(Ω),
whereA(t)mapsW∞{a α,2}(Ω)ontoW−∞{a α,2}(Ω)andtakestheaboveform.Then
π (t; y, φ)=(A(t)y, φ)L2(Ω)=
⎛
⎝∞
|α|=0
(−1)|α| a
α D2|α| y (x, t ), φ (x)
⎞
⎠
2
=
Ω
∞
|α|=0
a α D |α| y (x)D |α| φ (x)dx.
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Lemma 3.1. The bilinear form π (t;y, φ)is coercive on W∞{a α,2}(Ω) that is
π (t; y, y)≥λ ||y||2
Proof. ItiswellknownthattheellipticityofA(t)issufficientforthecoercivenessofπ (t;y, φ)onW∞{a α,2}(Ω).
π (t; φ, ψ)=
Ω
∞
|α|=0
a α D |α| φD |α| ψdx.
Then
π (t; y, y)=
Ω
∞
|α|=0
a α D |α| yD |α| ydx≥ ∞
|α|=0
a α ||D2|α| y (x)||2L2(Ω)≥λ ||y||2
W∞{a α,2}(Ω) , λ > 0.
Alsowehave:
∀y, φ∈W∞{a α ,2}(Ω)thefunctiont→π (t; y, φ)
iscontinuouslydifferentiablein(0, T)andπ (t; y, φ)=π (t; φ, y ). (6)
Eqs.(1)–(4)constituteaNeumannproblem.Thentheleft-handsideoftheboundarycondition(2)maybewritten
inthefollowingform:
∂y (u)
∂ν A = ∞
|ω|=0
where∂/∂ν AisanormalderivativeatΓ,directedtowardstheexteriorofΩ,andcos(n, x k)isthekthdirectioncosine
ofn,withnbeingthenormalatΓ exteriortoΩ
Then(2)canbewrittenas:
q (x, t)=
m
s=1
c s (x, t )y(x, t−k s (t))+v (x, t ), x∈Γ, t∈(0, T ). (8)
Lett→t−k s (t)beastrictlyincreasingfunctionon[0,T], k s (t)beingnon-negativein[0,T]andalsobeingaC1
function.Then,thereexisttheinversefunctionsoft→t−k s (t).
Letusdenoter s (t) =tˆ −k s (t),thentheinversefunctionsofr s (t)havetheformt=f s (r s)=r s+q s (r s),whereq s (r s)are time-varyingpredictions.Letf s (t)betheinversefunctionsoft→t−k s (t).Thus,wedefinethefollowingiterations:
ˆt0=0
ˆt1=min{f1(0), f2(0), , f m(0)}
ˆt2=min{f1(ˆt1), f2(ˆt1), , f m (ˆt1)}
ˆt j=min{f1(ˆt j−1), f2(ˆt j−1), , f m (ˆt j−1)}.
Remark 3.1. Weshallapplytheindicationq(x, t)appearingin(8)toprovetheexistenceofauniquesolutionfor
(1)–(4)
Weshallformulatesufficientconditionsfortheexistenceofauniquesolutionofthemixedinitial-boundaryvalue problem(1)–(4)forthecasewheretheboundarycontrolv∈L2(Σ).
