Asymptotic finite-dimensional approximations for a class of extensible elastic systems

1.   Introduction

Let $A^2$ be a diagonal, self-adjoint, strictly positive operator, densely defined on a real Hilbert space $({\mathcal{H}}, (\cdot, \cdot), {\lVert{\cdot}\rVert})$ and we consider the following nonlinear nonlocal evolution equation

where $\theta \in [0, 1]$, $\delta > 0$ and $g \in C^0(\mathbb{R}_+, \mathcal{H})$ is a given forcing term.

The purpose of the present paper is to give a rigorous finite-dimensional approximation of (1.1). To be more precise, we introduce the projection $P_N$ onto the space generated by the first $N$ modes, that is, by the first $N$ eigenvectors of the operator $A^2$ and we consider the approximated problem

We remark that, by taking $u(0)$ and $u_t(0)$ in $P_N\mathcal{H}$, Eq (1.2) can be interpreted as a system of $N$ ODEs. Therefore, Eq (1.2) actually provides a finite-dimensional approximation of equation (1.1). We aim to prove that any solution of (1.2) is asymptotically finite-dimensional and to estimate, for any $\varepsilon > 0$, the smallest $N = N(\varepsilon)$ such that the asymptotic distance in the phase space between the solution of (1.1) and the corresponding solution of (1.2) is less than $\varepsilon$. An improvement of the result will be studied for a particular class of forcing terms.

The reduction of infinite-dimensional dynamical systems to finite-dimensional systems of ODEs is a technique which has been widely used in the theoretical and numerical study of PDEs. The idea was first stated by Galerkin ^[28] and it has been used in many different applied frameworks as well as in the theory of finite-dimensional inertial manifolds (see ^{[15,19,21,52,54,55]} and the references therein). In particular, it is a fairly common procedure, which we aim to make rigorous, in the study of suspension bridges ^[3] to approximate the physical system with the dynamics finite number of modes in order to reduce the computational complexity of the model. This approach can be physically justified by observing that "the higher modes with their shorter waves involve sharper curvature in the truss and, therefore, grater bending moment at a given amplitude and accordingly reflect the influence of the truss stiffness to a greater degree than do the lower modes" [51,p.11], which means that the dynamics of the higher modes corresponds to a physically irrelevant phenomenon. We remark that our goal would not be achieved just by estimating the dimension of the inertial manifold of our system, since we are interested in providing a finite-dimensional approximation of its asymptotic behavior.

The problem of finding a finite number of natural parameters of a system that uniquely determine its asymptotic behavior was first discussed for the 2D Navier-Stokes equation ^[24,43] and to tackle it the concepts of finite-dimensional inertial manifold, determining modes and, later, determining nodes and determining local volume averages were introduced (see [16,Ch. 5], ^[18] and the references therein). Regarding our problem, Chueshov in [16,Ch. 5,Thm. 7.2] proved that the dynamics of the first $N$ modes of (1.1) completely determines the evolution of the system and Eden and Milani in ^[22] proved that if the forcing term is $N-$dimensional, then any solution is attracted to an $M-$dimensional manifold with $M\geq N$.

Some particular cases of the damped Eq (1.1) have been widely studied in mathematical literature. An ODE version of the problem was investigated by Loud in ^[44,45]. Fitouri and Haraux in ^[27] improved some of the previous results on the ODE case and in ^[26] they provided a close-to-optimal ultimate bound in the PDE version of the problem. More recently, some sharp stability criteria for the unimodal version of (1.1) and for a related evolution equation were obtained by Haraux in ^[37] in the case $g = 0$. The case when $\theta = 1$ was studied in a slightly different framework by Holmes and others in ^[40,47] as an example of chaotic dynamics (see also ^[34]) and some undamped versions of (1.1) were studied in the case $\theta = 0$ by Cazenave, Weissler and Haraux in ^{[11,12,13,14]} in order to obtain a description of the qualitative behavior of more complicated nonlinearities and by Gazzola and Garrione in ^[29] to study the dynamics of suspension bridges with multiple intermediate piers.

The considered abstract equation was analyzed by many other authors in an even more general framework. Biler ^[7] and de Brito ^[9] investigated the decay properties of the unforced problem with weak damping and a more general nonlinear nonlocal term. Da Silvia and Narciso ^[49,50] studied an extensible beam model subject to a nonlocal nonlinear parameter-dependent damping and a forcing term. A lot of different variations of (1.1) with a large variety of damping and nonlinear terms has been studied in mathematical literature (see ^[16,17,20] and the references therein).

In addition to its mathematical relevance, our study also presents a certain physical and engineering interest. In fact, the considered model is suitable to describe both mono-dimensional and multi-dimensional physical systems. More precisely, some particular cases of (1.1) concerning the dynamics of beams and plates was considered by Holmes and Marsden ^[38,39] in order to study the problem of flow-induced oscillations (see also ^[41,42]) and in order to provide some more information about the nonlinear structural behavior of suspension bridges. In particular, we expect our results to allow some progress in the study of the structural and torsional instability of plates, to which a vast literature is devoted ^{[1,2,4,5,31,32]}.

If we set $A^2 = \Delta^2$, $\theta = 1$ and $\mathcal{H} = L^2(\Omega)$, where $\Omega$ is a bounded domain in $\mathbb{R}^N$ ($N\geq 1$) with the smooth boundary $\partial\Omega$, we obtain the equation

This problem is a special case of the more general model

that was introduced in 1955 by Berger ^[6] as a simplification of the von Karman plate equation which describes large deflection of plate. Some related models were later applied to the study of the torsional instability of suspension bridges. In particular, our results apply also to the partially-hinged plate problem discussed in ^[8,25]

where $S > 0$ depends on the elasticity of the material of the deck of the bridge, $l > 0$ represents the width of the bridge and $\sigma > 0$ is the Poisson's ratio of the structure, which is assumed to be, in the case of suspension bridges, between $0$ and $0.5$. The term $P$ is called "prestressing constant" and it expresses the buckling loads on the plate. In the case of suspension bridges, the compressive forces along the edges are introduced in order to increase the stability of the structure. The abstract prestressed model reads

The study of this equation will not be discussed in detail since, under the hypothesis $P < \alpha_1^{1/2}$ (weak prestressing), the prestressing term does not modify the qualitative behavior of the system and in the case when $P\geq \alpha_1^{1/2}$ (strong prestressing) our results do not hold. In fact, in a strongly prestressed suspension bridge the linear part of (1.3), which is given by $A^2-PA$, is not a strictly positive operator anymore.

Concerning the case where the models describes the dynamics of a mono-dimensional structure, if we take $\mathcal{H} = L^2(I)$ (with $I = [-\pi, \pi]$) and $A = -\partial_{xx}$, we can distinguish three different physically significant cases: $\theta = 0$, $\theta = 1$ and $\theta = 2$.

In the first case, the considered model has been introduced by Garrione and Gazzola ^[29] in order to describe the behavior of the deck of suspension bridges with two intermediate piers. In the work of Garrione and Gazzola, the deck of the bridge is modeled by a degenerate plate consisting of a beam with a continuum of cross sections free to rotate around the beam. Therefore, the longitudinal dynamics of the bridge is modeled by a beam equation, whose nonlinear term can be interpreted as a representation of "a stiffened beam where the displacement behaves superquadratically and nonlocally: if the beam is displaced from its equilibrium position in some point, then this increases the resistance to further displacements in all the other points" ^[29]. The nonlocal nature of such term is due to the elastic behavior of the components of the bridge, the sustaining cables in particular. This choice of the nonlinear term follows from a comparison between the qualitative behavior of some possible models and the actual behavior of suspension bridges. If we consider $\mathcal{D}(A) = \{v\in {H}^2(I)\cap {H}^1_0(I):v(-\pi) = v(\pi) = v(-a\pi) = v(b\pi) = 0\}$ for $a, b\in (0, 1)$, where $a$ and $b$ model the position of the piers along the deck of the bridge, the system reads

An analogous equation, in a different functional framework, is involved in the study of the interaction between the cables and the deck of a suspension bridge in the case when the hangers are considered inextensible (see ^[29,46]).

