farhat shel - univ-smb.frgarnier/expose_faraht.pdf · introduction abstract setting exponential...

IntroductionAbstract setting

Exponential stability

Exponential stability of a networks of beams

Farhat Shel

Faculte des Sciences de Monastir

42e Congres National d’Analyse Numerique (CANUM 2014)Carry-le-Rouet Marseille

31 mars - 04 avril

Farhat Shel Exponential stability of a networks of beams



Model

elastic

thermoelastic

a1 e1

a3

a2a4

StabilityFarhat Shel Exponential stability of a networks of beams



Some definitions

A system is said to be

strongly stable if E (t) −→ 0,

polynomially stable if E (t) ≤ CE(0)tα ∀t > 0,

exponentially stable if E (t) ≤ CE (0)e−αt ∀t > 0.




Some definitions

A system is said to be

strongly stable if E (t) −→ 0,

polynomially stable if E (t) ≤ CE(0)tα ∀t > 0,

exponentially stable if E (t) ≤ CE (0)e−αt ∀t > 0.

à Frequency Domain Method

We will write the system (S) as first order evolution equation on aHilbert space H, {

d

dty = Ay ,

y(0) = y0,(1)




Stability

exponential stability ⇐⇒ S(t) = eAt is exponentially stable:

‖S(t)‖ ≤ Ce−wt ∀t > 0.




Outline

1 Introduction

2 Abstract setting

3 Exponential stability




First examples

Transversal displacement in a string

u(x,t)0 ℓ{

utt(x , t)− uxx(x , t) = 0,u(0, t) = 0, u(`, t) = 0

Transversal displacement in a beam

x

u(x,t)

0 ℓ{utt(x , t) + uxxxx(x , t) = 0,u(0, t) = 0, ux(`, t) = 0, M(0, t) = 0, Q(`, t) = 0.




First examples


u(x,t)0 ℓ

f(x,t): frictional force

{utt(x , t)− uxx(x , t)+f (x , t) = 0,u(0, t) = 0, u(`, t) = 0.


x

u(x,t)

0 ℓ{utt(x , t) + uxxxx(x , t) = 0,u(0, t) = 0, ux(`, t) = 0, M(0, t) = 0, Q(`, t) = 0.




First examples


u(x,t)0 ℓ




x

u(x,t)

0 ℓ

feedback

{utt(x , t) + uxxxx(x , t) = 0,u(0, t) = 0, ux(`, t) = 0, M(0, t) = 0, Q(`, t) = ut(`, t).

Polynomial or exponential stability.




First examples


u(x,t)0 ℓ




x

u(x,t)

0 ℓ

feedback

{utt(x , t) + uxxxx(x , t) = 0,u(0, t) = 0, ux(`, t) = 0, M(0, t) = 0, Q(`, t) = ut(`, t).

Polynomial or exponential stability.Farhat Shel Exponential stability of a networks of beams



More general examples

star or tree of stringsAmmari, Jellouli, Khenissi, 2005. Valein, Zuazua, 2009.

star or tree of beamsAmmari, 2007. Wang, Guo, 2008. Han, Q. Xu, 2010 .

strings and beamsAmmari et al, 2012.




Thermoelastic material

Thermoelastic string (Fourier’s law) Heat

utt − uxx + βθx = 0, in (0, `)× (0,∞),

θt + βuxt − κθxx = 0, in (0, `)× (0,∞),

Thermoelastic beam (Fourier’s law)

utt + uxxxx − γθxx = 0 in (0, `)× (0,+∞),

θt − θxx + γutxx = 0 in (0, `)× (0,+∞),

Exponential decay rate.




Transmission problem

Elastic Thermoelastic

or

ElasticThermoelastic Thermoelastic


Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.Revera, Oquendo, 2001.






or



Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.

Revera, Oquendo, 2001.






or



Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.Revera, Oquendo, 2001.




Model

elastic

thermoelastic

a1 e1

a3

a2a4




Associeted system SNetwork G: of N edges e1, ..., eN and n vertices a1, ..., an.Each edge ej is a curve, parameterized by

πj : [0, `j ] −→ ej , xj 7−→ πj(xj).

