onstar-shaped graphs Approximate controllability of coupled 1-d wave equations C.R. Acad. Sci. Paris, Ser.I

(1)

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C. R. Acad. Sci. Paris, Ser. I

www.sciencedirect.com

Partial differential equations/Optimal control

Approximate controllability of coupled 1-d wave equations on star-shaped graphs

Contrôlabilité approchée des équations d’ondes 1d couplées sur graphes étoiles

René Dáger

DepartamentodeMatemáticaAplicada,UPM,Av.PuertadeHierro4,28040Madrid,Spain

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received13March2016

Acceptedafterrevision11May2016 Availableonline25May2016 PresentedbyJean-MichelCoron

InthisNote,westudytheapproximatecontrollabilityofacascadesystemoftwo1-dwave equationsdefinedonastar-shapedplanargraph.Onlythefirstequationiscontrolled,with controlsappliedatthesimple vertices.Thecontrolsactonthesecondequationthrough thecouplingatthemultiplevertex.We giveanecessaryandsufficientconditionforthe approximatecontrollabilityofthesystem:alltheratiosofthelengthsofanytwodifferent edges of the graphshould be irrational numbers. The result implies that, under these conditions,theapproximatedesensitizingcontrollabilityofastar-shapednetworkofstrings holds,whenthedesensitizedfunctionalistheL²-normofthedisplacementofthemultiple node.

©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

ré s u m é

Danscette Note,onétudie lacontrôlabilitéapprochéed’unsystèmeencascadededeux équations d’ondes 1d déﬁnies sur un graphe planaire en forme d’étoile. Seulement la premièreéquationestcontrôlée,avecdescontrôlesquiagissentsurlessommetssimples.

Lescontrôlesagissentsurladeuxièmeéquationàtraverslecouplageausommetmultiple.

On donne une condition nécessaire et suﬃsante pour la contrôlabilité approchée du système :touslesrapportsdeslongueursdesarêtesdoiventêtredesnombresirrationnels.

Lerésultatimpliqueque,danscesconditions,lacontrôlabilitédésensibilisanteapprochée d’unréseaualieulorsquelafonctionnelledésensibiliséeestlanormeL²dudéplacement dunœudmultiple.

©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

E-mailaddress:[email protected].

http://dx.doi.org/10.1016/j.crma.2016.05.006

1631-073X/©²⁰¹⁶Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.Âll^rights^reserved.

(2)

1. Introduction

Letn

≥

^{2 be}^a^natural^number^and

1

, ...,

nbepositivenumbers.Considerthefollowingsystemof2ncoupled1-dwave equationscontrolledbymeansofncontrolfunctions:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

y_i_,_tt

−

^yi,xx

=

⁰ ⁱⁿ

R× (

0

,

_i

),

i

=

¹

, ...,

n yi

(·,

i

) =

vi in

R,

i

=

1

, ...,

n y_i

(·,

⁰

) =

^yj

(·,

⁰

)

in

R,

ⁱ

,

j

=

¹

, ...,

n y1,x

( · ,

0

) + · · · +

^yn,x

( · ,

0

) =

^{0 in}

R

y_i

(

0

, ·) =

^y⁰_i

,

y_i_,_t

(

0

,·) =

^y¹_i ⁱⁿ

(

0

,

i

),

i

=

¹

, ...,

n

,

(1)

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

qi,tt

−

qi,xx

=

0 in

R×(

0

,

i

)

i

=

1

, ...,

n

q

(·,

i

) =

⁰ ⁱⁿ

R,

ⁱ

=

¹

, ...,

n

qi

( · ,

0

) =

^qj

( · ,

0

)

in

R ,

i

,

j

=

¹

, ...,

n q₁_,_x

(·,

⁰

) + · · · +

^qn,x

(·,

⁰

) =

^yi

(·,

⁰

)

in

R

q_i

(

0

, · ) =

^q⁰_i ^qi,t

(

0

, · ) =

^q¹_i ⁱⁿ

(

0

,

_i

),

i

=

¹

, ...,

n

.

(2)

Thesystems(1)and(2)modelthemotionoftwonetworksofnhomogeneousvibratingstringsconnectedatonepoint inastar-shaped conﬁguration.Thescalarrealfunctions yi

(

t

,

x

)

,qi

(

t

,

x

)

,deﬁnedinR

× (

0

,

i

)

,denotethetransversaldis- placementofthei-thstringoflength

iofeachnetwork,respectively.Foreverystringthevalues0 and

iofthevariablex correspondtoitsends: 0 totheendwherethestringsarejoined(themultiplenode)and

i totheother one(thesimple node).

