Well Posedness for a Class of

Volume 2009, Article ID 358329,13pages doi:10.1155/2009/358329

Research Article

Well Posedness for a Class of

Flexible Structure in H ¨older Spaces

Claudio Cuevas

and Carlos Lizama

1Departamento de Matem´atica, Universidade Federal de Pernambuco, Av. Prof. Luiz Freire, S/N, Recife-PE, CEP. 50540-740, Brazil

2Departamento de Matem´atica, Facultad de Ciencias, Universidad de Santiago de Chile, Casilla 307-Correo 2, Santiago-Chile, Chile

Correspondence should be addressed to Carlos Lizama,[email protected] Received 4 December 2008; Accepted 6 April 2009

Recommended by J. Rodellar

We characterize well-posedness in H ¨older spaces for an abstract version of the equation∗u λu c²ΔuμΔu f which model the vibrations of flexible structures possessing internal material damping and external forcef. As a consequence, we show that in case of the Laplacian with Dirichlet boundary conditions, equation∗is always well-posed provided 0< λ < μ.

Copyrightq2009 C. Cuevas and C. Lizama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

During the last few decades, the use of flexible structural systems has steadily increased importance. The study of a flexible aerospace structure is a problem of dynamical system theory governed by partial differential equations.

We consider here the problem of characterize well posedness, for a mathematical model of a flexible space structure like a thin uniform rectangular panel. For example, a solar cell array or a spacecraft with flexible attachments. This problem is motivated by both engineering and mathematical considerations.

Such mechanical system was mathematically introduced in1 and consists of a short rigid hub, connected to a flexible panel of lengthl. Control torqueQtis applied to the hub.

The panel is made of viscoelastic material with internal Voigt-type damping with coefficient μ, that is, an ideal dashpot damping which is directly proportional to the first derivative of the longitudinal displacement, and opposing the direction of motion. The equation of motion of the panel is given by

u c²

ΔuμΔu

, 1.1

where c is the velocity of longitudinal wave propagation, c² Dp/ρJp, and Dp, ρ, Jp

are, respectively, torsional rigidity, density and radius of gyration about the central axis of the panel. Initial position and deflection angle are known. In1 exact controllability and boundary stabilization for the solution of 1.1was analyzed and in2, p. 188 , the exact decay rate was obtained.

More generally, the study of vibrations of flexible structures possessing internal material damping is modeled by an equation of the form

uλu^”’c²

ΔuμΔu

, 0< λ < μ, 1.2

in a bounded domainΩinRⁿwith smooth boundaryΓ, see3,4 .

In4 the explicit exponential energy decay rate was obtained for the solution of1.2 subject to mixed boundary conditions. However, consideration of external forces interacting with the system, which lead us naturally with the well posedness for the nonhomogeneous version of1.2, appears as an open problem.

In the first part of this paper we study well posedness of the following abstract version of1.2:

ut λut c²Aut c²μAut ft, 0< λ < μ, 1.3 whereAis a closed linear operator acting in a Banach spaceXandfis aX-valued function.

We emphasize that whenA Δ in general one cannot expect that1.3 is well posed due to the presence of the term u. In fact, it is well known that the abstract Cauchy problem associated with1.3is in general ill posed, see for example5 .

We are able to characterize well posedness, that is, temporal maximal regularity, of solutions of 1.3 solely in terms of boundedness of the resolvent set of A. This will be achieved in the H ¨older spacesC^αR, X,where 0< α <1.The methods to obtain this goal are those incorporated in6 where a similar problem in case of the first order abstract Cauchy problem has been studied.

2. Preliminaries

LetX, Y be Banach spaces, we writeBX, Yfor the space of bounded linear operators from XtoYand let 0< α <1.We denote by ˙C^αR, Xthe spaces

C˙^αR, X

f:R → X:f0 0,f

α<∞

2.1

normed by

f

αsup

t /s

ft−fs

t−s^α . 2.2

LetΩ⊂Rbe an open set. ByC^∞_c Ωwe denote the space of allC^∞-functions inΩ⊆Rhaving compact support inΩ.

