Well Posedness for a Class of Flexible Structure in Hölder Spaces

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 in 1 and consists of a short rigid hub, connected to a flexible panel of length l. Control torque Q t is 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


Introduction
During the last few decades, the use of flexible structural systems has steadily increased importance.The study of a flexible aerospace structure are problems 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 in [6] and consist of a short rigid hub, connected to a flexible panel of length l.Control torque Q(t) is applied to the hub.The panel is made of viscoelastic material with internal Voigt-type damping with coefficient µ, i.e. 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 where c is the velocity of longitudinal wave propagation, c 2 = D p /ρJ p , and D p , ρ, J p are respectively torsional rigidity, density and radius of gyration about the central axis of the panel.Initial position and deflection angle are known.In [6] exact controllability and boundary stabilization for the solution of (1.1) was analyzed and in [3, p.188] the exact decay rate was obtained.More generally, the study of vibrations of flexible structures possessing internal material damping is modeled by a equation of the form in a bounded domain Ω in R n with smooth boundary Γ, see [4] and [5].
In [5] the explicit exponential energy decay rate was obtained for the solution of (1.2) 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 of (1.2), appears as an open problem.
In the first part of this paper we study well-posedness of the following abstract version of (1.2) where A is a closed linear operator acting in a Banach space X and f is a X-valued function.We observe that when A = ∆ in general one cannot expect that (1.3) is wellposed due to the presence of the term u .We are able to characterize well posedness, i.e. 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 Hölder spaces C α (R, X), where 0 < α < 1.The methods to obtain this goal are those incorporated in [2] where a similar problem in case of the first order abstract Cauchy problem has been studied.

Preliminaries
Let X, Y be Banach spaces, we write B(X, Y ) for the space of bounded linear operators from X to Y and let 0 < α < 1.We denote by Ċα (R, X) the spaces Let Ω ⊂ R be an open set.By C ∞ c (Ω) we denote the space of all C ∞ -functions in Ω ⊆ R having compact support in Ω.
We denote by Ff or f the Fourier transform, i.e.
Define the space C α (R, X) as the set where k is a positive integer) be the Banach space of all u ∈ C k (R, X) such that u (k) ∈ C α (R, X), equipped with the norm Observe from Definition 2.1 and the relation  , B(X, Y )) be such that Then M is a Ċα -multiplier.

