Approximate local output regulation for a class of nonlinear fluid flows

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European Journal of Control

journalhomepage:www.elsevier.com/locate/ejcon

Approximate local output regulation for a class of nonlinear fluid flows ^R

Konsta Huhtala

^∗

, Lassi Paunonen

Faculty of Information Technology and Communication Sciences, Mathematics and Statistics, Tampere University, PO. Box 692, Tampere 33101, Finland

a rt i c l e i nf o

Article history:

Received 16 April 2021 Revised 19 June 2021 Accepted 25 June 2021 Available online 2 July 2021 Recommended by Prof. T Parisini Keywords:

Distributed parameter systems Nonlinear systems

Output regulation Fluid flow systems

a b s t r a c t

We consider output tracking foraclassof viscous nonlinearfluidflows includingthe incompressible 2DNavier–Stokesequations.Thefluidissubjecttoin-domaininputsanddisturbances.Weconstructan errorfeedbackcontrollerwhichguaranteesapproximatelocalvelocityoutputtrackingforaclassofrefer- enceoutputs.Thecontrolsolutioncoverspointvelocityobservationsandassumesasmoothenoughstate space.Efficacyofthecontrolsolutionisillustratedthroughanumericalexample.

© 2021TheAuthor(s).PublishedbyElsevierLtdonbehalfofEuropeanControlAssociation.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

In thiswork, weconsider an output trackingproblemforvis- cousnonlinearfluidflowsintheneighborhoodofa(locally)expo- nentiallystablesteadystatesolution.Weformulateourresultsfor theincompressibleNavier–Stokesequationsonasufficientlyregu- lardomain⊂R² withboundary^{.Moreprecisely,weconsider} controllinganoutputyoftheequations

∂

^w

∂

^{t =}

ν

^w−

(

^w·

∇ )

^w−

∇

^q+fw+fu+fd, (1a)

0=

∇

^{·w, w}

|

=0, (1b)

where w(

ξ

,t)^{is thefluid velocity, q}(

ξ

,t)^{is thefluid pressure,}

ν

is the kinematic viscosity, fw(

ξ

) ^{is a body force, f}u(

ξ

,t) ^{is the} control action and f_d(

ξ

,t) ^{is the} disturbance action. Our goal is to design a controller such that a chosen velocity output of (1) converges to a desired reference output approximately for initial states which are, in a certain sense, “close enough” to a steadystatesolutionof(1).

R The research was supported by the Academy of Finland Grant number 310489 held by L. Paunonen.

∗Corresponding author.

E-mail addresses: konsta.huhtala@tuni.fi(K. Huhtala), lassi.paunonen@tuni.fi(L.

Paunonen).

Theoryof output regulationfor nonlinear systems is still un- der development, especially forinfinite-dimensionalsystems.The finite-dimensionalresultsof[11]havebeenextendedtoaclassof infinite-dimensionalsystems in[5]and forco-locatedinputsand outputsin[6],basedonwhichseveralexamplecasesarepresented in[1].Inthiswork,we focusonoutputregulationinanapproxi- matesenseutilizingtheresultsof[13].In[13],theauthorsusean errorfeedback controller designed forrobust output regulation of exponentiallystableregularlinear systemsandshowthat thesame controllerachievesapproximatelocaloutputregulationforaclassof nonlinearsystemswhichthey callregularnonlinearsystems.Simi- larapproachofusinglinearcontrolsolutionsfornonlinearsystems hasbeenutilizedforlocalstabilizationofnonlinearfluid flowsin differentsetups,seee.g. [3]forin-domaininputs,[14]forbound- aryinputsand[10]forobserverdesign.