The second case ($\theta = 1$) was obtained by Woinowsky-Krieger ^[53] in 1950 and, independently, by Burgreen ^[10] in 1951. It models the physical phenomenon that "if the beam is stretched somewhere, then this increases the resistance to further stretching in all the other points" ^[29]. The system has been widely studied in both mathematical and engineering literature (see ^[22,33] and the references therein). If we choose $\mathcal{D}(A) = \{v\in {H}^2(I)\cap {H}_0^1(I): v(-\pi) = v(\pi) = v_{xx}(-\pi) = v_{xx}(\pi) = 0\}$, the model becomes

The case $\theta = 2$ was first introduced in ^[29]. If we consider $\mathcal{H} = L^2(I)$ and $A = -\partial_{xx}$ as we did before, the nonlinear term ${\lVert{u}\rVert}_\theta^2A^{\theta/2}u$ reads ${\lVert{u_{xx}}\rVert}^2_{L^2(I)}u_{xxxx}$ and the corresponding nonlinear equation can be interpreted as a model for "a stiffened beam with bending energy behaving superquadratically and nonlocally: this means that if the beam is bent somewhere, then this increases the resistance to further bending in all the other points" ^[29]. Despite the physical interest of the case $\theta = 2$, due to its technical difficulty, in this paper we decided to restrict ourselves to the cases where $\theta\in [0, 1]$.

The results of the paper are given in three main theorems. First, in Theorem 2.3, we prove that if the forcing term is finite-dimensional, i.e., if $g$ is a combination of a finite number $N$ of modes, then any solution is asymptotically finite-dimensional too in a sense that we specify in Definition 2.2. In the case of small oscillations or large damping, our result improves the one of Eden and Milani ^[22]. The proof is based on an application of a recent work of Haraux ^[37]. Next, in Theorem 2.4 we prove that, under suitable smallness conditions on the nonlinearity and on the forcing term, we are able to give an $M-$dimensional approximation of (1.1). More precisely, we prove that for any $\varepsilon > 0$ there exists $N\in \mathbb{N}$ such that the asymptotic distance between a solution of (1.1) and a solution of (1.2) is controlled by $\varepsilon$ in the phase space norm. The proof relies on a continuous dependence result and on Theorem 2.3. To conclude, in Theorem 2.5, fixed $\theta = 0$, we focus on a particular class of forcing terms and we refine the result of Theorem 2.4. In particular, under suitable smallness conditions on the solution, we improve the ultimate bounds previously given for general forcing terms in ^[8,26] and we estimate how much the dynamics changes as we eliminate a single mode from the dynamics. This latter result represents one of the main novelties of the paper since, to the author's knowledge, this is the first statement of this type present in literature.

The paper is organized as follows. In Section 2 we give some definitions and we state the main results of the paper. In Section 3, some technical results are given. The proofs of the main results are contained in Section 4, Section 5 and Section 6, which are devoted to the proof of Theorem 2.3, Theorem 2.4 and Theorem 2.5 respectively. In Section 7, we present some physical conclusions concerning the application of our results to suspension bridges with multiple intermediate piers.

2.   Statement of the main results

Let $({\mathcal{H}}, (\cdot, \cdot), {\lVert{\cdot}\rVert})$ be a Hilbert space and consider a diagonal, self-adjoint and positive-definite operator $A^2:\mathcal{D}(A^2)\subset {\mathcal{H}} \to {\mathcal{H}}$, with eigenvalues $0 < \alpha_1 < \dots < \alpha_j \nearrow \infty$ and eigenfunctions $e_n$, solutions of the problem

The sequence $(e_n)_{n\geq 1}$ is a complete orthonormal system of ${\mathcal{H}}$. For our convenience, we preferred to use $A^2$ instead of $A$ to build the functional framework of the problem. The operator $A^2$ defines a family of Hilbert spaces ${\mathcal{H}}^\sigma = \mathcal{D}(A^{\sigma/2})$ with $\sigma\geq 0$, endowed with the norms ${\lVert{\cdot}\rVert}_\sigma$ induced by the scalar products

where $u_n = (u, e_n)$ and $v_n = (v, e_n)$. In particular, ${\lVert{\cdot}\rVert}_0 = {\lVert{\cdot}\rVert}$. In the context of this work, we consider the cases when $\sigma \in [-2, 2]$, where for negative $s$ the space $\mathcal{H}^s$ is defined as the dual of $\mathcal{H}^{-s}$. Throughout the paper, we denote by $\langle \cdot, \cdot \rangle$ the duality product of ${\mathcal{H}}^2$. It possible to verify that ${\mathcal{H}}^\rho \hookrightarrow {\mathcal{H}}^\sigma$ densely whenever $0\leq \sigma \leq\rho$ and that

In this framework, for any family of indices $J = \{j_1, \dots, j_n\}$, we define the projection

In particular, we denote by $P_N$ and $Q_N: = I-P_N$ the orthogonal projections onto $\langle e_1, \dots e_N\rangle$ and onto $\langle e_{N+1}, \dots \rangle$ respectively. In addition, for any $k \in \mathbb{N}$ we introduce the projection $\sqcap_k$ onto the orthogonal complement of $e_k$ given by

Since $A$ is a diagonal operator, we remark that

Moreover, if $u = Q_Nu$ for some $N\in \mathbb{N}$, then the estimate (2.2) can be improved by

By using the notation in (2.1), problem (1.1) may be rewritten as

Let us make clear what is meant by weak solution of (2.5):

Definition 2.1. Assume that

A weak solution of (2.5) is a function

such that

We remark that by this definition it follows that $u(0) = u_0 \in \mathcal{H}^2$ and $u_t(0) = u_1\in \mathcal{H}$. Existence and uniqueness of weak solutions follows from an immediate adaptation of the result in [33,Theorem 2.1] (see Theorem 3.1).

First, we prove that if the forcing term if finite-dimensional, i.e. if $g = P_Ng$ for some $N\in \mathbb{N}$, then any weak solution of (2.5) is asymptotically finite-dimensional. Actually, we guarantee the validity of the result for a more general family of forcing terms. We introduce the notion of exponentially $N-$dimensional forcing term.

Definition 2.2. We say that $g \in C^0_b(\mathbb{R}_+, {\mathcal{H}})$ is exponentially $N-$dimensional if there exists $\eta > 0$ such that

In Section 4, we prove the following statement which describes the asymptotic behavior of the solution in the case when the forcing term is exponentially $N-$dimensional.

Theorem 2.3. Assume (2.6) and let $\delta > 0$. If $g$ is exponentially $N-$dimensional, there exists $M\geq N$ and $\tilde{\eta} > 0$, both depending on $\delta$, $\limsup_{t\to\infty}{\lVert{g(t)}\rVert}$, $\theta$, $N$, $\eta$ and $\alpha_1$, i.e., the first eigenvalue of $A^2$, such that

where $u$ is a weak solution of (2.5).