A characterization of well posedness in Hölder spaces
In this section we characterize C α -well-posedness.Given f ∈ C α (R, X), we consider in this section the linear problem where A is a closed linear operator in X and a, b, c > 0. In the case a = 0, solutions of (3.1) with periodic boundary conditions has been recently studied in [7].We denote by [D(A)] the domain of A considered as a Banach space with the graph norm.
In the next proposition, as usual we denote by ρ(T ), R(λ, T ) the resolvent set and resolvent of the operator T , respectively.
Then it is not difficult to see that u is a solution of (3.1) with f ≡ 0. Hence, by uniqueness, x = 0. Let y ∈ X and define f (t) = e iηt y.Let u = Lf.For fixed s ∈ R we define Then is easy to check that v 1 and v 2 are both solutions of (3.1) with f replaced by e iηs f.By uniqueness, u(t + s) = e iηs u(t) for all t, s ∈ R. In particular, it follows that u(s) = e iηs u(0) Taking t = 0 we conclude that (l(η) − A) is bijective and Define e η (t) = e iηt and (e η ⊗ y)(t) = e η (t)y.We have the identity Therefore, for > 0 we have On the other hand, since { 1 b+icη } η∈R is bounded and η → η 3 R(l(η), A) is continuous at η = 0, we obtain (ii) and the proof is complete.
In what follows, we denote by id k the function: s → (is) k for all s ∈ R, and k ∈ N. As before, we also use the notation .We first observe that the functions θ(s) := κ (s) κ(s) and ϑ(s) := l (s) l(s) have the property that sθ(s), sϑ(s), s 2 θ (s) and s 2 ϑ (s) are bounded on R. We next claim that M is a Ċα -multiplier.In fact, note that by hypothesis sup |s|> ||M (s)|| < ∞ for each > 0, and the function s → M (s) is continuous at t = 0 since b > 0. Hence M (s) is bounded.Moreover, defining ξ(s) := l(s) κ(s) = −s 2 − ias 3 we have where sξ(s) is of order s 4 and then Q(s) := ξ(s)[M (s)] 2 is bounded by (3.3).It follows that sM (s) is bounded.Next, we have the identity where the first three terms on the right hand side are bounded.For the last term, we have 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|> ||ξ(s)M (s)|| < ∞ for each > 0 and the function s → ξ(s)M (s) is continuous at s = 0. Hence ξ(s)M (s) is bounded.This completes the proof of the Lemma.
The following assertions are equivalent: The following theorem, which is one of the main result in this paper, shows that the converse of Proposition 3.2 is valid.Theorem 3.5.Let A be a closed linear operator defined on a Banach space X.Then the following assertions are equivalent: Proof.The implication (i) ⇒ (ii) follows by Proposition 3.2.We now prove the converse implication. Let for some y 1 ∈ X.Now we can choose φ 2 = id • φ 3 in (3.9), it follows that u 1 ∈ C α+2 (R, X) and (3.13) for some y 2 ∈ X.In a similar way, we can see that u 1 ∈ C α+3 (R, X) and for some y 3 ∈ X.We have the identity We multiply the above identity by φ, take Fourier transforms and integrate over R after taking the values at f (s), we obtain ).Using (3.12), (3.13) and (3.14) in the above identity we conclude that We define (3.17) Then, we can show that u solves (3.1) and that u In order to prove uniqueness, suppose that where where the hat indicates the Carleman transform (see e.g.[9]).By Proposition A.2 (i) in [8], we have that for all φ ∈ S(R), the Schwarz space of smooth rapidly decreasing functions on R. We will prove that the right term in (3.20) is zero, from which u ≡ 0 proving the theorem.In fact, by Proposition in [8] we have where Observe that l 0 (ρ) = l(ρ) ∈ ρ(A) for all ρ ∈ R. Therefore we have Multiplying by φ and integrating over R the above identity we obtain where We note that by Lemma A.4 in [8], It remains to prove that It is easy to check that We write H a,c (σ, ρ) = I 1 (σ, ρ) I 2 (σ, ρ) + I 3 (σ, ρ) + I 4 (σ, ρ).We first prove that (3.24) lim σ↓0 R N σ (ρ)I 1 (σ, ρ)dρ = 0.
In fact, define where C > 0. Therefore, we deduce (3.27).Proceeding in the same way we obtain This completes the proof of the assertion (3.23).
Corollary 3.6.The solution u of problem (3.1) given by Theorem 3.5 satisfies the following maximal regularity property: u, u ∈ C α (R; [D(A)]) and Au, Au , u , u ∈ C α (R; X).Moreover, there exists a constant C > 0 independent of f ∈ C α (R; X) such that The following of Theorem 3.5 is remarkable in the study of C α well posedness for flexible structural systems.We recall that l(η) := −η 2 1 + iaη b + icη .
Corollary 3.7.If A is the generator of a bounded analytic semigroup, then Equation (3.1) is C α -well posed.
For example, if A is a normal operator on a Hilbert space H satisfying σ(A) ⊂ {z ∈ C : arg(−z) < δ} for some δ ∈ [0, π/2), then A generates 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.
By putting µ t (x) = (4πt) −N/2 e −|x| 2 /4t , this can be written as Since it is also well known that the Dirichlet Laplacian ∆ generates a bounded analytic semigroup on L 2 (Ω), where Ω is a bounded domain with smooth boundary ∂Ω in R 3 , we obtain the following consequence for our initial problem.