As the main contribution of this paper, we show that the Eq. (1) can be formulated as a regular nonlinear system(in the sense of[13]) forawide rangeofvelocity observationsincluding thepointobservation.Toachieve this,we considertheEq.(1)on a “lifted” statespace, i.e.wedemand moresmoothness fromthe velocityandthepressurethan wouldtypically berequiredtoe.g.

solve similar control problems for linear systems. To formulate (1)as a regularnonlinear system,we rely onthe fluid being vis- cous and assume that the domain ^{together with the bound-} ary conditions are such that the system can be formulated on the “lifted” state space.These properties,together withthe type (^x·

∇

)^xofthenonlinearitytypicalforfluidflows,characterizethe fluidflowsforwhichtheresultscanbeapplied.

https://doi.org/10.1016/j.ejcon.2021.06.025

0947-3580/© 2021 The Author(s). Published by Elsevier Ltd on behalf of European Control Association. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

Usingtheresultsof[13],weshowthatintheneighborhoodof asteadystatesolution,velocity observationson(1)approximately converge toany desired“small enough” periodic referencesignal ofthetype

yr

(

^t

)

=a0+

qs

i=1

aicos

( ω

^it

)

+bisin

( ω

^it

)

inthesense thatforsmallenoughinitialdata,afinitenumberof chosen harmonicsof the systemoutput andthe referenceoutput are thesame.Here thecoefficient vectorsa_i,b_i∈R^py maybe un- known butwe expect toknowthe frequencies

ω

i. Thecontroller introducedin[13]alsorejectsperiodicin-domaindisturbancesig- nalsofthetype

u_d

(

^t

)

=c0+

qs

i=1

cicos

( ω

^it

)

+disin

( ω

^it

)

withsmallenough amplitude, whereagainc_i,d_i∈R^d are allowed to be unknown.Note thatseveralcontrollers havebeen designed for robust outputtracking ofsimilar signal classesin the caseof linearsystems,seee.g.[15,16].

Rest ofthepaperis organizedasfollows.InSection 2,we re- calltheconceptsofregularnonlinearsystemsandapproximatelo- caloutputregulation.InSection3,weshowthattheNavier–Stokes equationswithin-domaincontrolandpointobservationfitintothe framework ofregularnonlinearsystemsona suitablestate space.

InSection 4,weconstruct,basedon[13],a controllerforapprox- imate localoutputregulationfortheNavier–Stokesequationsand then illustrate the controller’s performance through a simulation exampleinSection5.Finally,thepaperisconcludedinSection6.

ThroughoutthepaperwedenotebyL(^X,Y)^{thesetofbounded} linear operators froma Hilbertspace X to a HilbertspaceY. For a linear operator A:D(^A)→X, D(^A) ^{is the domain ofA,}

ρ

(^A) ^is the resolvent set of A and TA is the strongly continuous semi- group generated by A on X. For a fixed s∈

ρ

(^A)^{, we denote by} X₋₁ thecompletionofX withrespecttothenorm

^x

X₋₁=

(^sI− A)^−1x

X and define X1=D(^A)^{, equipped with the norm}

^x

X₁=

(^sI−A)^x

X.Finally,theL²-innerproductoveradomain^isde- notedby

(·),(·)L²().

2. Regularnonlinearsystemsandoutputregulation

Output regulationforfluid flowsystems coveredby thiswork isbasedontheconceptsofregularnonlinearsystemsandapproxi- matelocaloutputregulation,whichwereintroducedin[13].These conceptsarepresentednext,withthedefinitionofregularnonlin- earsystemsformulatedinaslightlyrestrictedsettingbyexcluding partsthatarenotrelevanttothiswork.

Definition 1. Let X, U, Y and V be Hilbert spaces, and let C defined by Cx=lims→+∞Cs(^sI−A)^{−1x with D}(^C)=

{

^x∈ X

|

^{theabovelimitexists}

}

^{be the -extension of the} observation operatorC,see[21].Thesystem

x˙

(

^t

)

=Ax

(

^t

)

+Bu

(

^t

)

+Bdud

(

^t

)

+QF

(

^x

(

^t

))

, x

(

⁰

)

=x0∈X,

y

(

^t

)

=Cx

(

^t

)

,

whichwedenotebyF,iscalledaregularnonlinearsystemifthe followinghold.

(i) The operator A generates an exponentially stable strongly continuoussemigroupTA onX.

(ii) It holds that B∈L(^U,X₋1)^{, B}d∈L(^Ud,X₋1)^{, Q}∈L(^V,X₋1) and C∈L(^X1,Y)^{, and the triples} (^A,B,C)^, (^A,B_d,C) ^and (^A,Q,C)^{areregularlinearsystemsinthesenseof[21].}

(iii) The nonlinear map F:X→V satisfies F(⁰)=0 andis lo- cally Lipschitz. Thatis, forevery bounded set O⊂X, there existsaconstantL_Osuchthatforallx₁,x₂∈O

^F

(

^x1

)

−F

(

^x2

)

^{V ≤LO}

^x1−x2

^X.