Motivated by physical arguments (see Section 7), we now consider a "separated variables" forcing term such as $g(t) = \mathfrak{g}f(t)$, where $\mathfrak{g}\in \mathcal{H}$ and $f\in C_b^0(\mathbb{R}_+, \mathbb{R})$.

Let us consider a weak solution $u$ of (2.5). Numerical simulations show that for some $j$ we have $\limsup_{t\to\infty}|(u(t), e_j)|\ll\limsup_{t\to\infty}{\lVert{u(t)}\rVert}$, that is, we have that the asymptotic amplitude of some modes of $u$ seems to be negligible with respect to the overall dynamics (see Figure 3). Hence, we expect to be able to neglect such modes both from the forcing term $g$ and the solution $u$, thus reducing the numerical complexity of the model. Therefore, for any finite family of indices $J = \{j_1, \dots, j_m\}$, we consider the finite-dimensional approximation of (2.5) given by

We remark that in virtue of Theorem 2.3, any solution of (2.7) is exponentially finite-dimensional. We prove that under suitable smallness conditions on the forcing term, for an appropriate choice of $J$, (2.7) is a good approximation of (2.5), i.e., for any weak solution $u$ of (2.5), the weak solution $v$ of (2.7) provides a good exponentially finite-dimensional approximation of $u$. More precisely, in Section 5 we prove the following theorem:

Theorem 2.4. Assume $\delta > 0$ and $g(t) = \mathfrak{g}f(t)$ with $\mathfrak{g}\in \mathcal{H}$ and $f\in C^0_b(\mathbb{R}_+, \mathbb{R})$. There exists $\bar{g}_\infty = \bar{g}_\infty(\alpha_1, \delta, \theta) > 0$ such that, if

then for every $\varepsilon > 0$ there exists a finite family of indices $J = \{j_1, \dots j_m\}$ depending on $\alpha_1$, $\delta$, $g_\infty$ and $\varepsilon$ such that

where $u$ is a weak solution of (2.5) and $v$ is a weak solution of (2.7).

Moreover, if $g$ is exponentially $N-$dimensional, then there exist $M\geq N$ and $\tilde{\eta} > 0$, both depending on $\alpha_1$, $\delta$, $\limsup_{t\to\infty}{\lVert{g(t)}\rVert}$, $\theta$, $N$ and $\eta$, such that, if $J = \{1, \dots, M\}$, then

In Section 6 we further restrict ourselves to the case when the forcing term is sinusoidal in time and, for the sake of simplicity, we focus on the case when $\theta = 0$, i.e., we study the problem

For ${\lVert{\mathfrak{g}}\rVert}$ small enough, Theorem 2.4 states that if we replace $\mathfrak{g}$ with $P_M\mathfrak{g}$, we commit an error arbitrarily small as $M$ grows. This suggests to consider the case when $\mathfrak{g} = P_M\mathfrak{g}$ for some $M \in \mathbb{N}$. Let $v$ be a solution of

Let us now estimate the distance between $u$ and $v$. The following theorem holds:

Theorem 2.5. Assume $\delta > 0$ and let $g(t) = \mathfrak{g} \sin (\omega t)$ with $\mathfrak{g} = P_M\mathfrak{g}$ for some $M\in \mathbb{N}$. There exists $\bar{\mathfrak{g}} > 0$ depending on $\delta$, $\omega$ and $\alpha_j$ with $j = 1, \dots M$, such that, if ${\lVert{\mathfrak{g}}\rVert} < \bar{\mathfrak{g}}$, then, for any $k \in \{1, \dots, M\}$ and for any $u$ and $v$ weak solutions of (2.5) and (2.9),

where $C = C(\alpha_1, \dots, \alpha_M, \mathfrak{g}, \delta, \omega) > 0$.

The results involved in the proof of Theorem 2.5 are the most physically significant in the applications considered (see Section 7). In fact, Theorem 2.5 relies upon an estimate on the asymptotic amplitude of each mode, that allows us to study the distribution of the energy among the modes (see Figures 3 and 5) and to obtain a new bound on the asymptotic $\mathcal{H}^2-$norm of $u$ that improves the estimate given in [8,Lemma 22] (see Figure 2).

Theorems 2.4 and 2.5 are not perturbation statements. Indeed, for any fixed $\delta > 0$, an explicit expression of the smallness conditions on $g_\infty$ and ${\lVert{\mathfrak{g}}\rVert}$ required by the statements of Theorems 2.4 and 2.5 is obtained in Sections 5 and 6 respectively. Since the term $g$ models the action of the wind along the deck of the bridge, we physically interpret such smallness conditions on $g_\infty$ as requirements on the aerodynamic load on the structure. In particular, the conditions of Theorems 2.4 and 2.5 are equivalent to require that the speed of the wind $v$ is below a certain threshold $\bar{v}$. Moreover, we remark that such conditions can not be avoided since even in the ODE case large forcing terms lead to a chaotic dynamics ^[44,45] and the behavior of the solutions can be quite complicated, even where the forcing term is periodic in time ^[30,48].

Our results are adaptable to more general frameworks. In particular, exploiting the abstract results of Haraux ^[37] and Chueshov ^[16], the cases with strong damping terms and with more general nonlinearities such as $A^{\theta}u_t$ and $M({\lVert{u}\rVert}_\theta^2)A^{\theta/2}u$ with $0\leq \theta \leq 1$ appear to be treatable. On the other hand, our results can not be immediately generalized to evolution equations with nonlinear nonlocal damping terms such as $N({\lVert{u}\rVert}_1^2)g(u_t)$, since the linear analysis on which the proof of Theorem 2.5 is based seems not to be easily extendable to such case.

We notice that, if the initial states of (2.5) and (2.7) were close to each other, a uniform estimate on the distance in the phase space between the solutions of the approximated and the exact problem would be expected to hold for any $t\geq 0$. Unfortunately, we were not able to obtain such estimate and the techniques exploited in the proofs of Theorems 2.4 and 2.5 do not seem suitable to get this result.

3.   Preliminary results

We start by recalling some basic properties concerning well-posedness and regularity of the solutions.

Theorem 3.1. Let (2.6) hold. Then

1). (Weak solutions) If $u(0) = u_0\in \mathcal{H}^2$ and $u_t(0) = u_1\in \mathcal{H}$, problem (2.5) admits a unique global weak solution such that

2). (Regular solutions) If $u(0) = u_0\in \mathcal{H}^4$ and $u_t(0) = u_1\in \mathcal{H}^2$, problem (2.5) admits a unique regular solution, that is, a unique global weak solution such that

3). (Continuous dependence on initial data) Let $(u_{0n}, u_{1n})$ be any sequence with

and let $u_n(t)$ denote the weak solution of (2.5) with initial data $u_n(0) = u_n$ and $u_t(0) = u_{1n}$. Then for every $T > 0$ we have that

The proof follows from a standard applications of monotone operator theory with locally Lipschitz perturbations. We refer to [20,Theorem 1.5 and Proposition 1.15] and the references therein for a detailed discussion, that we decided to omit. For an alternative approach, see [33,Theorem 2.1] for the global existence and uniqueness of weak solutions and continuous dependence on initial data and [8,Theorem 5] for the global existence and uniqueness of regular solutions.

We remark that in Theorem 3.1 we did not introduce the concept of strong or classical solution. This choice is motivated by the fact that in some applications such formulations are not possible, as in the case of the multiple intermediate piers model discussed in the introduction (see [29,Section 4] for a more detailed discussion).

The following proposition gives some ultimate bounds on the Sobolev norms of $u$. Since the result comes from a straightforward generalization of the estimates proved in Section 7 of ^[8], we omit the proof.