Thenetworkgivenby(1)isacontrolledone.Ateverysimplenode,acontrolviisappliedthatregulatesitsdisplacement.

Atthe multiplenodes,the thirdandfourthequationsin(1)ensurethecontinuityofthe networkandthebalanceofthe tensions.

Thenetwork (2)iscontrolled inan indirectway:thesumofthetensions isequalto yi

( · ,

0

)

,thedisplacementofthe multiplenode.Throughthatcoupling,thesystemreceivestheactionofthecontrolmechanismappliedtotheﬁrstnetwork.

To simplify the notation we denote ^y

¯ := (

y1

, ...,

yn

)

, ^q

¯ := (

q1

, ...,

qn

)

,

¯

^y⁰

:= (

y⁰₁

, ...,

y_n⁰

)

, ^y

¯

¹

:= (

y¹₁

, ...,

y¹_n

)

, ^q

¯

⁰

:=

(

q⁰₁

, ...,

q⁰_n

)

,q

¯

¹

:= (

q¹₁

, ...,

q¹_n

)

,v

¯ := (

v1

, ...,

vn

)

.

Ourgoalistoinvestigatethecontrollabilitypropertiesof(1)–(2):givenT

>

0 andinitialstates^y

¯

⁰

, ¯

^y¹

,

^q

¯

⁰

,

^q

¯

¹^ﬁnd^controls

¯

v

∈

L²

(

0

,

T

)

n

suchthatbothnetworksareatrestintimeT,i.e., y

¯ (

T

) = ¯

^yt

(

T

) = ¯

^q

(

T

) = ¯

^qt

(

T

) = ¯

^0.

TobepreciseweintroducetheHilbertspaces V

:=

ψ ¯ ∈

n

i=¹

H¹

(

0

,

i

), ψ

i

(

i

) =

⁰

, ψ

i

(

0

) = ψ

j

(

0

),

i

,

j

=

¹

, ...,

n

,

H

:=

n

i=¹

L²

(

0

,

i

),

endowedwiththenaturalproductstructures.

Itisknown(see,e.g.,[6,p. 23]),thatsystem(1)–(2)iswellposedforinitialdata

¯

y⁰

, ¯

^y¹

∈

^H

×

^V^,

¯

q⁰

,

^q

¯

¹

∈

^V

×

^H^and

(¯

^q

,

^y

¯ ) ∈

^C

([

⁰

,

T

],

^V

×

^H

) ∩

^C¹

([

⁰

,

T

],

^H

×

^V

)

foranyT

>

0.

The controlled systems withthe type ofcoupling deﬁnedbetween(1) and(2) are calledcontrolled cascade systems.

Theirstudyismotivatedbythedesensitizing controllability,aconceptintroducedbyLionsin[8].

Indeed,considerthefollowingdesensitizingcontrollabilityproblemforthenetworkdeﬁnedby(1):giventheinitialdata

¯

y⁰

, ¯

y¹

∈

^H

×

^V^,^ﬁnd^controlsv

¯ ∈

L²

(

0

,

T

)

n

,suchthatthefunctional

Φ(

_h

¯

⁰

,

_h

¯

¹

) :=

T 0

(

yi

(

t

,

0

))

²dt

,

(3)

deﬁnedfor thesolutions to (1)withperturbed initial data ^y

¯

⁰

+ ¯

^h⁰

,

^y

¯

¹

+ ¯

^h¹^,^is ^locally^linearlyindependent ofthe small, unknown perturbation h

¯

⁰

∈

^H

,

h

¯

¹

∈

^V^, ^that ^is, ^D

Φ(

0

,

0

) =

^0.^Then, ^the ^controls v

¯

have that property ifand onlyif, the solutionto (1)–(2)withthosecontrolsandinitial data y

¯

⁰

,

y

¯

¹,q

¯

⁰

= ¯

^q¹

= ¯

^{0 satisﬁes} q

¯ (

T

) = ¯

^qt

(

T

) = ¯

^{0 (see,}^[2]). ^The ^latter propertyisclearlyimpliedbythecontrollabilityof(1)–(2).