Furthermore,foreach

γ

>0thereexistsa

ζ

>0suchthatif sup

^x

X

x∈O

<

ζ

^,thenLO<

γ

^.

Togeneratetheplantinput,weuseanerrorfeedbackcontroller oftheform

˙

z

(

^t

)

=G1z

(

^t

)

+G2e

(

^t

)

, z

(

⁰

)

=z0∈Z, (3a)

u

(

^t

)

=Kz

(

^t

)

, (3b)

wheree(^t)=y(^t)−y_r(^t)^{istheregulationerrorandZisaHilbert} space. Coupling the controller with a regular nonlinear system yieldstheclosed-loopsystemEdefinedby

˙

xe

(

^t

)

=Aexe+Bewext

(

^t

)

+QeF

(

^x

(

^t

))

, xe0∈Xe, e

(

^t

)

^=Cexe

(

^t

)

^+Deyr

(

^t

)

on the Hilbert space Xe=X×Z, where xe=[x, z]^T, wext= [u_d, yr]^T,

Ae=

A BK G2C G1

, Be=

B_d 0 0 −G2

, Qe=

Q 0

, Ce=

C 0 , De=

0 −I .

Before introducing theoutput tracking problem, we recall the concept ofharmonics ofa function. Consider a function f= fp+ f_e,wheref_p∈L²_loc(^[0,∞);Cⁿ)^{isT-periodicand f}e∈L²(^[0,∞);Cⁿ)^. Foranon-negativeintegerl,thel^thharmonicof f isthefunction

f_l

(

^t

)

⁼

α

lsin

₂

π

^lt

T

+

β

lcos

₂

π

^lt

T

, t≥0,

where

α

l= lim

k∈N,k→∞

2 kT

kT 0

f

(

^t

)

^sin

₂

π

^lt

T

dt,

β

l= lim

k∈N,k→∞

2 kT

kT 0

f

(

^t

)

^cos

₂

π

^lt

T

dt,

thusforfrequencies ofthe harmonics,we have

ω

i=2

π

^li/T. Now the problem of achieving approximate local output regulation is statedasfollows.

Problem 2. Let T>0 be a constant. Assume that yr and u_d are T-periodic functions and let V=

{

^l0,l₁,...,lnv

}

^{be a finite set of}

non-negativeintegers.Designanerrorfeedbackcontroller(3)such that:

1. Theclosed-loopsystemEisaregularnonlinearsystem.

2. There exist positive constants cy, c_d and ce such that if

^yr

L²[0,T]≤c_y,

^ud

L∞≤c_dand

^xe0

X_e≤c_e,thenx_econverges asymptoticallytoaT-periodicfunctionandthel^thharmonicof y−yris0foreachl∈V.Theoutputsatisfiesy=yp+ye,where ye∈L²(^[0,∞);Y) ^andyp∈L²_loc(^[0,∞);Y)^{is a T-periodicfunc-} tion.

Accuracy of output tracking by solving the above problem clearlydependsonhowdominanttheharmonicsincludedinVare.

In manycases,ensuring that the first few harmonics ofy match thoseofyr resultsinasmalltrackingerror,sincehigherharmon- icsoftheoutputaretypicallysmall.

3. TheincompressibleNavier–Stokesequationsasanabstract controlsystem

The goal of this section is to formulate the Navier–Stokes Eq.(1) asaregular nonlinear system. Westart by findinga suit- able state space forthe formulation andthen verify that there- quirementsofDefinition1arefulfilled.