Proposition 3.2. Assume (2.6) and let $u$ be a weak solution of (2.5). We introduce the quantities $g_\infty: = \limsup_{t\to\infty}{\lVert{g(t)}\rVert}$ and

Then, the following estimates on $u$ hold:

3.1.   Continuous dependence on the forcing term

We now prove the continuous dependence of the solutions on the forcing term under suitable smallness conditions on the parameters of the problem.

Proposition 3.3. Let $u$ and $v$ be weak solutions respectively of the problems

where $g_1, g_2\in C_b^0(\mathbb{R}_+, \mathcal{H})$. Let $\Upsilon_\mu: = \limsup_{t\to\infty}{\lVert{(u(t)+v(t))/2}\rVert}_\mu^2$ with $\mu$ in $[0, 2]$. There exists $\mathcal{F}_\theta(\alpha_1, \delta, \Upsilon_\theta, \Upsilon_{2\theta})$ such that, if $\mathcal{F}_\theta < 1$ holds, then there exists $C > 0$ depending on $\delta$ and $g_\infty$ such that

Moreover, if there exists $\eta > 0$ such that $\limsup_{t\to\infty}{\lVert{g_1(t)-g_2(t)}\rVert}e^{\eta t} = 0$, then there exists $\eta_1 > 0$ such that

In particular, we can take

Proof. The idea of the proof is standard but, for our purposes, it is mandatory to fully report it since we are interested in making the smallness conditions required from our results explicit.

Let $\alpha > 0$. We define

and let $E$ be the quantity

Remark that, by using the Cauchy-Schwarz inequality, the Young inequality and (2.2), we get

where $C_1$ and $C_2$ are positive numbers, obtainable for suitable choices of the values of $\alpha$, $\varepsilon_1$ and $\varepsilon_2$. In particular, to get $C_2$ we have to require

Hence, for every $\alpha$ such that $\alpha < \sqrt{\alpha_1}$ we can find $\varepsilon_2$ such that (3.5) holds.

We first consider $u$ and $v$ as regular solutions of the problems in (3.1). We define $w: = v-u$ and $r: = g_2-g_1$. The function $w$ is the regular solution of the problem

We remark that, if $\xi: = (u+v)/2$, we have

From the definition of $\Lambda_\alpha$, by using (3.6) and (3.7), since $u$ and $v$ are regular solutions we get

Let $C_\mu = \sup_{t\geq 0}{\lVert{\xi(t)}\rVert}_\mu^2$ for any $\mu \in[0, 2]$. For a suitable choice of $\alpha$, by using Cauchy-Schwarz and Young inequality we have that for some positive constants $\bar{\alpha}$ and $\widetilde{\alpha}$

In particular, we choose the parameter $\alpha$ so that

Hence, since $\alpha < \sqrt{\alpha_1}$, if

we can find values of $\alpha$ such that (3.9) holds. Therefore we can find $\alpha$ such that (3.9) is satisfied if

Now, for some positive $\widetilde{\alpha}$ and $\widetilde{C}$ we get, from (3.8) and (3.9),

By defining

from (3.11) we obtain

Hence, from the Gronwall inequality and from the fact that for any $\varepsilon > 0$ there exists $t_0 > 0$ such that $|\widetilde{f}(s)|\leq \widetilde{C}(\varepsilon+\limsup_{t\to\infty}{\lVert{r(t)}\rVert})$ for any $s \geq t_0$, we get

Since we can take $\varepsilon$ arbitrarily small as $t_0$ goes to infinity, from (3.12) we infer that there exists $C > 0$ such that

Moreover, if there exists $\eta > 0$ such that $\limsup_{t\to\infty}{\lVert{r(t)}\rVert}e^{\eta t} = 0$, then (3.12) yields that there exists $\eta_1 > 0$ such that

From (3.5), there exists a positive constant $C_2$ such that $\Lambda_\alpha(t) \geq C_2 E(t)$. Therefore, (3.13) and (3.14) imply (3.2) and (3.3) respectively.

We remark that

Hence, we can take $C_\mu = \Upsilon_\mu$. Therefore, from (3.10), we get that if

then the thesis holds for regular solution $u$ and $v$.

The same conclusions hold for $u$ and $v$ weak solutions of the problems in (3.1) by using a standard density argument. Indeed, since $\mathcal{H}^4$ is dense in $\mathcal{H}^2$ and $\mathcal{H}^2$ is dense in $\mathcal{H}$, setting $(u(0) = u^0, u_t(0) = u^1)$ and $(v(0) = v^0, v_t(0) = v^1)$, there exists two sequences $(u_n^0, u_n^1)$ and $(v_n^0, v_n^1)$ in $\mathcal{H}^4\times \mathcal{H}^2$ such that

Hence, from Theorem 3.1 we have the two sequences of regular solutions $u_n$ and $v_n$ with $(u_n(0) = u_n^0, u_{n, t}(0) = u_n^1)$ and $(v_n(0) = v_n^0, v_{n, t}(0) = v_n^1)$ such that, for any $T > 0$,

Therefore, since all the calculations hold for $u_n$ and $v_n$ (and the difference $w_n: = u_n-v_n$), we get the thesis for the weak solutions $u$ and $v$ passing to the limit when $n\to \infty$.

3.2.   Some general stability results

In order to prove Theorem 2.3, we give a reformulation of Theorem 4.1 of ^[37] adapted to our framework.

Proposition 3.4. Let $(H, (\cdot, \cdot), |\cdot|)$ be a Hilbert space and let $A^2$ be a self-adjoint and strictly positive linear operator on $H$ with dense domain $\mathcal{D}(A)$. We introduce the Hilbert space $V: = \mathcal{D}(A)$ endowed with the norm ${\lVert{\cdot}\rVert}^2: = (A\cdot, A\cdot)$ and we identify the unbounded operator $A^2$ with its extension in $\mathcal{L}(V, V')$. The duality pairing in $V'\times V$ will be denoted in the same way as the inner product in $H$.

We consider $B(t)\in C^1(\mathbb{R}_+, \mathcal{L}(V, H))$ such that for any $v\in V$

for some positive numbers $\lambda$ and $\lambda'$.

Let $u$ be a bounded solution of

where $\delta > 0$, $g\in C(\mathbb{R}_+, H)$ and $\lim_{t\to\infty}|g(t)|e^{c_0 t} = 0$ for some positive constant $c_0$.

If

then there exists $c > 0$ such that

Proof. We proceed as in the proof of Theorem 4.1 of ^[37] and we define the quadratic form on $V\times H$ given by

For any fixed $t_0 > 0$ we have, if $t\geq t_0$,

for some positive constant $K$. Hence, for $t_0$ large enough

Therefore, if $\lambda' < \delta$ we get, for some positive $\alpha$,

for any $t\geq t_0$ and from Gronwall lemma we get the thesis.

We recall a further stability result due to Haraux for an ODE related to our problem.

Proposition 3.5. [Theorem 2.1 of ^[37]] Let $\lambda, \delta > 0$, $a\in L^{\infty}(\mathbb{R}_+)$ with $a(t)\geq 0$ for any $t\geq 0$. Let $x\in C^2(\mathbb{R}_+)$ be a solution of

Assume

There there are $\eta_1 > 0$ and $M > 0$ such that any bounded solution $x$ of (3.15) satisfies

for any $s\leq t$.

With minimal effort, the same statement can be proven for $x$ solving

where $\widetilde{g}\in C(\mathbb{R}_+)$ satisfies $\lim_{t\to\infty}\widetilde{g}(t)e^{\eta t} = 0$ for some $\eta > 0$.

