A SEMIGROUPS THEORY APPROACH TO A MODEL OF SUSPENSION BRIDGES

In this paper we study the existence and uniqueness of the weak solution of a mathematical model that describes the nonlinear oscillations of a suspension bridge. This model is given by a system of partial differential equations with damping terms. The main tool used to show this is the C0-semigroup theory extending the results of Aassila [1].


Introduction
Since the collapse of the Tacoma Narrows Bridge on November 7, 1940, several mathematical models have been proposed to study the oscillations of the bridge.Lazer and McKenna proposed a model governed by a coupled system of PDEs which takes into account the coupling provided by the stays (ties) connecting the suspension (main) cable to the deck of the road bed.In this model the coupling is nonlinear (for more details see [12]).
In [3] Ahmed and Harbi used the model proposed by Lazer and McKenna to do a detailed study of various types of damping.Also, they presented an abstract approach which allows the study of the regularity of solutions of these models.
The model of suspension bridges is given by the system of partial differential equations x ∈ Ω, t 0, m c y tt − βy xx + F (y − z) = f 2 (y t ), x ∈ Ω, t 0, z(0, t) = z(l, t) = 0, z x (0, t) = z x (l, t) = 0, y(0, t) = y(l, t) = 0, z(x, 0) = z 1 (x), z t (x, 0) = z 2 (x), x ∈ Ω, y(x, 0) = y 1 (x), y t (x, 0) = y 2 (x), x ∈ Ω. (1.1) Here we denote by Ω the interval (0, l).See [3] and [12] for the physical interpretations of the parameters α, β, the variables y, z and the boundary conditions respectively.As described in [3] the function F represents the restraining force experienced by both the road bed and the suspension cable as transmitted through the tie lines (stays), thereby producing the coupling between these two.The functions f 1 and f 2 represent external forces as well as non-conservative forces, which generally depend on time, the constants m b , m c , α, β are positive and F : R → R is a function with F (0) = 0 (F can be linear or not), see [3].The interested reader is also refereed to the works of Drábek et.al [6,7] and Holubová [11] where other models for the oscillations of the bridge are studied.
The aim of this work is to study the existence and uniqueness of weak solutions for (1.1).
To do this we make use of the semigroup theory.This allows us to do it in a much simpler way without using maximal monotone operators theory as in [10,Theorem 1] or the Galerkin approach as in [3,Theorem 4.4].
In Section 2, we study the existence and uniqueness of weak solutions of the linear model of suspension bridges, i.e., when F (ξ) = kξ and f 1 = f 2 = 0 (see, for instance, [3]).The case F (ξ) = kξ and f 1 = f 2 = 0 was considered by Aassila in [1].We consider the nonlinear model in Section 3.

Linear abstract model
The linear model is obtained through the bed support bridge tied with cords connected to two main cables placed symmetrically (suspended), one above and one below the bed of the bridge.In the absence of external forces (f 1 = f 2 = 0), the linear dynamic of suspension bridge around the equilibrium position can be described by the following system of linear coupled EDP's Here, F (ξ) = kξ, where k denotes the stiffness coefficient of the cables connecting the bridge to the bed and suspended cable.

Existence and uniqueness of solution. Let us denote for
the Hilbert spaces endowed with scalar products and with their respective norms W defined in W is equivalent to the norm of H 4 (Ω)×H 2 (Ω).Therefore, by the Sobolev embeddings in [5, p. 23], we have the embeddings dense and compact W ⊂ V ⊂ H. Identifying H with its dual H , we obtain W ⊂ V ⊂ H = H ⊂ V ⊂ W with embeddings dense and compact.
Let the bilinear form a : V × V → R be given by where u = (u 1 , u 2 ), ũ = (ũ 1 , ũ2 ) ∈ V .To simplify the notation, we use The bilinear form a is continuous, symmetric and coercive. Proof.
and this shows that a is continuous.The symmetry of a is immediate.For the last, a(u, u) V , for all u = (u 1 , u 2 ) ∈ V , and thus we have the coercivity of a.
From Lemma 2.1, there exists a linear operator For u = (z, y), the system (2.1) can be written as where With this notation we can see the problem (2.3) as second order ODE in H, where the operator C : D(C) ⊂ H → H has domain D(C) given by Consequently, we have for the operator C, Notice that in the equation (2.4) we are looking for u as a function of t taking values on The problem (2.4) can be written as a first order EDO abstract with the boundary condition v = u t = (z t , y t ) = 0 on ∂Ω × (0, ∞).
Let us denote for H = V × H the Hilbert space endowed with the inner product For U = (u, v) the system (2.7) can be written as an abstract Cauchy problem in H U + AU = 0, t ∈ (0, ∞) EJQTDE, 2013 No. 51, p. 4 where U 0 = (u 0 , v 0 ), A : D(A) ⊂ H → H is given by AU = (−v, Cu) and Lemma 2.4.The operator −A is the infinitesimal generator of a C 0 -semigroup of contractions in H.

Thus, (−A)
Analogously to what we did before, we get −AU, U H = 0. Therefore, −A and (−A) * are dissipative.Now, let Now, by Lemma 2.3 and Corollary 4.4 [13, p. 15] it follows that −A is infinitesimal generator of a C 0 -semigroup of contractions in H.
On the other hand, if U 0 ∈ D(A) = W × V and −A is infinitesimal generator of a C 0 -semigroup contractions in H then we have a unique solution (Proposition 6.2 in [8, p. 110]) This proves the second part of the theorem.

Nonlinear abstract model
In this section we consider the general problem (1.1) which can be seen as an abstract ODE in a suitable Hilbert space.The abstract setting has many advantages as we can see below.We first write the equation of the problem (1.1) as follows z tt + a 2 ∆ 2 z = F 1 (t, x, y, z), x ∈ Ω, t > 0, y tt − b 2 ∆y = F 2 (t, x, y, z), x ∈ Ω, t > 0. (3.1) Here Let H be the Hilbert space as before and consider V given by V = H 2 (Ω) × H 1 0 (Ω) endowed with the inner product and norm given by Ω) are equivalents and thus V is a Hilbert space.Note that the embedding V → H is continuous, dense and compact.If V denotes the dual topological of V and identifying H with its dual we have the inclusions , where H −s (Ω), s > 0, denotes the Sobolev's space with negative exponent, for more details see [2].EJQTDE, 2013 No. 51, p. 6 Consider the bilinear form c : Proof.
for all u, v ∈ V , and thus we have that c is continuous.The symmetric property is obvious.Finally, denoting V , ∀ u ∈ V ; i.e., c is coercive.
From Lemma 3.1, there exists a linear operator A ∈ L (V, V ) such that c(u, v) = Au, v V ,V , for all u, v ∈ V .

Lemma 2 . 3 .
For the operator A holds that D(A) = W × V and D(A) is dense in H. Proof.See the details in [1, Lemma 2.3].