3.1. Choosingthestatespace

Translating the Eq.(1) to the vicinity of a steady state solu- tion(

v

e,pe)^{usingthechangeofvariables}

v

₍

ξ

,t)=w(

ξ

,t)−

v

e(

ξ

)^, p(

ξ

,t)=q(

ξ

,t)−pe(

ξ

)^yields

∂ v

∂

^{t =}

ν v

−

( v

e·

∇ ) v

−

( v

·

∇ ) v

e (4a)

−

( v

·

∇ ) v

−

∇

^p+fu+f_d, (4b)

0=

∇

^·

v

,

v |

=0 (4c)

withthe initialstate

v

(

ξ

,0)=

v

0(

ξ

)^{.We assumethat thecontrol} andthedisturbancearedefinedby

fu

( ξ

^,t

)

=

g1

( ξ )

^g2

( ξ )

· · · gm

( ξ )

^u

(

^t

)

, (5a)

f_d

( ξ

^,t

)

⁼

g₁

( ξ )

^g₂

( ξ )

· · · g_d

( ξ )

^ud

(

^t

)

^{, (5b)}

where each g₁,...,gm and g₁,...,g_d is an R²-valued function on ^{, u}(^t)∈U:=C^m is the finite-dimensional control input and u_d(^t)∈U_d:=C^d is the finite-dimensional disturbance input. Ad- ditionally, we assume that the output space Y is also finite- dimensionalandY=C^py withp_y≤m.

Forsimplernotation,wedefinethespaces X˜=

v

∈

(

^L2

())

²

∇

·

v

=0,

( v

·n

) |

=0

, H˜ =

v

∈

(

^H1

())

²

∇

^·

v

=0,

v |

=0

andthebilinearandtrilinearforms

a

( v

,

ψ )

⁼²

ν ( v ₎

,

( ψ )

L²()

∀ v

,

ψ

^∈H˜,

b

( v

,

φ

^,

ψ )

=

( v

·

∇ ) φ

^,

ψ

L²()

∀ v

,

φ

^,

ψ

^∈H˜,

where

(

v

)=1/2(

∇ v

+(

∇ v

)^T)^.

Assumption3. Weassumethefollowing:

(i) Theboundary^{isofclassC3and f}w∈(^H1())^2. (ii) Thelinearizationof(4)isexponentiallystable.

The first part ofthe assumption guarantees sufficient regular- ityofthesolutionsoftheNavier–StokesEq.(1),whilethesecond partisrequiredforDefinition1.(i)toholdandissatisfiedforlarge enough

ν

^,c.f.[3].

As the first steptowards choosing thestate spaceX, we con- sidersemigroupgenerationforthelinearizedversionof(4)onX˜.A weak formulationforthestationary,linearizedversion of(4)sub- jecttozerocontrolanddisturbanceinputsisgivenby

0=−a

( v

,

ψ )

−b

( v

,

v

e,

ψ )

−b

( v

e,

v

,

ψ ) ∀ ψ

∈H˜.

Lemma4. TheoperatorA˜definedby A˜=A˜₂+A˜₁,

^A˜2x,

ψ

L²()=−a

(

^x,

ψ )

^,

^A˜1x,

ψ

L²()=−b

(

x,

v

e,

ψ )

−b

( v

e,x,

ψ )

, D

(

^A˜

)

=D

(

^A˜2

)

=

x∈H˜

∀ ψ

^∈H˜,

ψ

^→a

(

x,

ψ )

^{is X˜}-continuous

generatesanexponentiallystableanalyticsemigrouponX˜.

Proof. We start by showing that a(·,·) ^{is H˜-bounded and H˜-} coercive, i.e.H˜ can be continuously and densely embedded inX˜ andthereexistc₁,c₂,

λ

>0suchthatforevery

φ

,

ψ

∈H˜

|

^a

( φ

,

ψ ) |

≤c₁

φ

H˜

ψ

H˜, (6a)

a

( φ

^,

φ )

≥c2

φ

²H˜−

λ φ

²X˜. (6b) Sincethenorm

(·)

X˜ isequivalenttothe

·

H˜ normthrough Korn’sandPoincare’sinequalities,weimmediatelyhaveforacon- stantc₁>0andforany

v

_∈H^˜

a

( v

,

v )

=2

ν ( v )

²X˜≥c1

v

²H˜.

Similarly,foraconstantc₂>0andany

v

_,

φ

∈H˜,

|

^a

( v

,

φ ) |

≤2

ν ( v )

X˜

( φ )

X˜ ≤c2

v

H˜

φ

H˜.

Assuch,a(·,·)^{isH˜-boundedandH˜-coercive,whichimpliesthatA˜}2

generatesananalyticsemigroupTA˜₂ onX˜ [2,Section2].