3.3.   Linear analysis

Some preliminary results on the behavior of a damped and forced harmonic oscillator are useful in order to simplify the following study. In particular, we study the equation

where we require $\Psi$ to be antiperiodic. We recall that a function $f:\mathbb{R}\to \mathbb{R}$ is said to be antiperiodic of antiperiod $\tau$ (i.e. $\tau-$antiperiodic) if

Proposition 3.6. Let us consider $\Psi\in L^2_{loc}(\mathbb{R_+})$ antiperiodic of anti-period $\pi/\omega$. We suppose that $\lambda > 0$ and $\delta > 0$. Then there exists an antiperiodic solution $z$ of anti-period $\pi/\omega$ of (3.16) and we have that for some $\eta > 0$, for any $y(t)$ solution of (3.16),

Proof. Let us consider $\mathcal{A}_\omega\subset L^2([0, \pi/\omega])$ the space of the locally square-integrable antiperiodic functions with anti-period $\pi/\omega$, endowed with the standard $L^2$ norm on the interval $[0, \pi/\omega]$. The family $\{e_n = \sqrt{\omega/\pi}e^{(2n+1)i\omega t}\}_{n\in\mathbb{Z}}$ is an orthonormal basis of this space. Hence, we write

Setting

it is immediate to verify that $z(t)$ is an antiperiodic solution of (3.16). The thesis now follows from the standard theory of ODEs. Indeed, any solution of (3.16) is given by the sum of $z(t)$ with a general solution $y_g$ of the associated homogeneous equation

which is given by

with

where the arbitrary constants $S$ and $\varphi$ are dependent from the initial conditions. We notice that

for some constants $C > 0$ and $0\leq\mu < \delta/2$. Therefore, since $y(t) = z(t)+y_g(t)$, we get that for a suitable choice of $\eta > 0$

which is the thesis.

Proposition 3.7. Let us consider $\Psi\in L^2_{loc}(\mathbb{R_+})$ antiperiodic of anti-period $\pi/\omega$ and let $y(t)$ satisfy (3.16). We suppose $\lambda, \delta > 0$ and $2\sqrt{\lambda}\neq\delta$. We introduce the quantities

where, for any $w\in \mathbb{C}$, $\sqrt{w}$ is the complex number $z$ such that

Then the following estimate holds

Moreover, if $\Psi\in C^2(\mathbb{R}_+)$, then

Proof. From Proposition 3.6, Eq (3.16) admits an antiperiodic solution $z(t)$ and any solution of $y(t)$ of (3.16) converges exponentially to $z(t)$, which yields that $\limsup_{t\to\infty}y(t) = \limsup_{t\to\infty}z(t)$. Hence, since from the antiperiodicity of $z(t)$ we have that $\limsup_{t\to\infty}z(t) = {\lVert{z}\rVert}_{\infty}$, in order to get the result it suffices to estimate the $L^\infty-$norm of $z(t)$. In the notation of Proposition 3.6, we have that

Then, if $c_n = \sqrt{(-\omega^2(2n+1)^2+\lambda)^2+\delta^2\omega^2(2n+1)^2}$, from Cauchy-Schwarz inequality we obtain

Moreover, if $\Psi\in C^2(\mathbb{R}_+)$, we have

First, we remark that from Parseval's theorem

Then, to conclude the proof, we compute a closed form for the serie

We observe that (3.21) becomes

We now recall that the Mittag-Leffler expansion for the cotangent function gives

Some straightforward computations give

Thus, we can infer that

Hence, from (3.22) we can conclude that

By using (3.20) and (3.23) in (3.18) and (3.19), we obtain the thesis.

In [36,Theorem 2.1], a result similar to Proposition 3.7 is proven. In particular, the maximum value of $\limsup_{t\to\infty}y(t)$ as the forcing term $\Psi$ varies in the unitary ball of $L^{\infty}(\mathbb{R})$ is determined. On the other hand, for any fixed antiperiodic forcing term $\Psi$ in $C^2(\mathbb{R})$, in Proposition 3.7 we estimated $\limsup_{t\to\infty}y(t)$ and $\limsup_{t\to\infty}\dot{y}(t)$. As Figure 1 shows, Proposition 3.7 almost always gives a better estimate on $\limsup_{t\to\infty}y(t)$.

3.4.   Structure of the paper

The remainder of the paper is organized as follows. First, in Section 4 we apply the results of Subsection 3.2 in order to prove Theorem 2.3. In particular, we apply Proposition 3.4 to prove that for $N$ large enough, if $g$ is exponentially $N-$dimensional, then there exists $\bar{N}\geq N$ such that any solution $u$ of (2.5) is exponentially $\bar{N}-$dimensional (see Lemma 4.1). After that, fixed $n > N$, we study the asymptotic amplitude of $u_n(t) = (u(t), e_n)$ for any $u$ solution of (2.5) and in Lemma 4.2 we determine whether $u_n(t)$ decays exponentially as $t$ goes to infinity. In subsection 4.2 we exploit Lemma 4.1 and Lemma 4.2 in order to get Theorem 2.3. We remark that, even though the thesis of Theorem 2.3 follows from Lemma 4.1, Lemma 4.2 is necessary in order to improve the result of Lemma 4.1. More precisely, Lemma 4.2 provides an improvement of the smallest number $M\geq N$ obtained in Lemma 4.1 such that if $g$ is exponentially $N-$dimensional then any solution $u$ is exponentially $M-$dimensional.

Next, by exploiting the continuous dependence of the solution from the forcing term, that is, Proposition 3.3, and Theorem 2.3, in Section 5 we give the proof of Theorem 2.4.

In Section 5, by proceeding as in a result of Bonheure, Gazzola and Moreira dos Santos [8,Theorem 6], we show that (2.8) admits an antiperiodic solution $p$. In Lemma 6.2 we use Proposition 3.7 to estimate, for any $n\in \mathbb{N}$, the asymptotic amplitude of $p_n(t): = (p(t), e_n)$. Such result yields an estimate on the $H^s-$norms of $p$ (see Lemma 6.3) which we numerically verified to be better than the a-priori estimates obtained in ^[8] (see Figure 2). From Proposition 3.3, we have that under suitable smallness conditions on $\limsup_{t\to\infty}{\lVert{g(t)}\rVert}$, any solution $u$ of (2.8) converges to $p$ in the phase space norm. Hence, from Lemma 6.2 and Lemma 6.3, in Lemma 6.4 we get an estimate on the asymptotic amplitude of $u_n(t) = (u(t), e_n)$ and on the $H^s-$norms of $u$ for any $u$ solution of (2.8). Finally, in Lemma 6.5, we exploit the previous results of Section 6 in order to get a results for finite-dimensional systems of ODEs and in Subsection 2.5 we apply Lemma 6.5 and Lemma 6.4 to get Theorem 2.5.

4.   Proof of Theorem 2.3

4.1.   Stability of the higher modes

We now apply the results of the previous section to our framework in order to prepare the proof of Theorem 2.3.

Lemma 4.1. Let $u$ be a weak solution of (2.5). Let $g$ be exponentially $N-$dimensional. If there exists $\bar{N}\geq N$ such that

then there exists $\widetilde{\eta} > 0$ such that

Proof. Fix $\bar{N}\geq N$ and, for any $s\in [0, 2]$, let $\Upsilon_s: = \limsup_{t\to\infty}{\lVert{u(t)}\rVert}_s^2$. We introduce the operator-valued function $B(t): = {\lVert{u(t)}\rVert}_\theta^2A^\theta$. By using (2.3), we get that $w = Q_{\bar{N}}u$ solves

By using (2.4) we remark that for any $v\in \mathcal{H}^2$ such that $Q_{\bar{N}}v = v$

We introduce

By applying Proposition 3.4 to (4.1), from (4.2) we get that if

then $\varphi(t)\to 0$ exponentially as $t$ goes to infinity. This yields that there exists $\tilde{\eta} > 0$ such that

Therefore, since ${\lVert{{A}w}\rVert}^2 = {\lVert{w}\rVert}_2^2$, we get the thesis.