Additionally, the weakerproperty that forﬁxed

¯

y⁰

,

y

¯

¹

∈

^H

×

^V ^and^every

ε >

0 it holdsthat

^D

Φ(

0

,

0

)

V

< ε

for some controls v

¯

ε (known as approximatedesensitizing controllability, see[1]) coincides withthe fact that the set ofall ﬁnal states

(¯

^q

(

T

),

^q

¯

t

(

T

))

of solutionsto (1)–(2)with^q

¯

⁰

= ¯

^q¹

= ¯

^{0 as}^v

¯

^ranges^over

L²

(

0

,

T

)

n

isdense in V

×

^H,^that ^is, withaweakenedversionoftheapproximatecontrollabilityof(1)–(2).

The main resultof this Note related to the approximate controllability of (4)–(5) is givenin Theorem 1. Besides, in Theorem 5, a weighted observability inequality is obtained that allows usto provide in Corollary 6 an alternative more precisedescriptionoftheapproximatedesensitizingcontrollabilityforthefunctional(3).

(3)

2. Approximatecontrollabilityresults

We should mention that, as it was proved in [9], the control mechanism in (1) is strong enough to guarantee the controllability ofall initial states in H

×

^V^,êven îfône ôf ^the^controls îs^kept ^switched ôff^(e.g., ^v1

=

^0). ^However, ^the couplingmechanismbetween(1)and(2)isveryweak.Indeed,astar-shapednetworkofstringscontrolledfromthemultiple node failstobecontrollableinthenaturalenergyspaceH

×

^V^and^even^theapproximatecontrollabilitymayfailinsome circumstances, namely,whenthe ratioofthelengthoftwo ofthestringsis arationalnumber(see[4]). Aswe shallsee, thatisalsothecaseinthepresentproblem.

We willsaythat thepositivenumbers

1

, ...,

n satisfytheD-conditionifalltheratios

i

/

j fori

,

j

=

¹

, ...,

n, i

=

^j^,^are irrational numbers.Aswe shallseebelow,theD-conditionallows ustocharacterizethenetworksthatare approximately controllable.

Besides,wedenote

_∗

:=

^max

{

i

:

ⁱ

=

¹

, ...,

n

}

^and^T0

:=

²

(

1

+ · · · +

n

) +

²

_∗. ThefollowingisthemainresultofthisNote:

Theorem1.System(1)–(2)isapproximatelycontrollableintimeT

≥

^T0if

1

, ...,

nsatisfytheD-condition.Inaddition,whenthe D-conditionisnotsatisﬁed,theapproximatecontrollabilityof(1)–(2)failsforeveryT

>

0.

Toprovethistheoremweconsidertheauxiliaryadjointsystem

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

pi,tt

−

^pi,xx

=

⁰ ⁱⁿ

R× (

0

,

i

),

i

=

¹

, ...,

n p_i

(·,

i

) =

⁰ ⁱⁿ

R,

ⁱ

=

¹

, ...,

n p_i

( · ,

0

) =

^pj

( · ,

0

)

in

R ,

i

,

j

=

¹

, ...,

n p1,x

(·,

0

) + · · · +

pn,x

(·,

0

) =

0 in

R

p_i

(

T

, · ) =

^p⁰_i

,

p_i_,_t

(

T

, · ) =

^p¹_i ⁱⁿ

(

0

,

_i

),

i

=

¹

, ...,

n

,

(4)

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

z_i_,_tt

−

^zi,xx

=

⁰ ⁱⁿ

R× (

0

,

_i

)

i

=

¹

, ...,

n

z

(·,

i

) =

⁰ ⁱⁿ

R,

ⁱ

=

¹

, ...,

n

z_i

(·,

⁰

) =

^zj

(·,

⁰

)

in

R,

ⁱ

,

j

=

¹

, ...,

n z1,x

( · ,

0

) + · · · +

^zn,x

( · ,

0

) =

^pi

( · ,

0

)

in

R

z_i

(

T

, ·) =

^z⁰_i ^zi,t

(

T

, ·) =

^z¹_i ⁱⁿ

(

0

,

i

),

i

=

¹

, ...,

n

.