Regarding the trilinearform b(·,·,·)^{, Assumption 3.(i) guaran-} tees that

v

e∈H˜, c.f. [12, Ch. 5]. We have forconstants c₃,c₄>0 usingintegrationbypartsandSobolevembeddings

|

^b

( v

1,

v

2,

ψ ) |

^≤

| v

1,

( v

2·

∇ ) ψ

|

⁺

| v

1·n,

v

2·

ψ

|

≤c3

v

1

L⁴()

v

2

L⁴()

ψ

H^˜

≤c4

v

1

H^˜

v

2

H^˜

ψ

H^˜

∀ v

1,

v

2,

ψ

^∈H˜.

NowA˜1∈L(^H˜,X˜)^,thusperturbationtheoryofsemigroups,seee.g.

[7,Ch.III],impliesthatA˜generatesananalyticsemigroupTA˜ onX˜. ByAssumption3.(ii),TA˜ isexponentiallystable.

The fact that we may choose

λ

=0 in (6b) implies that the semigroupTA˜₂ isexponentiallystableforany

ν

>0.Furthermore, A˜₂ is self-adjoint andthe fractional powers (−A˜₂)^{δ are well de-} fined. Domains of the fractional powers are defined by, c.f. [18, Ch.2],[14],

D

((

−A˜2

)

^δ

)

=

v

∈

(

^H2δ

())

²

∇

·

v

=0,

( v

·n

) |

=0

, 0≤

δ

^<1₄^,

D

((

−A˜2

)

^δ

)

=

v

∈

(

^H2δ

())

²

∇

^·

v

=0,

v |

=0

, 1

4<

δ

^≤1.

Thenorms correspondingto domainsofthefractional powersfor thefullrange

δ

∈Raregivenby

^x

D((−A˜2)^δ)=

(

^−A˜2

)

^δx

X˜.

Wenextutilizedomains ofthefractionalpowerstofinda“lifted”

statespaceX suchthatinparticularDefinition1.(iii)issatisfiedby (4).

ForthetranslatedNavier–StokesEq.(4),nonlinearityintheab- stractframeworkF isdescribedby

F

( v )

=−P

( v

·

∇ ) v

, Q=I, (7)

wherePistheLerayprojector,seee.g.[8,14].Thedomainsofdef- initionforFandQ aredictatedbythefollowingLemma.

Lemma5. For a (small)

δ

>0,choose X=D((−A˜₂)^1/2+δ) ^andV= D((−A˜₂)^δ)^{.ThenDefinition1.(iii)holdsfor}F.

Proof. The proof is based on the “propertiesof multipliers”, see [17,Ch.4.6.1,Thm.1],[4,Lemma5.4],whichstatethatif

s2>s1, s2> d

2 , (8)

wheredisthespatialdimension,then H^s1·H^s2→H^s1

isacontinuousembedding,where H^s1·H^s2:=

f g

f ∈H^s1, g∈H^s2

.

Since d=2, we choose s₁=2

δ

^{and s}2=1+2

δ

^{, and apply the}

aboveresult.Nowforaconstantc₁>0and

φ

,

ψ

∈X

φ

i

∂

j

ψ

k

H^s1()≤c₁

φ

i

H^s2()

∂

j

ψ

k

H^s1() (9a) fori,j,k∈

{ ξ

1,

ξ

2

}

^{,thusforsomeconstantsc}2,c₃>0also

( φ

·

∇ ) ψ

V≤c2

φ

X

∇ψ

V

≤c₃

φ

X

(

−A˜₂

)

^1/2

ψ

V

=c3

φ

X

ψ

X. (9b) Utilizing (9), for

v

1,

v

2∈X and some constants c₄,c₅>0 we have

^F

( v

1

)

^−F

( v

2

)

V

=

^−P

( v

1·

∇ ) v

1−

( v

2·

∇ ) v

2

V

=

P

(( v

1−

v

2

)

·

∇ ) v

1+

( v

2·

∇ )( v

1−

v

2

)

V

≤c4

( v

1−

v

2

)

^X

∇ v

1

^V+

v

2

^X

∇ ( v

1−

v

2

)

^V

≤c5

( v

1

X+

v

2

X

) v

1−

v

2

X,

thusF islocallyLipschitz.ClearlyF(⁰)=0,andif

v

1

X,

v

2

X<

γ

c for a large enough constant c>0, then (

v

1

X+

v

2

X)<

γ

^,

whichcompletestheproof.