We now apply Proposition 3.5 to the projection of (2.5) on the $n-$th mode. The following lemma holds.

Lemma 4.2. Let $g$ be exponentially $N-$dimensional. For any weak solution $u$ of (2.5), if

then for any $M\ge n$ there exists $\tilde{\eta} > 0$ such that for any $n\le\bar{N}\leq M$

Proof. Fixed $n\geq N+1$, we consider the projection of $u$ on the $n-$th mode, i.e., $u_n: = (u, e_n)$. The function $u_n$ satisfies

Since $n\geq N+1$, for some $\eta > 0$, $\lim_{t\to\infty}(g(t), e_n)e^{\eta t} = 0$. Let us suppose that $\limsup_{t\to\infty}{\lVert{u(t)}\rVert}_\theta^2 < \delta\max(2^{\theta}\delta^{1-\theta}, 2\alpha_n^{(1-\theta)/2})$. Since

we have that

which yields that, from Proposition 3.5,

Since $(\alpha_j)_j$ is strictly increasing, $\max(2^{\theta}\delta^{1-\theta}, 2\alpha_n^{(1-\theta)/2})$ is an increasing sequence. Hence, if (4.3) holds, then for any $\bar{N}\geq n$

that implies that for any $M\ge n$ there exists $\tilde{\eta} > 0$ such that for any $n\le\bar{N}\leq M$

that is the thesis.

4.2.   Completion of the proof of Theorem 2.3

Let $g$ be exponentially $N-$dimensional and let $u$ be a weak solution of (2.5). We recall that, from Proposition 3.2, we have

We introduce the quantity $\bar{N}$ defined as the smallest integer number greater than $N$ such that

From (4.4), (4.5) implies

Hence, from Lemma 4.1, if (4.5) holds then there exists $\eta_1 > 0$ such that

We introduce the set

and we define

From Proposition 3.2 we have that $\limsup_{t\to\infty}{\lVert{u(t)}\rVert}_\theta^2\leq\Phi_\theta$. Hence, from Lemma 4.2, if $\underline{N}\neq +\infty$, there exists $\eta_2 > 0$ such that

for any $n\in [\underline{N}, \bar{N}]\cap \mathbb{N}$, which yields

Hence, if we set $P_{\infty}: = I$, $Q_{\infty}: = 0$ and $M: = \min\{\underline{N}, \bar{N}\}$, for some $\tilde{\eta} > 0$

This concludes the proof of Theorem 2.3.

5.   Proof of Theorem 2.4

Let us suppose that

where $\Phi_\theta$ and $\Phi_2$ are defined in Proposition 3.2. Since $\Phi_\theta$ and $\Phi_2$ depend on $g_\infty$ and $\delta$, we get that, for any fixed $\delta$, (4.5) translates into $F_\theta(\alpha_1, \delta, g_\infty) < 1$ for some $F_\theta$. Therefore, for any fixed $\delta > 0$, there exists $\bar{g}_\infty > 0$ such that if $g_\infty < \bar{g}_\infty$, then (4.5) holds. We remark that, since the term $g$ models the action of the wind along the deck of the bridge, we physically interpret (4.5) as a requirement on the load exerted on the structure by the wind. In particular, since $\bar{g}_\infty$ in engineering applications (see ^[23]) is proportional to the speed of the wind $v$, the relation (4.5) is equivalent to require that $v < \bar{v}$ for some $\bar{v} > 0$.

Let $u$ be a weak solution of (2.5) and for any $J = \{j_1, \dots, j_m\}$ let $v^J$ be a weak solution of the problem

We introduce the quantities $\Upsilon_\mu = \limsup_{t\to\infty}{\lVert{(u(t)+v^J(t))/2}\rVert}_\mu^2$, where $\mu\in [0, 2]$. From Proposition 3.3 with $g_1 = P_{J}g$ and $g_2 = g = P_{J}g+Q_{J}g$, there exists a function $\mathcal{F}_\theta = \mathcal{F}_\theta(\alpha_1, \delta, \Upsilon_\theta, \Upsilon_{2\theta})$, given by (3.4), such that if $\mathcal{F}_\theta < 1$ then there exists a constant $C > 0$ such that

Since $g = \mathfrak{g}f(t)$, for a suitable choice of $J$, we have that $C\limsup_{t\to\infty}{\lVert{Q_Jg(t)}\rVert} < \varepsilon$. Hence we can conclude that, for a suitable choice of the family $J$, (5.2) gives

From Proposition 3.2 and (2.2), we have that $\Upsilon_\theta\leq\Phi_\theta$ and $\Upsilon_{2\theta} < \alpha_1^{\theta-1}\Phi_2$. Hence, $\mathcal{F}_\theta < 1$ is implied by (4.5). Therefore, fixed $\delta$, if $g_\infty < \bar{g}_\infty$ for some positive constant $\bar{g}_\infty$, where $\bar{g}_\infty$ does not depend by $J$, then (5.3) holds. This proves the first part of Theorem 2.4.

Let now $g$ be exponentially $N$-dimensional and let $M\geq N$ be obtained from Theorem 2.3, i.e., let $M\geq N$ be such that for some $\eta > 0$

Let $u$ and $v$ be, respectively, weak solutions of (2.5) and

We remark that $u$ is solution of the following problem

Since we supposed $g$ to be exponentially $N$-dimensional and $M\geq N$, there exists $\eta > 0$ such that

Therefore, from Proposition 3.3 with $g_1 = P_{M}g$ and $g_2 = g = P_{M}g+Q_{M}g$ we have that, fixed $\delta$, if $g_\infty$ is sufficiently small, then there exists $\eta_1 > 0$ such that

Since $v = P_{M}v$, from (5.4) we get that for some $\tilde{\eta} > 0$

This concludes the proof of Theorem 2.4.

6.   Proof of Theorem 2.5

6.1.   Some preliminary results

In Theorem 2.5, we restrict ourselves to the case when the forcing term is antiperiodic in time due to the engineering interest of this case (see Section 7). Moreover, for the sake of simplicity, we consider the case $\theta = 0$. The antiperiodicity of the forcing term allows us to provide some more information about the solution of (2.8). In particular, proceeding as in Theorem 6 of ^[8], where the result was proven in the periodic framework, by using Proposition 3.6, we obtain the following statement:

Proposition 6.1. If $g(t)$ is a continuous antiperiodic function of anti-period $\tau$, then there exists a solution of (2.5) antiperiodic of anti-period $\tau$.