(5)

System(4)–(5)iswellposedfor

¯

p⁰

,

^p

¯

¹

∈

^H

×

^V^,

¯

z⁰

, ¯

^z¹

∈

^V

×

^H^and

( ¯

^z

,

^p

¯ ) ∈

^C

( [

⁰

,

T

] ,

V

×

^H

) ∩

^C¹

( [

⁰

,

T

] ,

H

×

^V

)

for anyT

>

0.

Itisknown(see[6,p.142]or[5,p.15])thattheeigenvalueproblemassociatedwith(4)

−

^wi,xx

= μ

w_i

,

w_i

∈

^H²

(

0

,

_i

),

w_i

(

0

) =

^wj

(

0

),

w1,x

+ · · · +

^wn,x

(

0

) =

⁰

,

w_i

(

_i

) =

⁰

,

i

,

j

=

¹

, ...,

n

,

has a positive unbounded increasing sequence ofeigenvalues

( μ

k

)

_k_∈N andthat the corresponding eigenfunctions ^w

¯

k

:=

(

w₁_,_k

, ...,

w_n_,_k

)

,k

∈

N^may^be^chosen^so^that^the^sequence

(

^w

¯

k

)

k∈N isanorthonormalbasisofH.Further,denote

λ

k

:= √ μ

k. InthespaceV_∗ ofalltheﬁnitelinearcombinationsoftheeigenfunctions

(

w

¯

k

)

_k_∈N,wedeﬁnethenorms

·

α for

α ∈

R as

ψ ¯

_α

:=

k∈N

λ

^α_k

ψ

_k

²

¹₂

,

for

ψ ¯ ∈

^V∗ givenby

ψ ¯ =

ψ

_k^w

¯

k,andVα asthecompletionof V_∗ withrespectto

·

α.Thespaces V, H andVcan be identiﬁedwithV¹,V⁰andV⁻¹,respectively.

Besides,thesolutionto(4)withinitial(ﬁnal)data

¯

p⁰

=

k∈N

p⁰^,^kw

¯

_k

,

p

¯

¹

=

k∈N p¹^,^kw

¯

_k

att

=

^t0maybewrittenas

¯

p

(

t

) =

k∈N

p⁰^,^kcos

λ

_k

(

t

−

^t0

) +

^p¹^,^k

λ

_k ^sin

λ

_k

(

t

−

^t0

)

¯

w_k andits

α

-energy

(4)

E^α_p_¯

(

t

) := ¯

^p

(

t

)

²_α₊₁

+ ¯

^pt

(

t

)

²_α

=

k∈N

λ

_kp⁰^,^k

2

+

p¹^,^k

2

(λ

_k

)

²^α

isconserved.

For

(θ

1

, θ

2

) ∈

^H

×

^V^and

(ϑ

1

, ϑ

2

) ∈

^V

×

^H^we^denote

(θ

1

, θ

2

), (ϑ

1

, ϑ

2

) := θ

1

, ϑ

2

H

+ θ

2

, ϑ

1

V,V. Proposition2.ForanyT

>

0andanycontrolsv

¯ ∈

L²

(

0

,

T

)

n

thefollowingdualityidentityholdsforthesolutionsto(1)–(2)and (4)–(5)

n i=1

T 0

vi

(

t

)

zi,x

(

t

,

i

) =

( ¯

y

(

T

), − ¯

yt

(

T

)) , ¯

z⁰

, ¯

z¹

+

¯

y⁰

, ¯

y¹

, (¯

z

(

T

), − ¯

zt

(

T

))

+

(6)

+

¯

p⁰

,

^p

¯

¹

, (¯

^q

(

T

), − ¯

^qt

(

T

)) +

(

^p

¯ (

T

), − ¯

^pt

(

T

)) ,

^q

¯

⁰

,

^q

¯

¹

.

Fromidentity(6),wecanprove Proposition3.Forevery

¯

y⁰

, ¯

^y¹

∈

^H

×

^V^,

¯

q⁰

,

^q

¯

¹

∈

^V

×

^{H the}^sets

(¯

q

(

T

),

q

¯

_t

(

T

)) : ¯

^v

∈

L²

(

0

,

T

)

_n

,

(

y

¯ (

T

),

y

¯

_t

(

T

)) : ¯

^v

∈

L²

(

0

,

T

)

_n

aredenseinV

×

^{H and}^H

×

^Vrespectively,ifandonlyif,thefollowinguniquecontinuationpropertyholdsforallsolutionsto(4)–(5) with

¯

p⁰

,

p

¯

¹

∈

^H

×

^V^,

¯

z⁰

, ¯

z¹

∈

^V

×

^{H :}

z_i_,_x

(

t

,

i

) =

⁰

,

i

=

¹

, ...,

n

,

a.e. in

(

0

,

T

)

implies

p

¯

⁰

,

p

¯

¹

=

¯

z⁰

, ¯

z¹

= (

0

¯ ,

0

¯ ).