Forthispurpose,weintroducetheSobolevspaceW∞,1(Q)([47],vol.2,p.6)definedby:
whichisaHilbertspacenormedby
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||y|| W ∞,1 (Q) = T
0
||y||2
W∞{a α,2}(Ω) dt+||y||2
W1(0,T ;L2(Ω))
1/2
=
⎡
⎣
Q
⎛
⎝∞
|α|=0
a α |D α y|2+
∂y
∂t
2
⎞
⎠ dxdt
⎤
⎦
1/2
=
⎡
⎣
Q
⎛
⎝a0|y|2+
∞
|α|=1
a α |D α y|2+
∂y ∂t 2
⎞
⎠ dxdt
⎤
⎦
1/2
wherethespaceW1(0,T;L2(Ω))denotestheSobolevspaceoforder1offunctionsdefinedon(0,T)andtakingvalues
inL2(Ω).([47],vol.1)
Theexistenceofauniquesolutionforthemixedinitial-boundaryvalueproblem(1)–(4)onthecylinderQcanbe provedusingaconstructivemethod,i.e.,solvingatfirstEqs.(1)–(4)onthesub-cylinderQ1andinturnonQ2etc., untiltheprocedurecoversthewholecylinderQ.Inthisway,thesolutioninthepreviousstepdeterminesthenextone Forsimplicity,weintroducethefollowingnotation:
E j =(ˆtˆ j−1, ˆt j ), Q j=Ω×E j , Σ j=Γ ×E j
Q0=Ω×[−Δ(0),0), Σ0=Γ ×[−Δ(0),0) forj=1, 2, 3,
Itcanbeproved,usingTheorem3.1of[47],and[40],thatiftheinitialstatey(x,0)isanarbitraryfixedfunction, thenthefollowingresultholds
Theorem 3.1. Let y(0), Φ0,v and u be given with y(0)∈W∞{a α,2}(Ω), Φ0∈L2(Σ0),v∈L2(Σ) and u∈(W ∞,1 (Q)).
Then, there exists a unique solution y∈W∞,1(Q) for the mixed initial-boundary value problem(1)–(4). Moreover,
y (., ˆt j)∈W∞{a α ,2}(Ω)forj=1, 2, 3,
4 Problem formulation-optimization theorems
Now,weformulatetheoptimalcontrolproblemfor(1)–(4)inthecontextofTheorem3.1,thatisv∈L2(Σ).
LetusdenotebyU=L2(Σ)thespaceofcontrols.ThetimehorizonTisfixedinourproblem
Theperformancefunctionalisgivenby
I (v)=λ1
Q
[y(x, t;v)−z d]2dxdt+λ2
Σ
whereλ i≥0,andλ1+λ2>0,z disagivenelementinŁ2(Q); NisapositivelinearoperatoronL2(Σ)intoL2(Σ).
Control constraints:WedefinethesetofadmissiblecontrolsU adsuchthat
Lety (x, t;v)denotethesolutionofthemixedinitial-boundaryvalueproblem(1)–(4)at(x, t)correspondingtoa givencontrolv∈U ad.Wenotefromtheorem3.1thatforanyv∈U adtheperformancefunctional(11)iswell-defined sincey (v)∈W ∞,1 (Q)⊂L2(Q).
Making useof theLoins’sschemewe shallderivethe necessaryandsufficientconditionsof optimality forthe optimization problem(1)–(4), (11),(12) The solvingof the formulatedoptimalcontrol problem is equivalentto seekingav∗∈U ad suchthat
I (v∗)≤I (v), ∀v∈U ad
FromtheLion’sscheme(Theorem1.3of[46],p.10),itfollowsthatforλ2>0auniqueoptimalcontrolv∗exists.
Moreover,v∗ischaracterizedbythefollowingcondition:
Fortheperformancefunctionalofform(11)therelation(13)canbeexpressedas
λ1 (y(v∗)−z d )[y(v)−y (v∗)]dxdt+λ2 Nv∗(v−v∗)dΓ dt≥0, ∀v∈U ad (14)
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Inordertosimplify(14),weintroducetheadjointequation,andforeveryv∈U ad,wedefinetheadjointvariable
p=p (v)≡p (x, t;v)asthesolutionoftheequations:
−∂p (v)
∂p (v)
∂ν A∗(x, t)=0, x∈Γ, t∈(T −Δ (T ), T ), (16)
∂p (v)
∂ν A∗(x, t)=
m
s=1
c s (x, t+q s (t))p(x, t+q s (t); v)(1+q
s (t)), x∈Γ, t∈(0, T−Δ (T )), (17)
where
Δ (T)=max{k1(T ), k2(T ), , k m (T)},
∂p (v)
∂ν A∗(x, t)= ∞
|ω|=0
(D ω p (v)) cos(n, x ω ),
A∗(t)p(v)=
∞
|α|=0
Asintheabovesectionwithchangeofvariables,i.e.withreversedsenseoftime,i.e.,t=T−t,forgivenz d∈L2(Q)
andanyv∈L2(Σ),thereexistsauniquesolutionp (v)∈W ∞,1 (Q)forproblem(15)–(18).