We denote byFforfthe Fourier transform, that is, Ff

s:fs :

Re^−istftdt 2.3

s∈R, f ∈L¹R;X.

Definition 2.1. LetM:R\ {0} → BX, Ybe continuous. We say thatMis a ˙C^α-multiplier in BX, Yif there exists a mappingL: ˙C^αR, X → C˙^αR, Ysuch that

R

Lf s

Fφ sds

R

F

φ·M

sfsds 2.4

for allf∈C^αR, Xand allφ∈C^∞_cR\ {0}.

HereFφ·Ms _Re^−istφtMtdt∈ BX, Y.Note thatLis well defined, linear and continuouscf.6, Definition 5.2 .

Define the spaceC^αR, Xas the set C^αR, X

f :R → X:f

C^α <∞

2.5 with the norm

f

C^α f

αf0. 2.6

LetC^αkR, X wherek is a positive integerbe the Banach space of allu ∈ C^kR, Xsuch thatu^k∈C^αR, X, equipped with the norm

u_C^αk u^k

C^αu0. 2.7

Observe from Definition2.1and the relation

R

F

φM

sds2π φM

0 0, 2.8

that forf ∈C^αR, Xwe haveLf ∈C^αR, X. Moreover, iff∈C^αR, Xis bounded thenLf is bounded as wellsee6, Remark 6.3 . The following multiplier theorem is due to Arendt, Batty and Bu6, Theorem 5.3 .

Theorem 2.2. LetM∈C²R\ {0},BX, Ybe such that

sup

t /0

Mtsup

t /0

tMtsup

t /0

t²Mt<∞. 2.9 Then M is a ˙C^α-multiplier.

Remark 2.3. IfXisB-convex, in particular ifXis aUMDspace, Theorem2.2remains valid if condition2.2is replaced by the following weaker condition:

sup

t /0

Mtsup

t /0

tMt<∞, 2.10

whereM∈C¹R\ {0},BX, Y cf.6, Remark 5.5 .

3. A Characterization of Well Posedness in H ¨older Spaces

In this section we characterizeC^α-well posedness. Givenf ∈ C^αR, X,we consider in this section the linear problem

ut aut bAut cAut ft, t∈R, 3.1

whereAis a closed linear operator inXanda, b, c >0.Note that the solution of3.1does not have to satisfy any initial condition. In the casea 0, solutions of3.1 with periodic boundary conditions has been recently studied in7 . On the other hand, well posedness of the homogeneous abstract Cauchy problem has been observed recently in8 fora0 and allb∈Cunder certain assumptions onA. See also9 for related maximal regularity results in the case of a damped wave equation.

We denote byDA the domain ofAconsidered as a Banach space with the graph norm.

Definition 3.1. We say that3.1isC^α-well posed if for eachf ∈C^αR, Xthere is a unique functionu∈C^α3R, X∩C^α1R,DA ∩C^αR,DA such that3.1is satisfied.

In the next proposition, as usual we denote by ρT, Rλ, T the resolvent set and resolvent of the operatorT, respectively.

Proposition 3.2. Assume that3.1isC^α-well-posed. Then

ilη:−η²1iaη/bicη∈ρA for allη∈Rand, iisup_η∈R||η³/bicηRlη, A||<∞.

Proof. Denote byL:C^αR, X → C^α3R, Xthe bounded operator which associates to each f∈C^αR, Xthe unique solutionuof3.1. Letη∈R.Letx∈DAbe such thatAx−lηx 0.Defineut e^iηtx.Then it is not difficult to see that uis a solution of 3.1withf ≡ 0.

Hence, by uniqueness,x0.