(7)

From Proposition 3,itfollowsthat theapproximatecontrollabilityof(1)–(2)for

¯

y⁰

, ¯

^y¹

∈

^H

×

^V^,

¯

q⁰

,

^q

¯

¹

∈

^V

×

^H ^is equivalenttotheuniquecontinuationproperty(7).

Proposition4.Thefollowingholds:

1)theuniquecontinuationproperty(7)holdswithT

≥

^T0ifandonlyifthelengths

1

, ...,

nofthestringssatisfytheD-condition;

2)ifthelengths

1

, ...,

nofthestringsdonotsatisfytheD-conditiontheuniquecontinuationproperty(7)failsforanyT

>

0.

Proof. 1)Recallthe followingfactderivedfromtheD’Alembertformula(see[5,p. 26]):ifu

(

t

,

x

)

solvesthewaveequation utt

−

^uxx

=

^{0 in}R

× (

0

, )

andsatisﬁestheDirichletboundarycondition u

(

t

, ) =

^{0 for}^t

∈ (

t1

,

t2

)

witht2

−

^t1

≥

²

then,for everyt

∈ (

t1

+ ,

t2

− )

,

u_x

(

t

,

0

) =

¹

2

(

u_x

(

t

+ , ) +

^ux

(

t

− , )) .

(8)

Nowassumethat zi,x

(

t

,

i

) =

0 foreveryi

=

1

, ..,

n,a.e.in

(

0

,

T

)

.Applying(8)ineachstring,weconcludethat

¯

z

(

t

) = ¯

0 in

[

∗

,

T₀

−

_∗

]

^.Consequently,p_i

(

t

,

0

) =

^{0 for}^every ⁱ

=

¹

, ...,

n,in

[

∗

,

T₀

−

_∗

]

^and^system⁽⁴⁾^becomes

⎧ ⎨

⎩

p_i_,_tt

−

^pi,xx

=

⁰ ⁱⁿ

[

∗

,

T₀

−

_∗

] × (

0

,

i

),

i

=

¹

, ...,

n p_i

( · ,

0

) =

^pi

( · ,

_i

) =

⁰ ⁱⁿ

[

_∗

,

T0

−

_∗

] ,

i

=

¹

, ...,

n

p1,x

(·,

0

) + · · · +

pn,x

(·,

0

) =

0 in

[

_∗

,

T0

−

_∗

].

If the lengths

1

, ...,

n satisfy the D-condition, we can apply the unique continuation property associated with the simultaneous controlofn strings proved in[4]toconcludethat p

¯ (

t

) = ¯

^{0 in}

[

_∗

,

T0

−

_∗

]

^.^Then, ^as^the

( −

¹

)

-energyof p

¯

is conserved,

¯

p⁰

,

p

¯

¹

= (

0

¯ ,

0

¯ )

.Plugging this informationinto (5), the 0-energy of

¯

z is also conserved. Thus,

¯

z

(

t

) = ¯

^{0 in}

[

_∗

,

T0

−

_∗

]

^implies

¯

z⁰

, ¯

z¹

= (

0

¯ ,

0

¯ )

andtheuniquecontinuationproperty(7)istrue.

2) AssumetheD-conditionis notveriﬁed andtake ^p

¯ = ¯

^{0 and} ^zi

=

^{0 for} ⁱ

=

^s

,

r with

s

/

r

∈

Q^.^Choose^m

,

n

∈

N^such that

s

/

r

=

ⁿ

/

manddeﬁne

z_s

(

t

,

x

) =

smcos

(

ⁿ

π(

^T

−

^t

)

s

)

sin

(

ⁿ

π

^x

s

),

z_r

(

t

,

x

) = −

rncos

(

^m

π(

^T

−

^t

)

r

)

sin

(

^m

π

^x

r

).