Sometimes we identify ej with the interval (0, `j).Incidence matrix: D = (dkj)n×N :

dkj =

−1 if πj(`j) = ak ,1 if πj(0) = ak ,0 otherwise.

Vext : boundary (external) vertices, Vext 6= ∅.Vint : internal vertices,I (ak): indices of edges incident to ak .Ie , Ite , Ie(ak), Ite(ak).




Equations

Every thermoelastic edge ej satisfies the following equations:

ujtt + ujxxxx − γjθjxx = 0 in (0, `j)× (0,+∞), (2)

θjt − θjxx + γjujtxx = 0 in (0, `j)× (0,+∞), (3)

with

uj(x , 0) = uj0(x), ujt(x , 0) = uj1(x), θj(x , 0) = θj0(x).

Every elastic edge ej satisfies the following equation:

ujtt − ujxxxx = 0 in (0, `j)× (0,+∞), (4)

withuj(x , 0) = uj0(x), ujt(x , 0) = uj1(x). (5)




Boundary and transmission conditions,at ak ∈ Vext

uj(ak , t) = 0, j ∈ I (ak),

ujxx(ak , t) = 0, j ∈ I (ak),

at ak ∈ V ′ext ,

θj(ak , t) = 0, j ∈ Ite(ak),

at ak ∈ Vint ,

uj(ak , t) = ul(ak , t) j , l ∈ I (ak),

θj(ak , t) = θl(ak , t) j , l ∈ Ite(ak),

ujxx(ak , t) = ulxx(ak , t) j , l ∈ I (ak),




∑j∈I (ak )

dkjujx(ak , t) = 0,

∑j∈Ite(ak )

dkj(ujxxx(ak , t)− γjθjx(ak , t)

)+

∑j∈Ie(ak )

dkjujxxx(ak , t) = 0,

∑j∈Ite(ak )

dkj

(γju

jxt(ak , t)− θjx(ak , t)

)= 0.




Dissipative system

Energy

E (t) =1

2

N∑j=1

∫ `j

0

(∣∣∣ujt∣∣∣2 +∣∣ujxx ∣∣2) dx +

1

2

∑j∈Ite

∫ `j

0

∣∣θj ∣∣2 dx .d

dt(E (t)) = −

∑j∈Ite

∫ `j

0

∣∣θjx ∣∣2 dx ≤ 0.

The system is dissipative

Exponential stability ?




Notations

For j in Ie ,let γj = 0, Vj = {0}, V k

j = {0}, k = 1, 2.

For j in Ite ,let Vj = L2(0, `j) and V k

j = Hk(0, `j), k = 1, 2.

L2(G) =N∏i=1

L2(0, `j), V (G) =N∏j=1

Vj , Vk(G) =

N∏j=1

V kj




Energy space

F (G) = {f = (f 1, ..., f N) ∈ H2(G) satisfying (6)-(8)},

f j(ak) = f l(ak) j , l ∈ I (ak), ak ∈ Vint , (6)

f j(ak) = 0 j ∈ I (ak), ak ∈ Vext , (7)∑j∈I (ak )

dkj∂x fj(ak) = 0 j ∈ I (ak), ak ∈ Vint . (8)

Energy space:H = F (G)× L2(G)× V (G),

〈y1, y2〉H =N∑j=1

(∫ `j

0∂xx f

j1 (x)∂xx f

j2 (x)dx +

∫ `j

0g j

1(x)g j2(x)dx

+

∫ `j

0hj1(x)hj2(x)dx

): Hilbert.