Lety∈Xand defineft e^iηty.LetuLf.For fixeds∈Rwe define

v1t uts, v2t e^iηsut. 3.2

Then is easy to check thatv1andv2are both solutions of3.1withf replaced bye^iηsf.By uniqueness,uts e^iηsutfor allt, s∈R.In particular, it follows thatus e^iηsu0for alls∈R.Letxu0∈DA.Replacingut e^iηtxin3.1we obtain

−η²−iaη³ ut

bicη

Aut e^iηty. 3.3

Takingt0 we conclude thatlη−Ais bijective and

ut 1

bicηR l

η , A

e^iηty. 3.4

Defineeηt e^iηtandeη⊗yt eηty.We have the identity||eη⊗x||αKα|η|^α||x||where Kα2 sup_t>0t^−αsint/2 see6, section 3 . Hence

Kαη^α

η³ bicηR

l η

, A y

eη⊗ η³

bicηRlη, Ay

α

u

α

≤ u_α3Lf_α3≤ Lf

α

≤ Lf

αf0Leη⊗y

αy

≤ L

Kαη^α1y.

3.5

Therefore, forε >0 we have sup

|^η|^>ε η³

bicηR l

η , A

y

≤ Lsup

|^η|^>ε

1 1

Kαη^α

y<∞. 3.6

On the other hand, since{1/bicη}_η∈Ris bounded andη → η³Rlη, Ais continuous at η0, we obtainiiand the proof is complete.

In what follows, we denote byid^kthe function:s → is^kfor alls∈R,andk∈N.As before, we also use the notation

ls:−s²1ias bics,

Ms: 1

bicsRls, A, ∀s∈R.

3.7

Lemma 3.3. Assume that

sup

s∈R

s³Ms<∞, 3.8 thenid²·Mandid³·Mare ˙C^α- multipliers inBX.MoreoverMandid·Mare ˙C^α-multipliers in BX, DA.

Proof. Defineκs : 1/bias. We first observe that the functionsθs : κs/κsand ϑs : ls/lshave the property thatsθs, sϑs, s²θsands²ϑsare bounded onR.

We next claim thatMis a ˙C^α-multiplier. In fact, note that by hypothesis sup_|s|>εMs<∞ for eachε >0,and the functions → Msis continuous att0 sinceb >0.HenceMsis bounded. Moreover, definingξs:ls/κs −s²−ias³we have

Ms θsMs−ϑsξsMs ², 3.9

wheresξsis of orders⁴ and thenQs: ξsMs ²is bounded by3.8. It follows that sMsis bounded. Next, we have the identity

s²Ms s²θsMs sθssMs−s²ϑsQs−s²ϑsQs. 3.10

where the first three terms on the right hand side are bounded. For the last term, we have s²ϑsQs sϑs ²Qs−sϑssθsQs 2sϑsξsMssMs. 3.11 It is clear that the first two terms on the right hand side are bounded. We observe that the last term also is bounded. In fact, note that by hypothesis sup_|s|>εξsMs < ∞for each ε > 0 and the function s → ξsMsis continuous ats 0.HenceξsMsis bounded.

This completes the proof of the Lemma.

Lemma 3.4. Let 0< α <1, k∈Nandu, v∈C^αR, X.The following assertions are equivalent:

iu∈C^αkR, Xandu^k−vis constant.

ii _RvsFϕsds _RusF id^k·ϕsdsfor allϕ∈ DR\ {0}.

Proof. i ⇒ ii. Let Φ ∈ DR{0}. Then _RvsFϕsds _Ru^ksFϕsds

−1^k _RusFϕ^ksds _RusFid^k·ϕsds.

ii⇒i. LetΦ∈ DR\ {0}andψs ϕs/s^k. Thenψ∈ DR\ {0}andFϕ Fψ^k. Let wt ^t₀t−s^k−1vsds. Then integration by parts and assumption give _RwsFϕsds

RusFϕsds. It follows from10, Theorems 4.8.2 and 4.8.1 thatw−uis a polynomial.

Since wt ≤ c1 |t|^αk it follows that ut wt y0 ty1 t²y2 · · ·t^k yk

t

0 t−s^k−1 vsdsy0ty1t² y2· · ·t^k ykfor some vectorsy0, y1, . . . , yk ∈X. Thus u^kvxfor some vectorx∈X.

The following theorem, which is one of the main results in this paper, shows that the converse of Proposition3.2is valid.