Proof. The proof proceeds as in [8,Theorem 6]. First, we fix $n\geq 1$ and we prove the existence of a $\tau-$antiperiodic solution for the problem

Hence, we seek a $\tau-$antiperiodic solution $u^n$ in the form

We consider the spaces $C^2_\tau(\mathbb{R})$ and $C^0_\tau(\mathbb{R})$ of $C^2$ and $C^2$ $\tau-$antiperiodic functions and in the same notations of [8,Theorem 6] we have that (6.1) is equivalent to

where $\mathit{\boldsymbol{h}}: = (h^n_1, \dots, h^n_n)$, $\mathit{\boldsymbol{g}}: = (g_1, \dots, g_n)$, $L_n$ is a diagonal operator such that

and

We observe that for any $\mathit{\boldsymbol{q}}\in (C^0_\tau(\mathbb{R}))^n$ from Proposition 3.6 there exists a unique $\mathit{\boldsymbol{h}}\in (C^2_\tau(\mathbb{R}))^n$ such that $L_n(\mathit{\boldsymbol{h}}) = \mathit{\boldsymbol{q}}$. Thanks to the compact embedding $(C^2_\tau(\mathbb{R}))^n\subset(C^0_\tau(\mathbb{R}))^n$, we have that the nonlinear map $\Gamma_n:(C^0_\tau(\mathbb{R}))^n\times [0, 1]\to (C^0_\tau(\mathbb{R}))^n$ defined by

is compact. Moreover, from Proposition 3.2 we have that there exists $H_n > 0$ (independent of $\nu$) such that if $\mathit{\boldsymbol{h}}\in (C^0_\tau(\mathbb{R}))^n$ solves $\mathit{\boldsymbol{h}} = \Gamma_n(\mathit{\boldsymbol{h}}, \nu)$, then

Hence, since the equation $\mathit{\boldsymbol{h}} = \Gamma_n(\mathit{\boldsymbol{h}}, 0)$ from Proposition 3.6 admits a unique $\tau-$antiperiodic solution, the Leray-Schauder principle ensures the existence of a solution $\mathit{\boldsymbol{h}}\in(C^0_\tau(\mathbb{R}))^n$ of $\mathit{\boldsymbol{h}} = \Gamma_n(\mathit{\boldsymbol{h}}, 1)$. This proves the existence of a $\tau-$antiperiodic solution of (6.1). The proof the result follows from the existence of a $\tau-$antiperiodic solution of (6.1) exactly as in [8,Theorem 6] by showing that the sequence $(u^n)$ converges to a $\tau-$antiperiodic solution $u$ of (2.8).

In this section we use the quantities

obtained by replacing $\lambda$ by $\alpha_j$ in Proposition 3.7.

We now apply Proposition 3.7 in order to get an estimate on the $j-$th mode of the antiperiodic solution $p$ of (2.8), which we proved to exist in Proposition 6.1. In the following, whenever a real-valued function $f(t)$ will be antiperiodic, we will write interchangeably $\limsup_{t\to\infty}f(t)$ and ${\lVert{f}\rVert}_\infty$.

Lemma 6.2. Let $p$ be an antiperiodic solution of (2.8). If

where $\Omega_j$ is defined in (6.2), then, if we set $\Upsilon_0: = \limsup_{t\to\infty}{\lVert{p(t)}\rVert}^2$ and $\Upsilon_v: = \limsup_{t\to\infty}{\lVert{p_t(t)}\rVert}^2$,

where $p_j: = (p, e_j)$ and $g_{j}: = \limsup_{t\to\infty}(g(t), e_j) = (\mathfrak{g}, e_j)$.

Proof. We study the $j-$th component of the problem (2.8), namely

We consider the antiperiodic solution $v$ of the problem

It is possible to verify that the general solution of (6.5) is given by

where the constants $S$ and $\varphi$ are determined by the initial data of (6.5). Hence, it follows that, for any choice of the initial data of (6.5),

If we subtract (6.5) from (6.4), if $w: = p_j-v$ we get

Hence, from Proposition 3.7 we get, if $\mathfrak{p}^{(0)}_j: = \limsup_{t\to\infty} p_j(t)$, $\mathfrak{p}^{(1)}_j: = \limsup_{t\to\infty} \dot{p}_j(t)$, $\Upsilon_0: =$ $\limsup_{t\to\infty}{\lVert{p(t)}\rVert}^2$ and $\Upsilon_v: = \limsup_{t\to\infty}{\lVert{{p}_t(t)}\rVert}^2$,

Since $p$ and $v$ are both antiperiodic, $w$ is antiperiodic and (6.7) gives

We get then

Hence, from (6.6) we get, since hypothesis (6.3) holds,

which yields

that is the thesis.

We now apply the results of Lemma 6.2 in order to get an estimate on the $\mathcal{H}-$norm and $\mathcal{H}^2-$norm of an antiperiodic solution $p$ of (2.8).

Lemma 6.3. Let $p$ be an antiperiodic solution of (2.8). Let us suppose that

where $\Phi_0$ is defined in Proposition 3.2. Then the following estimates hold:

Proof. We prove (6.10) only, since the proofs of (6.8) and (6.9) are completely analogous. From Lemma 6.2, by using that from Proposition 3.2 $\Upsilon_0: = \limsup_{t\to\infty}{\lVert{p(t)}\rVert}^2\leq \Phi_0$,

We recall that the sequence $(\alpha_j)_j$ is divergent. Therefore, for $j$ large enough, $w_j^- = \overline{w_j^+}$ and $|w_j^+-w_j^-| = 2\pi^2\delta\sqrt{\alpha_j-\delta^2/4}/\omega^2\geq \pi^2\delta\sqrt{\alpha_j}/\omega^2$. Hence

We remark that

Moreover, from the definition of $w_j^+$ (see (6.2)), we have that $\Im(w_j^+)\to+\infty$. Hence, we conclude that $\lim_{j\to\infty}|\tan(\sqrt{w_j^+}/2)| = 1$ and consequently

Then, since $\lim_{j\to\infty}\alpha_j = +\infty$ and $\max_{j}\Omega_j\Phi_0 < 1$, we have that, for some positive constant $C$, for any $j\in \mathbb{N}$

Therefore, by using that

we get that

that is the thesis.

We observe that, from Proposition 3.3, any solution $u$ of (2.8) exponentially converges to $p$ under suitable smallness conditions on ${\lVert{\mathfrak{g}}\rVert}$. Hence, Lemma 6.2 and Lemma 6.3 hold for any weak solution $u$ of (2.8). More precisely, the following lemma holds.

Lemma 6.4. Let $u$ be a weak solution of (2.8). If

where $F(\xi) = 3\xi\max(1/\delta, 1/(2\sqrt{\alpha_1}))/\sqrt{\alpha_1}$ and $\xi_\infty: = \left((\sqrt{\Phi_0}+\sqrt{\varphi})/2\right)^2$, then

and

where $\varphi, \varphi_v$ and $\varphi_2$ are defined in (6.8), (6.9) and (6.10) respectively.

Proof. Let $p$ be an antiperiodic solution of (2.8). We define $w = p-u$. The function $w$ solves

We proceed as in Proposition 3.3 and we get that if

where $\xi = (u+p)/2$, then

Since

from the monotonicity of $F$ we get that $F(\xi_\infty) < 1$ implies (6.11). Hence, the thesis follows from Lemma 6.2 and Lemma 6.3.

6.2.   The role of a single mode in the dynamics

Let us consider the finite-dimensional problem

where $\underline{x}(t) = (x_1(t), \dots, x_n(t)) \in \mathbb{R}^n$, $\underline{g}(t) = (g_1(t), \dots, g_n(t))$, $\Lambda = \rm{diag}(\alpha_j)_{j = 1}^n$ and ${\lVert{\cdot}\rVert}$ is the Euclidean norm in $\mathbb{R}^n$. This problem is a finite-dimensional approximation of (2.8).