DuetoLemma5,wechooseforafixed(small)

δ

>0 X=D

((

−A˜2

)

^1/2+δ

)

as the state space for our abstract system presentation and de- note

Xs=D

((

−A˜₂

)

^1/2+δ+s

) ∀

^s∈R,

withthecorrespondingnormsdefinedaccordinglyby

^x

Xs=

(

−A˜2

)

^sx

X =

(

−A˜2

)

^1/2+δ+sx

X˜. NowV=X_−1/2,F:X→V andQ=I_V ∈L(^V)^. 3.2. Theabstractsystemformulation

Wedefinetheoperators

A=A2+A1:D

(

^A

)

→X, (10a)

A2=

ν

^P

, A1

v

=−P

( v

e·

∇ ) v

+

( v

·

∇ ) v

e

, (10b)

D

(

^A

)

=D

(

^A2

)

=D

((

−A˜₂

)

^3/2+δ

)

. (10c) Now A₂

v

=A˜₂

v

and A₁

v

=A˜₁

v

for

v

∈D(^A)^{. To verify that} Definition 1.(i) holds on the state space X, we note that A gen- eratesastronglycontinuoussemigroupTAonX,c.f.[7,Ch.5].The semigroupTAisexponentiallystable,sinceforx∈X

TAx

X=

(

−A˜2

)

^1/2+δTAx

X˜

=

TA˜

(

−A˜2

)

^1/2+δx

X˜

≤

TA˜

_L(X˜)

^x

X.

We still need to verifyDefinition 1.(ii).We do so forcontrols anddisturbancesoftheform(5)andobservationsuptothe“level of unboundedness” of a point observation. Using integration by parts,wehavefortheX-adjointofA₁

A^∗₁

φ

^=P

( v

e·

∇ ) φ

⁻

( ∇ v

e

)

^T

φ

.

Properties ofmultipliers withthechoicess1=1+2

δ

^,s2=2+2

δ

tosatisfy(8)imply,similarlyto(9),forany

φ

,

ψ

∈X_1/2andacon- stantc>0

^P

( φ

^·

∇ ) ψ

X≤c

φ

2

X1/2

ψ

2

X1/2,

^P

( ∇φ )

^T

ψ

^X≤c

φ

²

^X1/2

ψ

²

^X1/2.

SinceAssumption3.(i)implies

v

e∈X1/2⊂(^H2+2δ())^{2 [12,Ch. 5],} wehaveA₁,A^∗₁∈L(^X1/2,X)^.

Assuch,theoryofadmissiblecontrolandobservationoperators, see[20,Ch.4–5],nowstatesthat

• LetY˜ beaHilbertspace.IfC˜∈L(^X1/2,Y˜)^{,thenC˜isanadmissi-} bleobservationoperatorforTA₂ anditsadjointC˜^∗∈L(^Y˜,X_−1/2) isanadmissiblecontroloperatorforTA₂.

• The sets of admissible control (observation) operators for TA

andTA₂ arethesame.

NotethataboveweassumedforY˜tobeself-dual.

Wefirstsearchforadmissibleobservationsfor(4)onthestate spaceX byconsidering observationssuchthatC∈L(^X1/2,Y)^.Typ- icallythe“mostunbounded” observationofinterestwouldbe the pointobservation

Cpx

( ξ

,t

)

=x

( ξ

p,t

)

⁽¹¹⁾

forsome

ξ

^p∈^.BySobolevembeddings,when ⊂R²,H^s()⊂ C()^{¯ for s}>1,thus Cp∈L(^X,C)^{. Thatis, all the} observations of interestfor(4)areboundedoperatorsfromX toY.Assuch,C=C andifU,U_d andQ are admissiblecontrol operatorsforTA₂,then Definition1.(ii)holds.

