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
IntroductionAbstract setting
Exponential stability
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Model
elastic
thermoelastic
a1 e1
a3
a2a4
StabilityFarhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Stability
exponential stability ⇐⇒ S(t) = eAt is exponentially stable:
‖S(t)‖ ≤ Ce−wt ∀t > 0.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Outline
1 Introduction
2 Abstract setting
3 Exponential stability
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Outline
1 Introduction
2 Abstract setting
3 Exponential stability
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Outline
1 Introduction
2 Abstract setting
3 Exponential stability
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
First examples
Transversal displacement in a string
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.
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
First examples
Transversal displacement in a string
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.
Transversal displacement in a beam
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
IntroductionAbstract setting
Exponential stability
First examples
Transversal displacement in a string
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.
Transversal displacement in a beam
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
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Transmission problem
Elastic Thermoelastic
or
ElasticThermoelastic Thermoelastic
Exponential decay rate.
Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.Revera, Oquendo, 2001.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Transmission problem
Elastic Thermoelastic
or
ElasticThermoelastic Thermoelastic
Exponential decay rate.
Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.Revera, Oquendo, 2001.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Transmission problem
Elastic Thermoelastic
or
ElasticThermoelastic Thermoelastic
Exponential decay rate.
Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.
Revera, Oquendo, 2001.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Transmission problem
Elastic Thermoelastic
or
ElasticThermoelastic Thermoelastic
Exponential decay rate.
Marzocchi, Rivera, Nazo, 2002. Racke, Revera, Sare, 2008.Revera, Oquendo, 2001.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Model
elastic
thermoelastic
a1 e1
a3
a2a4
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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).
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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),
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
∑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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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 ?
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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 ?
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
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
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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
IntroductionAbstract setting
Exponential stability
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}
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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).
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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).
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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).
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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).
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Exponential stability
elastic
thermoelastic
a1 e1
a3
a2a4
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Exponential stability
a1 e1a2
a4
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Exponential stability
a1 e1
a3
a2a4
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Exponential stability
a3
a2a4
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Exponential stability
a1 e1
a3
a2a4
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
Exponential stability
Theorem
The semigroup S(t), generated by the operator A is exponentiallystable.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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).
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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)
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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}.
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
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
IntroductionAbstract setting
Exponential stability
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),
Farhat Shel Exponential stability of a networks of beams
IntroductionAbstract setting
Exponential stability
∑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.
Farhat Shel Exponential stability of a networks of beams
Perspectives
1 More general cases.
2 Infinite cardinal of edges or infinite lengths of edges.
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Farhat Shel Exponential stability of a networks of beams
Farhat Shel Exponential stability of a networks of beams
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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Farhat Shel Exponential stability of a networks of beams