Theexistenceof auniquesolution fortheproblem(15)–(18) onthecylinderΩ×(0,T) canbeprovedusinga constructivemethod.Itiseasytonoticethatforgivenz dandu,theproblem(15)–(18)canbesolvedbackwardsintime startingfromt=T,i.e.firstsolving(15)–(18)onthesub-cylinderQ KandinturnonQ K−1,etc.untiltheprocedurecovers thewholecylinderΩ×(0,T).Forthispurpose,wemayapplyTheorem3.1(withanobviouschangeofvariables) Hence,usingTheorem3.1,thefollowingresultcanbeproved
Lemma 4.1. Let the hypothesis of Theorem3.1be satisfied Then for given z d∈L2(Ω, R∞)and any v∈L2(Σ), there exists a unique solution p (v)∈W ∞,1 (Q) for the adjoint problem(15)–(18).
Wesimplify (14)using the adjointEq (15)–(18).For thispurpose denoting byp(0)≡p (x,0;v)and p (T)≡
p (x, T;v)respectively,settingv=v∗in(15)–(18),multiplyingbothsidesof(15)byy (v)−y (v∗),thenintegrating
overQ,andthenaddingbothsidesof(15)–(18),weget
λ1
Q
(y(T;v∗)−z d )[y(T;v)−y (T;v∗)]dxdt= T
−∂p (v∗)
∂t +A∗(t)p(v∗)
×[y(v)−y (v∗)]dxdt
=−
Ω
p (x, T;v∗)[y(x, T;v)−y (x, T;v∗)]dx
+
Ω
p (x,0;v∗)[y(x,0;v)−y (x,0;v∗)]dx
p (v∗)∂
∂t [y(v)−y (v∗)]dxdt
+ T A∗(t)p(v∗)[y(v)−y (v∗)]dxdt. (20)
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UsingGreen’sformula,thelastcomponentin(20)canbewrittenas
T
A∗(t)p(v∗)[y(v)−y (v∗)]dxdt= T
p (v∗)A(t)[y(v)−y (v∗)]dxdt
p (v∗)
∂y (v)
∂ν A −∂y (v∗)
∂ν A
dΓ dt
∂p (v∗)
Usingtheboundarycondition(2),onecantransformthesecondintegralontheright-handsideof(21)intotheform:
T
p (v∗)
∂y (v)
∂ν A −∂y (v∗)
∂ν A
dΓ dt
=
m
s=1
T
p (x, t;v∗)c
s (x, t)×[y(x, t −k s (t); v)−y(x,t −k s (t); v∗)]dΓ dt+ T
p (x, t;v∗)(v−v∗)dΓ dt
=
m
s=1
T −k s (T)
p (x, t+g s (t s);v∗)c
s (x, t+g s (t s))(1+g
s (t s))×[y(x, t s;v)−y (x, t s;v∗)]dΓ dt
Thelastcomponentin(21)canberewrittenas
T
∂p (v∗)
∂ν A∗ [y(v)−y (v∗)]dΓ dt
= T −Δ(T)
∂p (v∗)
∂ν A∗ [y(v)−y (v∗)]dΓ dt+ T
T −Δ(T) Γ
∂p (v∗)
∂ν A∗ [y(v)−y (v∗)]dΓ dt. (23)
Substituting(22)and(23)into(21),andthentheresultsinto(20),weobtain
λ1
Ω
(y(T;v∗)−z d )[y(T;v)−y (T;v∗)]dx
=−
Ω
p (x, T;v∗)[y(x, T;v)−y (x, T;v∗)]dx+
Ω
p (x,0;v∗)[y(x,0;v)−y (x,0;v∗)]dx
p (v∗)
∂
∂t +A (t)
[y(v)−y (v∗)]dxdt+ T
p (x, t;v∗)(v−v∗)dΓ dt
+
m
s=1
0
p (x, t+q s (t); v∗)c