.Farhat Shel Exponential stability of a networks of beams



Operator

A : D(A) ⊆ H → H,

A =

0 A01 0

−A41 0 A2

γ

0 −A2γ A2

1

,

where Akγ = diag(γ1∂

kx , ..., γN∂

kx ), k ∈ N, and ∂0

x = I , and whosedomain is given by

D(A) ={

(u, v , θ) ∈(F (G) ∩ H4(G)

)× F (G)× V 2(G)

satisfying (9)-(14) below}




Operator

∂2xu

j(ak) = ∂2xu

l(ak), j , l ∈ I (ak), ak ∈ Vint , (9)

∂2xu

j(ak) = 0, j ∈ I (ak), ak ∈ Vext , (10)

θj(ak) = θl(ak), j , l ∈ Ite(ak), ak ∈ Vint , (11)

θj(ak) = 0, j ∈ I (ak), ak ∈ V ′ext , (12)

∑j∈I (ak )

dkj(∂3xu

j(ak)− γj∂xθj(ak))

= 0, ak ∈ Vint , (13)

∑j∈I (ak )

dkj(γj∂xv

j(ak)− ∂xθj(ak))

= 0, ak ∈ Vint , (14)




Evolution equation

Then the system (S) may be rewritten as the first order evolutionequation on H,

{d

dty(t) = Ay(t), t > 0,

y(0) = y0

(15)

where y = (u, ut , θ), y0 = (u0, u1, θ0).




A is a dissipative operator on H.

Moreover, by the Lax-Milgram’s lemma (complex version),

1 ∈ ρ(A): the resolvent set of A, and (I − A)−1 is compact.




Lemma (Lumer-Phillips theorem)

B is the generator of a C0-semi-group of contraction if and only ifB is m-dissipative.

then

Theorem

The operotor A generates a C0-semigroup S(t) = eAt ofcontraction on H.

For an initial datum y0 ∈ H there exists a unique solution

y ∈ C ([0,+∞),H)

of the Cauchy problem (15).Moreover if y0 ∈ D(A), then

y ∈ C ([0,+∞),D(A)) ∩ C 1([0,+∞),H).






then

Theorem



y ∈ C ([0,+∞),H)

of the Cauchy problem (15).

Moreover if y0 ∈ D(A), then

y ∈ C ([0,+∞),D(A)) ∩ C 1([0,+∞),H).






then

Theorem



y ∈ C ([0,+∞),H)

of the Cauchy problem (15).Moreover if y0 ∈ D(A), then

y ∈ C ([0,+∞),D(A)) ∩ C 1([0,+∞),H).





elastic

thermoelastic

a1 e1

a3

a2a4





a1 e1a2

a4





a1 e1

a3

a2a4





a3

a2a4





a1 e1

a3

a2a4





Theorem

The semigroup S(t), generated by the operator A is exponentiallystable.




Proof

We will use the frequency domain characterization due toGearhard Pruss and Huang,

Lemma [Gearhard-Pruss-Huang]

A C0-semigroup of contraction etB is exponentially stable if, andonly if,

iR = {iβ | β ∈ R} ⊆ ρ(B) (16)

andlim sup|β|→∞

∥∥(iβ − B)−1∥∥H <∞. (17)

© Note that, etB is strongly stable if, and only if, it satisfies (16).




I The operator A satisfies condition (16).

I The operator A satisfies condition (17).Suppose that (17) is not true, then there exists a sequence(βn) of real numbers, with |βn| −→ +∞ and a sequence ofvectors (yn) = (un, vn, θn) in D(A) with ‖yn‖H = 1, such that

‖(iβn −A)yn‖H −→ 0.

We prove that this condition yields the contradiction‖yn‖H −→ 0 as n −→ 0.




iβnujn − v jn = f jn −→ 0, in H2(0, `j), (18)

iβnvjn + ∂4

xujn − γj∂2

xθjn = g j

n −→ 0, in L2(0, `j), (19)

iβnθjn − ∂2

xθjn + γj∂

2xv

jn = hjn −→ 0, in L2(0, `j). (20)

Step 1 : we prove that θjn → 0 in H1(0, `j), ∂2xu

jn → 0 and v jn → 0

in L2(0, `j) for j in Ite .

Step 2 : we prove that ∂2xu

jn → 0 and v jn → 0 in L2(0, `j) for j in Ie .