Theorem 3.5. LetA be a closed linear operator defined on a Banach spaceX. Then the following assertions are equivalent:

iEquation3.1isC^α-well posed;

iilη:−η²1iaη/bicη∈ρAfor allη∈Rand

sup

η∈R

η³ bicηR

l η

, A

<∞. 3.12

Proof. The implication i ⇒ ii follows by Proposition 3.2. We now prove the converse implication.

Let f ∈ C^αR, X. By Lemma3.3 there exists u1, u2 ∈ C^αR,DA and u3, u4 ∈ C^αR, Xsuch that

Ru1s Fφ1

sds

RF φ1·M

sfsds, 3.13

Ru2s Fφ2

sds

RF

φ2·id·M

sfsds, 3.14

Ru3s Fφ3

sds

RF

φ3·id²·M

sfsds, 3.15

Ru4s Fφ4

sds

RF

φ4·id³·M

sfsds 3.16

for all Φi ∈ C₀^∞R\ {0} i 1,2,3,4. Choosing Φ1 id · Φ2 in 3.13, it follows from Lemma3.4that u1∈C^1αR, Xand

u₁u2y1, 3.17

for somey1∈X.Now we can chooseφ2id·φ3in3.14, it follows thatu1 ∈C^α2R, Xand

u^”₁u3y2, 3.18

for somey2∈X.In a similar way, we can see thatu1∈C^α3R, Xand

u^”’₁ u4y3, 3.19

for somey3 ∈X.From the definition ofMswe obtainbicsls−AMs I. Taking into account the definition oflswe get−s²1ics−bicsA Ms I.Then we deduce the identity

is²Ms ais³Ms bAMs icsAMs I. 3.20

We multiply the above identity byφ, take Fourier transforms and then integrate overRafter taking the values atfs, we obtain

RF

φ·id²·M

sfsdsa

RF

φ·id³·M

sfsds b

RAF φ·M

sfsdsc

RAF

φ·id·M

sfsds

RF φ

sfsds 3.21

for allφ∈C^∞₀ R\ {0}.Using3.17,3.18and3.19in the above identity we conclude that

Ru₁s Fφ

sdsa

Ru₁s Fφ

sds b

RAu1s Fφ

sdsc

RAu₁s Fφ

sds

RF φ

sfsds

3.22

for allφ∈C^∞₀ R\ {0}.By Lemma3.4there existsz∈Xsuch that

u₁t au₁t bAu1t cAu₁t ft z, t∈R. 3.23

We define

ut u1t 1

bA⁻¹z. 3.24

Then, we can show that u solves 3.1 and that u ∈ C^α3R, X ∩ C^α1R,DA ∩ C^αR,DA .

In order to prove uniqueness, suppose that

ut aut bAut cAut, t∈R. 3.25

whereu∈C^α3R, X∩C^α1R,DA ∩C^αR,DA .Letσ >0.We defineLσuby Lσu

ρ u

σiρ

−u

−σiρ

, ρ∈R 3.26

where the hat indicates the Carleman transformsee e.g.11 . By 12, PropositionA.2i , we have that

Ru ρ

Fφ ρ

dρlim

σ↓0

RLσu ρ

φ ρ

dρ 3.27

for allφ ∈ SR,the Schwartz space of smooth rapidly decreasing functions onR.We will prove that the right term in3.27is zero, from whichu≡0 proving the theorem. In fact, by 12, PropositionA.2iii we have

bc

σiρ lσ

ρ

−A Lσu

ρ

2σcAu

−σiρ

− 2σ

σiρ 2σa

σiρ₂ u

−σiρ

−

2σ2σa

σiρ u

−σiρ

−2σau

−σiρ :Ha,c

σ, ρ ,

3.28

where

lσ

ρ

σiρ₂1a σiρ bc

σiρ. 3.29

Observe thatl0ρ lρ∈ρAfor allρ∈R.Therefore we have lσ

ρ

−l

ρ

lρ−A₋₁ Lσu

ρ

Lσu ρ

1 bc

σiρ

lρ−A₋₁ Ha,c

σ, ρ . 3.30

Letφ∈C^∞₀ R.Multiplying byφand integrating overRthe above identity we obtain

RLσu ρ

φ ρ

dρ

RNσ

ρ Ha,c

σ, ρ dρ−

RMσ

ρ Lσu

ρ

dρ 3.31

where

Nσ

ρ

1 bc

σiρφ ρ

lρ−A₋₁ Mσ

ρ φ

ρ lσ

ρ

−l

ρ

lρ−A₋₁ .