s (x, t+q s (t))(1+q
s (t))×[y(x, t;v)−y (x, t;v∗)]dΓ dt
+
m
s=1
T −k s (T)
p (x, t+q s (t); v∗)c
s (x, t+q s (t))(1+q
s (t))×[y(x, t;v)−y (x, t;v∗)]dΓ dt
− T −Δ(T)
∂p (v∗)
∂ν A∗ [y(v)−y (v∗)]dΓ dt− T
T −Δ(T) Γ
∂p (v∗)
∂ν A∗ [y(v)−y (v∗)]dΓ dt
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=
Ω
p (x,0;v∗)[y(x,0;v)−y (x,0;v∗)]dx+ T
p (x, t;v∗)(v−v∗)dΓ dt
=
Ω
p o (0)[y(0)−y o (0)]dx+ T
p (v∗)(v−v∗)dΓ dt. (24)
Substituting(24)into(14)gives
Ω
p o (0)[y(0)−y o (0)]dx+ T
(p(v∗)+λ2Nv∗)(v−v∗)dΓ dt≥0∀v∈U ad (25) Theforegoingresultisnowsummarized
Theorem 4.1. For the problem(1)–(4), with the performance functional(11)with z d∈L2(Q) and λ2>0and with conditions(4),(12), there exists a unique optimal control v∗which satisfies the maximum condition(25).
Wecanalsoconsiderananalogousoptimalcontrolproblemwheretheperformancefunctionalisgivenby
ˆI(v)=λ1
Σ
[y(x, t;v)|Σ−z d]2dΓ dt+λ2
Σ
wherez d∈L2(Σ).
Fromtheorem3.1andtheTraceTheorem([47],vol.2,p.9),foreach v∈L2(Σ),thereexistsauniquesolution
y (v)∈W ∞,1 (Q)withy| Σ∈L2(Σ) Thus, ˆI(v)iswelldefined.Then,theoptimalcontrolv∗ischaracterizedby
λ1
Σ
(y(v∗)|Σ−z d )[y(v)| Σ−y (v∗)|Σ ]dΓ dt+λ2
Σ
Nv∗(v−v∗)dΓ dt≥0 ∀v∈U ad (27)
Wedefinetheadjointvariablep=p (v∗)=p (x, t;v∗)asthesolutionoftheequations:
−∂p (v∗)
∂p (v∗)
∂ν A∗ (x, t)=λ1(y(v∗)|Σ (x, t)−z Σd ), x∈Γ, t∈(T −ΔT, T ), (29)
∂p (v∗)
∂ν A∗ (x, t)=
m
s=1
c s (x, t +q s (t))p(x, t+q s (t); v∗)(1+q
s (t))+λ1(y(v∗)|Σ (x, t)−z Σd ), x∈Γ, t∈(0, T −ΔT ),
(30)
Asintheabovesection,wehavethefollowingresult
Lemma 4.2. Let the hypothesis of Theorem3.1be satisfied Then, for given z Σd∈L2(Σ) and any v∈L2(Σ), there exists a unique solution p (v∗)∈W ∞,1 (Q) to the adjoint problem(28)–(31).
UsingtheadjointEqs.(28)–(31)inthiscase,thecondition(27)canalsobewritteninthefollowingform
Ω
p o (0)[y(0)−y o (0)]dx+ T
(p(v∗)+λ2Nv∗)(v−v∗)dΓ dt≥0 ∀v∈U ad (32) Thefollowingresultisnowsummarized
Theorem 4.2. For the problem(1)–(4)with the performance function(26)with z Σd∈L2(Σ) and λ2>0, and with constraint(12), and with adjoint Eqs.(28)–(31), there exists a unique optimal control v∗which satisfies the maximum
condition(32).