Consider next admissible control operators for TA₂, thus also for TA. We start with the operator Q=I_V =I_X

−1/2. Note that in this case the “input space” V is not self-dual, but instead the correct dual is the X-dual of X_−1/2, i.e. V=X_1/2. Thus we have Q^∗∈L(^X1/2,V) ^{and Q}∈L(^V,X_−1/2)^{, i.e. the triple} (^A,Q,C) ^{is a} regularlinearsystem.

Forasinglecontrolinputofthetype(5a),wehaveB=g(

ξ

)^.If g∈Xs,thenB∈L(C,Xs)^.Thatis,if

g∈X₋1/2=D

((

^−A˜2

)

^δ

)

^,

thenBisanadmissiblecontroloperatorforTA₂.

We concludethe section by gatheringourfindings inthe fol- lowingresult.

Theorem6. GivenAssumption3,assumethatthecontrolshapefunc- tionsg_iandthedisturbanceshapefunctionsg_jsatisfyg_i,g_j∈X_−1/2= D((−A˜₂)^δ)^{foreach i}=1,2,...,m, j=1,2,...,d andasmall

δ

>0. ThenthetranslatedNavier–StokesEq.(4)withthedynamicsoperator (10), thenonlinearity(7),the control(5a),thedisturbance (5b)and py point observations (11) form a regular nonlinear system on the statespaceX=D((−A˜₂)^1/2+δ)^.

4. Thecontroller

Weusealow-gain-typecontrollerdesignintroducedin[13]to solveProblem2.Theonlysysteminformationrequiredtoconstruct thecontrolleristhetransferfunctiongains

G

(

±i

ω

k

)

=C

(

±i

ω

kI−A

)

^−1B

ofthe linearizedsystem(^A,B,C) ^forthefrequencies

ω

k=2

π

^lk/T foreach l_k∈V.A goodestimate for thesegainsof thelinearized systemcanbeobtainedexperimentallyfromthegainsofthenon- linearsystemN,see[13],androbustnessofthecontrollermeans that the approximate gains can be used to achieve the output trackinggoal.

The controller consistsoftwo finite-dimensional systems. The firstsystemF isdescribedbythetransferfunction

GF

(

^s

)

=IY−

n_v

k=0

s²+

ω

²k

s²+

ε

^s+

ω

²k

IY,

Fig. 1. The closed-loop system.

where

ε

>0 is the control tuning parameter. The second system Risdescribedbythetransferfunction

GR

(

^s

)

=

n_v

k=0

s²+

ω

²k

s²+2s+

ω

²k

×

n_v

k=0

ρ

kR_k s−i

ω

k

+

ρ

−kR_−k s+i

ω

k

,

where

R_k=G^∗

(

ⁱ

ω

k

)(

^G

(

ⁱ

ω

k

)

^G∗

(

ⁱ

ω

k

))

⁻¹, R_−k=G^∗

(

−i

ω

k

)(

^G

(

−i

ω

k

)

^G∗

(

−i

ω

k

))

⁻¹,

ρ

k=

n_v

j=k,j=0

ω

²j−

ω

²_k+2i

ω

k

ω

²j−

ω

k²

,

ρ

−k=

nv

j=k,j=0

ω

²j−

ω

²k−2i

ω

k

ω

²j−

ω

_k^{2 .}

WedenoteastatespacerealizationofG_F onX_F=C^nF by

˙

xF

(

^t

)

=AFxF

(

^t

)

+BFuF

(

^t

)

, xF

(

⁰

)

=xF0∈XF, yF

(

^t

)

=CFxF

(

^t

)

,

andastatespacerealizationofG_RonX_R=C^nR by

˙

xR

(

^t

)

=ARxR

(

^t

)

+BRuR

(

^t

)

, xR

(

⁰

)

=xR0∈XR, y_R

(

^t

)

^=CRx_R

(

^t

)

^+DRu_R

(

^t

)

^.

After coupling the two subsystems of the controller as depicted in Fig.1,i.e.bysettingu_F =y−yr+y_F andu_R=y_F,we havethe structure ofan errorfeedback controller (3) withz=[x_F, x_R]^T∈ XF×XR,

G1=

A_F+B_FC_F 0 BRCF AR

, G2=

−B_F 0

, (12a)

K=

DRCF CR . (12b)

The followingresultisobtainedin[13]fortheclassofregular nonlinear systemsandwe formulateit fortheincompressible2D Navier–Stokesequations.