Clearly,

z_s_,_x

(

t

,

0

) +

^zr,x

(

t

,

0

) =

^mn

π

cos

(

ⁿ

π(

^T

−

^t

)

s

) −

^cos

(

^m

π(

^T

−

^t

)

r

)

=

⁰

(5)

and then, z1,x

(

t

,

0

) + · · · +

^zn,x

(

t

,

0

) =

^{0 for} ^any^t

∈

R^. ^Besides, ^since ^zr

(

T

,

x

) =

^sin

(

ⁿ^π^x

r

)

,

¯

z⁰

= ¯

^0. Consequently, p,

¯ ¯

z are solutionsto(4)–(5)forwhichtheuniquecontinuationproperty(7)isfalsewithanyvalue T

>

0.

Now,Theorem 1isaconsequenceofthePropositions 3 and4.

3. Desensitizingcontrollability

Asmentionedbefore,theapproximatecontrollabilityofsystem(1)–(2)impliestheapproximatedesensitizingcontrolla- bilityofthefunctional

deﬁnedby (3)alongthesolutions to(1),thus,inview ofTheorem 1,itholdsfor T

≥

^T0 ifthe lengthsofthestringssatisfytheD-condition.However,thedesensitizingcontrollabilityisimpliedbyanapparentlyweaker version oftheuniquecontinuationproperty(7).Itisenoughthat,foreverysolutionto(4)–(5)withﬁnaldata p

¯

⁰

,

p

¯

¹

∈

^V∗,

¯

z⁰

, = ¯

^z¹

= ¯

^0,

zi,x

(

t

,

i

) =

0

,

i

=

1

, ...,

n

,

a.e. in

(

0

,

T

)

imply

p

¯

⁰

,

p

¯

¹

= (

₀

¯ ,

₀

¯ ),

(9)

toguaranteethedesensitizingcontrollability.

Usingthetechniquedevelopedin[5,Chapter5],thelatteruniquecontinuationpropertymaybequantiﬁedbymeansof thefollowingweightedobservabilityinequality.

Theorem5.LetT

≥

^T0.Then,thereexistsapositiveconstantC suchthat,foranyﬁnaldata^p

¯

⁰

=

p⁰^,^k^w

¯

k,^p

¯

¹

=

p¹^,^k^w

¯

k,^z

¯

⁰^,^z

¯

¹^, thatbelongtoV_∗,thecorrespondingsolutionto(4)–(5)satisﬁes

C

n

i=1

T 0

zi,x

(

t

,

i

)

2

dt

≥

k∈N

p⁰^,^k

2

+ λ

⁻_k¹p¹^,^k

2

ω

²_k

,

where

ω

k

=

n

i=1

sin

(λ

k

i

).

(10)

Besides,allthenumbers

ω

karedifferentfromzerowheneverthelengthsofthestringssatisfytheD-condition.Therefore, Theorem 5impliestheuniquecontinuationproperty(9).Unfortunately,foranyvaluesofthelengths

1

, ...,

n,thesequence

( | ω

k

| )

_k_∈N is notuniformlyboundedfrombelowwithapositive bound,so(10) leadsto theobservationofnorms weaker than

·

H×V.

Additionally, basedontheHilbertUniquenessMethod(J.-L.Lions[7]),Theorem 5allowsustogiveanalternative,more precisecharacterizationoftheapproximatedesensitizingcontrollabilityofsystem(1):

Corollary6.If

1

, ...,

nsatisfytheD-condition,thefunctional

Φ

deﬁnedby(3)canbedesensitizedalonganysolutionto(1)with initialdata

¯

^y⁰

, ¯

^y¹^that^are^ﬁnite^linearcombinationsoftheeigenfunctions

(

^w

¯

k

)

k∈N.

Remark7.AlltheresultsinthisNoteremaintruewhenn

=

^1.^We^have^excluded^this^value^ofⁿ^to^guarantee^that^the^terms multipleandsimplenodesmakesense.

Remark8.Asfar aswe know,desensitizingcontrollabilityproblemsonnetworksofstrings havenot beenstudiedbefore.

A similarapproach,basedonthetechnique developedin[3]andinchapters4and5of[5],should workaswellwhena smallernumberofcontrollersisusedorwhenthenetworkissupportedonatree-shapedgraph.

Acknowledgement

ThisworkwaspartiallysupportedbyProjectMTM2013-42907DGICT,Spain.

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