Trang 10G.M Bahaa / Journal of Taibah University for Science 7 (2013) 146–161 155
Example 4.1 Case: u∈L2(Q).Wecanalsoconsiderananalogousoptimalcontrolproblemwheretheperformance functionalisgivenby
ˆˆI(u)=λ1
Q
[y(x, t;u)−z d]2dxdt+λ2
Q
wherez d∈L2(Q).
FromTheorem3.1andtheTraceTheorem([47],vol.2,p.9), foreach u∈L2(Q), thereexistsauniquesolution
y(u)∈W∞,1(Q). Thus, ˆˆI iswelldefined.Then,theoptimalcontrolu*ischaracterizedby
λ1
Q
(y(u∗)−z d )[y(u)−y (u∗)]dxdt+λ2
Q
Wedefinetheadjointvariablep=p(u*)=p(x, t;u*)asthesolutionoftheequations:
−∂p (u∗)
∂t +A∗(t)p(u∗)=λ1(y(u∗)(x, t)−z d ), x∈Ω, t∈(0, T ), (35)
∂p (u∗)
∂ν A∗ (x, t)=0, x∈Γ, t∈(T−ΔT, T ), (36)
∂p (u∗)
∂ν A∗ (x, t)=
m
s=1
c s (x, t+q s (t))p(x, t+q s (t); u∗)(1+q
s (t)), x∈Γ, t∈(0, T−ΔT ), (37)
Asintheabovesection,wehavethefollowingresult
Lemma 4.3. Let the hypothesis of Theorem3.1be satisfied Then, for given z d∈L2(Q) and any u∈L2(Q), there exists
a unique solution p(u*)∈W∞,1(Q) to the adjoint problem(35)–(38).
UsingtheadjointEqs.(35)–(38)inthiscase,thecondition(34)canalsobewritteninthefollowingform
Ω
p o (0)[y(0)−y o (0)]dx+ T
(p(u∗)+λ2Nu∗)(u−u∗)dxdt≥0 ∀u∈U ad (39) Thefollowingresultisnowsummarized
Theorem 4.3. For the problem (1)–(4) with the performance function(33) with z d∈L2(Q) and λ2>0, and with constraint(12), and with adjoint Eqs.(35)–(38), there exists a unique optimal control u*which satisfies the maximum condition(39).
5 Generalization
Theoptimalcontrolproblemspresentedhercanbeextendedtocertaindifferenttwocases.Case1:Optimalcontrol for2×2coupledinfiniteorderparabolicsystemswithmultipletime-varyinglags.Case2:Optimalcontrolforn×n
coupledinfinite orderparabolicsystems withmultipletime-varyinglags.Suchextensioncanbeappliedtosolving manycontrolproblemsinmechanicalengineering
5.1 Case 1: Optimal control for 2×2 coupled infinite order parabolic systems with multiple time-varying lags
Wecanextendthediscussionstostudytheoptimalcontrolfor2×2coupledinfiniteorderparabolicsystemswith multipletime-varyinglags.Weconsiderthecasewherev=(v1, v2)∈L2(Σ)×L2(Σ),theperformancefunctionalis givenby:[25,26]
I (v)=
2
i=1
λ1
Q
[y i (x, t;v)−z id]2dxdt+λ2
Σ (N i v i )v i dxdt
wherez =(z , z )∈(L2(Q))2
... infinite order parabolic systems with multiple time- varying lags< /i>Wecanextendthediscussionstostudytheoptimalcontrolfor2×2coupledinfiniteorderparabolicsystemswith multipletime-varyinglags.Weconsiderthecasewherev=(v1,... Neumann problem for infinite order parabolic system with multiple time- varying lags< /b>
TheobjectofthissectionistoformulatethefollowingmixedinitialboundaryvalueNeumannproblemforinfinite orderparabolicsystemwithmultipletime-varyinglagswhichdefinesthestateofthesystemmodel[3,5,6,27,35–45]...
coupledinfinite orderparabolicsystems withmultipletime-varyinglags.Suchextensioncanbeappliedtosolving manycontrolproblemsinmechanicalengineering
5.1 Case 1: Optimal control for 2×2