Theorem7. AssumethatG(ⁱ

ω

k)^{issurjectiveforeachk}=1,2,...n_v andtheassumptionsofThm.6hold.Thereexists

ε

^∗>0suchthatan error feedback controller(3)with theoperatorschosenas (12)with 0<

ε

≤

ε

^{∗solvesProblem2forthesystem(10),(11).}

Proof. The proof follows directly from Theorem 6 and [13, Sec- tion5.2].

5. ANumericalExample

Let ^{be the unit disk and consider the} Navier–Stokes Eq.(1) with

ν

=1/25around a steadystate solutioncorrespond- ingtothebodyforce

fw

( ξ

1,

ξ

2

)

=

ξ

2

(

¹−

ξ

1²−

ξ

2²

)

, −

ξ

1

(

¹−

ξ

1²−

ξ

2²

)

^T∈H˜ and fu=0, f_d=0.Our output trackinggoal isto have thepoint observation

y

(

^t

)

=C

v

1

( ξ

^,t

) v

2

( ξ

,t

)

=

v

2

0.4, −0.4 ,t

∈L

(

X,R

)

Fig. 2. The steady state velocity field ve(ξ), where color depicts speed of the fluid.

trackthereferencesignal

yr

(

^t

)

^=0.5sin

(

^2t

)

⁽¹³⁾

despitethedisturbance f_d

( ξ

,t

)

=B_du_d

(

^t

)

=P

0,

χ

d

( ξ )

^T

(

¹+cos

(

^2t

))

,

where

χ

_d is the characteristicfunction and d=[−0.4,−0.1]× [−0.4,−0.1].The outputtrackingistobeachieved,approximately andlocally,byusingthecontrol

fu

( ξ

^,t

)

=Bu

(

^t

)

=P

χ

u

( ξ )

, 0 ^Tu

(

^t

)

where ^u=[−0.6,−0.3]×[0.1,0.4]. Now U=U_d=Y=R and since

χ

u,

χ

_d∈H^s()^{for any s}<1/2 [19], also B∈L(^U,X_−1/2) andB_d∈L(^Ud,X₋1/2)^{.Assuch,ifthesteadystate}(

v

e,pe)^islocally exponentiallystable,thenthetranslatedsystem(4)formsaregular nonlinearsystemonthestatespaceX.

WeusetheTaylor–Hoodfiniteelementspatialdiscretizationfor theNavier–Stokes equations.Withthe help offunctionsincluded inthe MatlabPDEtoolbox, theunit diskis approximatedby 694 triangleswiththemaximumedge lengthof≈0.1,whichleadsto approximationorderof1453 foreach ofthevelocity components and380forthe pressure.The steadystate solution (

v

e,pe)^,with thesteadystate velocitydepictedinFig.2,iscalculatedusingthe Newton’s method, and we assume p_e(⁰)=0 to obtain a unique steadystatepressure.

We check numericallythat linearizationof the translatedsys- tem(4)isexponentiallystable. Thenwe designanerrorfeedback controller(12)withV=

{

⁰,1,2,3

}

^{andchooseasthecontroltun-}

ingparameter

ε

=0.095toroughlymaximizethestabilitymargin ofthelinearizedclosed-loopsystem. Forthesimulation,werelax the incompressibility condition by using a penalty method with thepenaltyparameter

^p=10⁻⁵,seee.g.[9,Ch.5.2],todecouple the fluid pressurefromthe fluid velocity.As the initial state, we use

x_e0=

v

e−

v

e1/2, 0 ^T∈X×Z,

where

v

e1/2(

ξ

) ^{is the steady state velocity} corresponding to the body force f_w1/2=0.5f_w and f_u=0, f_d=0. Evolution of the closed-loop system is then solved using Crank–Nicolson method withthetimestep^t=0.01togetherwithNewtoniteration.

Output tracking performance of the controller is depicted in Fig.3 anda snapshotof thefluid velocity at thetime t=120 is showninFig.4.

Thecontrollerachievesoutputtrackingof(13)withsatisfactory performance forthe chosen initial state despite the disturbance.

The effectofthedisturbance isnot clearlyvisibleinFig. 3,since