3.32

We note that in12, Lemma A.4 ,

limσ↓0

RMσ

ρ Lσu

ρ

dρ0. 3.33

It remains to prove that

limσ↓0

RNσ

ρ Ha,c

σ, ρ

dρ0. 3.34

In fact, sinceLσuρ _Re^−σ|t|e^−iρtutdt, we have

RMσ

ρ Lσu

ρ dρ

R

Re^−iρtMσ

ρ

dρe^−σ|t|utdt

RFMσte^−σ|t|utdt.

3.35

Then

RMσ

ρ Lσu

ρ dρ

≤

RFMσtutdt.

≤2C

Mσ_L¹M_σ

L¹

.

3.36

It is easy to check thatMσ_L¹M_σL1 → 0 asσ → 0,proving3.34.

We write

Ha,c

σ, ρ I1

σ, ρ I2

σ, ρ I3

σ, ρ I4

σ, ρ

. 3.37

We first prove that

limσ↓0

RNσ

ρ I1

σ, ρ

dρ0. 3.38

In fact, we apply Fubini’s theorem to obtain

RNσ

ρ I1

σ, ρ

dρ2σc

RNσ

ρAu

−σiρ dρ −2σc

₀

−∞

Re^−iρtNσ

ρ dρ

e^σtAutdt −2σc

₀

−∞FNσte^σtAutdt

3.39

It follows from12 , Lemma A.3 that

₀

−∞FNσte^σtAutdt ≤2C

Nσ_L¹N_σ

L¹

, 3.40

whereCis a positive constant. Taking into account3.39and3.40we deduce3.38.

We next prove that

limσ↓0

RNσ

ρ I2

σ, ρ

dρ0. 3.41

In fact, defineN_σ^aρ 1aσiρ σiρNσρ.Then

RNσ

ρ I2

σ, ρ

dρ−2σ

R

₀

−∞e^σ−iρsN_σ^a ρ

usdsdρ −2σ

₀

−∞FN_σ^ase^asusds.

3.42

By12, LemmaA.3 , we have for 0≤σ≤ε,

₀

−∞FN^a_σse^asusds

≤2Csup

0≤σ≤ε

N^a_σL1N^a_σ^”

L¹

, 3.43

whereC >0.Therefore, we deduce3.41. Proceeding in the same way we obtain

limσ↓0

RNσ

ρ Ij

σ, ρ

dρ0, j3,4. 3.44

This completes the proof of the assertion3.34.

Corollary 3.6. The solutionuof problem3.1given by Theorem3.5satisfies the following maximal regularity property:u, u∈C^αR;DA andAu, Au, u, u”∈C^αR;X.Moreover, there exists a constantC >0 independent off ∈C^αR;Xsuch that

u_αu

αu

αu

αAu_αAu

α≤Cf

α. 3.45

The following consequence of Theorem 3.5 is remarkable in the study of C^α well posedness for flexible structural systems. We recall thatlη:−η²1iaη/bicη.

Corollary 3.7. IfAis the generator of a bounded analytic semigroup, then3.1isC^α-well posed.

Proof. SinceAgenerates a bounded analytic semigroup, we have that{τ :Re τ >0} ⊆ρA and there is a constantM >0 such thatττ−A⁻¹ ≤MforRe τ >0.Note that

η³ bicηR

l η

;A

−η 1iαηl

η R

l η

;A

3.46

and that

Re l

η

bacη²

b²c²η² >0 for eachη∈R. 3.47 We conclude thatlη∈ρAand

sup

η∈R

η³

bicηR l

η , A

<∞. 3.48

The conclusion follows by Theorem3.5.