Let j in {1, ...,N}. Combining (18) and (19) we obtain

−β2nu

jn + ∂4

xujn − γj∂2

xθjn = g j

n + iβnfjn . (21)




Let q(x) = x or `j − x on [0,`j ] and taking the inner product in

L2(0, `j) of (21) with q∂xujn :

−1

2β2n

[∣∣ujn(x)∣∣2 q(x)

]`j0− 1

2

[∣∣∂2xu

jn(x)

∣∣2 q(x)]`j

0

+Re

([∂3xu

jn(x)q(x)∂xu

jn(x)

]`j0

)+[Re(iβnf

jn (x)q(x)ujn(x)

)]`j0

−γjRe([∂xθ

jn(x)q(x)∂xu

jn(x)

]`j0

)+

1

2

∫ `j

0β2n

∣∣ujn∣∣2 ∂xqdx+

3

2

∫ `j

0

∣∣∂2xu

jn

∣∣2 ∂xqdx + Re

([∂2xu

jn(x)∂xu

jn(x)∂xq(x)

]`j0

)−→ 0.

(22)




I If ej is thermoelastic, the last term in (22) converge to zero

and for every interior end ak of ej , β1/2n ∂xu

jn(ak), ∂

3xu

jn(ak )

β1/2n

,

βjnujn(ak) and ∂2

xujn(ak) tend to zero.

• Let j in {1, ...,N} and ak an inner node of ej

1

β1/2n

dkj∂3xu

jn(ak)+ε∂2

xujn(ak)+β

1/2n dkj∂xu

jn(ak)+εβnu

jn(ak)→ 0.

(23)with ε ∈ {−1, 1}.




I For every elastic edge ej attached to only thermoelastic edgesat an internal node ak we have,∫ `j

0

(∣∣∂2xu

jn(x)

∣∣2 + 3β2n

∣∣ujn(x)∣∣2) dx → 0.

and β1/2n ∂xu

jn(as), ∂

3xu

jn(as)

β1/2n

, βjnujn(as) and ∂2

xujn(as) tend to

zero. where as is the second end of ej .

I We iterate such procedure in each maximal subgraph ofelastic edges of G




Comments

If we replace the continuity condition of θ at inner nodes,

θj(ak) = θl(ak) j , l ∈ Ite(ak), ak ∈ Vint

and the condition∑j∈Ite(ak )

dkj(γjuxt(ak , t)− θjx(ak , t)) = 0, ak ∈ Vint

by the following

θj(ak) = 0 j ∈ Ite(ak), ak ∈ Vint

and Kirchhoff’s law,∑j∈Ite(ak )

dkjθjx(ak , t) = 0, ak ∈ Vint

then we obtain the same results.Farhat Shel Exponential stability of a networks of beams



Furthermore, If we consider the following boundary conditions

uj(ak , t) = 0, ujx(ak , t) = 0, j ∈ I (ak), ak ∈ Vext ,θj(ak , t) = 0, j ∈ I (ak), ak ∈ V ′ext ,

and for ak in Vint ,

uj(ak , t) = ul(ak , t) j , l ∈ I (ak),

θj(ak , t) = θl(ak , t) j , l ∈ Ite(ak),

ujx(ak , t) = ulx(ak , t) j , l ∈ I (ak),




∑j∈Ite(ak )

dkj(ujxx(ak , t)− γjθj(ak , t)

)= 0,

∑j∈Ite(ak )

dkj(ujxxx(ak , t)− γjθjx(ak , t)

)+

∑j∈Ie(ak )

dkjujxxx(ak , t) = 0,

∑j∈Ite(ak )

dkjθjx(ak , t) = 0.

we can prove that the energy of the system decay exponentially tozero.


Perspectives

1 More general cases.

2 Infinite cardinal of edges or infinite lengths of edges.

THANKS!


Heat

θt + γqx = 0

Fourier

q + κθx = 0 I θt − γκθxx = 0.

Cattaneo

τqt + q + κθx = 0 I τθtt + θt − γκθxx = 0.

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