For example, ifAis a normal operator on a Hilbert spaceHsatisfying σA⊂

z∈C: arg−z< δ

3.49

for some δ ∈ 0, π/2, then Agenerates a bounded analytic semigroup. In particular, the semigroup generated by a self-adjoint operator that is bounded above is analytic of angle π/2.Another important class of generators of analytic semigroups is provided by squares of group generators.

Example 3.8. Since the Laplacian Δ is the generator a bounded analytic semigroup the diffusion semigroupinX L^pR^N 1≤p <∞, we obtain that for eachf∈C^αR, L^pR^N the problem

uttt, x λutttt, x c²

Δut, x μΔutt, x

ft, x 3.50

has a unique solutionu∈C^α3R, L^pR^N∩C^α1R, W^2,pR^N∩C^αR, W^2,pR^N.

Since it is also well known that the Dirichlet LaplacianΔgenerates a bounded analytic semigroup onL²Ω,whereΩis a bounded domain with smooth boundary ∂ΩinR³, we obtain the following consequence for our initial problem.

Corollary 3.9. If Ω is a bounded domain with boundary of class C² in R³ then for each f ∈ C^αR, L²Ω, the problem 3.50 is C^α-well posed, that is, has a unique solution u ∈ C^α3R, L²Ω∩C^α1R, H²Ω∩H₀¹Ω∩C^αR, H²Ω∩H₀¹Ω.

We note that the same assertion remain true for allp∈1,∞.

Acknowledgment

The first author is partially supported by CNPQ/Brazil. The second author is partially financed by Laboratorio de An´alisis Estoc´astico, Proyecto Anillo ACT-13.

References

1 S. K. Bose and G. C. Gorain, “Exact controllability and boundary stabilization of torsional vibrations of an internally damped flexible space structure,” Journal of Optimization Theory and Applications, vol.

99, no. 2, pp. 423–442, 1998.

2 A. B´atkai and S. Piazzera, Semigroups for Delay Equations, vol. 10 of Research Notes in Mathematics, A.

K. Peters, Wellesley, Mass, USA, 2005.

3 S. K. Bose and G. C. Gorain, “Exact controllability and boundary stabilization of flexural vibrations of an internally damped flexible space structure,” Applied Mathematics and Computation, vol. 126, no.

2-3, pp. 341–360, 2002.

4 S. K. Bose and G. C. Gorain, “Stability of the boundary stabilised internally damped wave equation yλyc²ΔyμΔyin a bounded domain inRⁿ,” Indian Journal of Mathematics, vol. 40, no. 1, pp.

1–15, 1998.

5 T.-J. Xiao and J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations, vol. 1701 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1998.

6 W. Arendt, C. Batty, and S. Bu, “Fourier multipliers for H ¨older continuous functions and maximal regularity,” Studia Mathematica, vol. 160, no. 1, pp. 23–51, 2004.

7 V. Keyantuo and C. Lizama, “Periodic solutions of second order differential equations in Banach spaces,” Mathematische Zeitschrift, vol. 253, no. 3, pp. 489–514, 2006.

8 D. Mugnolo, “A variational approach to strongly damped wave equations,” in Functional Analysis and Evolution Equations: The G ¨unter Lumer Volume, pp. 503–514, Birkh¨auser, Basel, Switzerland, 2008.

9 R. Chill and S. Srivastava, “L^p-maximal regularity for second order Cauchy problems,” Mathematische Zeitschrift, vol. 251, no. 4, pp. 751–781, 2005.

10 W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, vol. 96 of Monographs in Mathematics, Birkh¨auser, Basel, Switzerland, 2001.

11 J. Pr ¨uss, Evolutionary Integral Equations and Applications, vol. 87 of Monographs in Mathematics, Birkh¨auser, Basel, Switzerland, 1993.

12 V. Keyantuo and C. Lizama, “H ¨older continuous solutions for integro-differential equations and maximal regularity,” Journal of Differential Equations, vol. 230, no. 2, pp. 634–660, 2006.