Here, we estimate how much the evolution of the system changes as we eliminate a single mode from the dynamics. For the sake of simplicity, in the following we consider the case when the higher mode is the one we choose to neglect. We observe that

where $P_{n-1}(a_1, \dots, a_n) = (a_1, \dots, a_{n-1})$, $\Lambda_{n-1} = \rm{diag}(\alpha_j)_{j = 1}^{n-1}$. We consider now the function $\underline{y}(t)$, solution of

At this point, the question is reduced to estimate the (asymptotic) distance between the solution $\underline{x}$ of (6.12) and the solution $\underline{y}$ of (6.14). To this end, with a slight abuse of notations, we introduce the $\mathbb{R}^n-$norms ${\lVert{\cdot}\rVert}_1$ and ${\lVert{\cdot}\rVert}_2$ defined by ${\lVert{\underline{x}}\rVert}_1 = |x_1|+\dots +|x_n|$ and ${\lVert{\underline{x}}\rVert}_2 = \sqrt{\alpha_1|x_1|^2+\dots +\alpha_n|x_n|^2}$. We remark that the result is completely independent of the choice of the mode neglected. The following lemma holds.

Lemma 6.5. Let $\underline{x}$ and $\underline{y}$ be solutions of Eqs (6.12) and (6.14) respectively. Let $g = \underline{g}\sin(\omega t)$ with $\underline{g}\in \mathbb{R}^n$ and we suppose that $F(\xi_\infty) < 1,$ where $\xi_\infty$ is defined in Lemma 6.4 and $F(\xi) = 3\xi\max(1/\delta, 1/(2\sqrt{\alpha_1}))/\sqrt{\alpha_1}$. Moreover, we suppose that

Then there exists a function $S$ of the parameters of the problem such that if $S < 1$ then we have that

where $\underline{\chi} = (\chi_1, \dots \chi_n)$, $\chi_j: = \limsup_{t\to\infty}\max(|x_j(t)|, |y_j(t)|)$, $\underline{\chi}_v = (\chi_{1, v}, \dots \chi_{n, v})$ and $\chi_{j, v}: = \limsup_{t\to\infty}\max(|\dot{x}_j(t)|, |\dot{y}_j(t)|)$.

Proof. First, we remark that as in Lemma 6.4, since $F(\xi_\infty) < 1$, we have that there exist two antiperiodic functions $\underline{p_1}\in C^2(\mathbb{R}_+, \mathbb{R}^n)$ and $\underline{p_2}\in C^2(\mathbb{R}_+, \mathbb{R}^{n-1})$ such that

Therefore, since we are interested in the asymptotic behavior of our system, we can restrict ourselves to the case when $\underline{x}$ and $\underline{y}$ are both antiperiodic without loss of generality.

Let us consider the difference between Eqs (6.13) and (6.14). If we set $\underline{w}: = P_{n-1}\underline{x}$ and $\underline{z}: = \underline{w}-\underline{y}$, we get

where $\underline{\Psi} = -x_n^2 \underline{w} - ({\lVert{\underline{w}}\rVert}^2-{\lVert{\underline{y}}\rVert}^2)\underline{y}- {\lVert{\underline{w}}\rVert}^2\underline{z}$ and for the sake of simplicity, abusing the notations, we wrote ${\lVert{\underline{w}}\rVert}$ and ${\lVert{\underline{y}}\rVert}$ instead of ${\lVert{(\underline{w}, 0)}\rVert}$ and ${\lVert{(\underline{y}, 0)}\rVert}$ respectively.

We focus on one component, say $j$, in order to treat only scalar quantities. Hence, we consider the equation

where $\Psi_j = -x_n^2x_j-({\lVert{\underline{w}}\rVert}^2-{\lVert{\underline{y}}\rVert}^2)y_j-{\lVert{\underline{w}}\rVert}^2z_j = -x_n^2x_j-(\underline{w}-\underline{y}, \underline{w}+\underline{y})y_j-{\lVert{\underline{w}}\rVert}^2z_j$. The fact that $\underline{x}$ and $\underline{y}$ are antiperiodic implies that $\underline{\Psi}$ is antiperiodic too. Hence, we can apply Proposition 3.7 to (6.15) and, if we introduce the quantities

then, set $\mathcal{Z}: = \max_{t\geq 0}{\lVert{\underline{z}(t)}\rVert}$, we have

Therefore, set $Z_j: = {\lVert{z_j}\rVert}_\infty$ and $C_j: = \Omega_j\varphi$, by requiring that $C_j < 1$ for any $j = 1, \dots n$ we get

We define the quantity

and we suppose $S < 1$.

We remark that for any $\underline{x}\in \mathbb{R}^n$, ${\lVert{\underline{x}}\rVert}\leq {\lVert{\underline{x}}\rVert}_1: = |x_1|+\dots |x_n|$ and, for any bounded function $\underline{f}:\mathbb{R}\to\mathbb{R}^n$, $\sup_{t}{\lVert{\underline{f}(t)}\rVert}_1\leq {\lVert{f_1}\rVert}_\infty+\dots+ {\lVert{f_n}\rVert}_\infty$. Hence we have that $\mathcal{Z}\leq \sum_{j = 1}^{n-1} Z_j$. Therefore, by summing (6.16) over $j$ and solving in $\mathcal{Z}$ we get

Next, we remark that for any bounded function $\underline{f}:\mathbb{R}\to\mathbb{R}^n$ we have that $\sup_{t}{\lVert{\underline{f}(t)}\rVert}_2\leq \sqrt{\alpha_1}{\lVert{f_1}\rVert}_\infty+\dots+ \sqrt{\alpha_n}{\lVert{f_n}\rVert}_\infty$. Hence $\mathcal{Z}_2: = \max_{t\geq 0}{\lVert{\underline{z}(t)}\rVert}_2\leq \sum_{j = 1}^{n-1} \sqrt{\alpha_j}Z_j$ and from (6.16) and (6.17) it follows that

In particular, from (6.17) and (6.18) we conclude that there exist two positive constants $b$ and $c$ such that

Moreover, from (6.19) and (6.16), there exist constants $a_j$ such that

We now define $Z_j^{(1)}: = {\lVert{\dot{z}_j}\rVert}_\infty$ and $\mathcal{Z}^{(1)}: = \max_{t\geq 0}{\lVert{\dot{\underline{z}}(t)}\rVert}$. By applying Proposition 3.7 to (6.15) we get

Since ${\lVert{\underline{w}}\rVert}^2-{\lVert{\underline{y}}\rVert}^2 = (\underline{w}+\underline{y}, \underline{w}-\underline{y}) = (\underline{w}+\underline{y}, \underline{z})$, we have

Therefore from (6.22) and (6.21) we get

Hence, by using (6.19) and (6.20), if $L_j: = \chi_{j, v}+2\sqrt{\varphi_v\varphi}a_j+2(\sqrt{\varphi}\chi_{j, v}+\sqrt{\varphi_v}\chi_{j})b$ and $C_j$ is defined as before, then

By reasoning as before we conclude that, if $S < 1$, then

where $L$ is a suitable constant.

We are now able to estimate the asymptotic distance between $\underline{x}$ and $\underline{y}$, since

We remark that, since we can estimate $\varphi$ and $\varphi_v$ in function of $\underline{\chi}$ and $\underline{\chi}_v$, $S$ and $L$ are dependent by $\chi_1, \dots \chi_n$ and $\chi_{v, 1}, \dots \chi_{v, n}$ only. Therefore, from (6.23) we get the thesis.

6.3.   Completion of the proof of Theorem 2.5

Since $\mathfrak{g} = P_M\mathfrak{g}$, from Lemma 6.4 we get that, if $F(\xi_\infty) < 1$,

Therefore, we can rewrite (2.8) and (2.9) as finite-dimensional dynamical systems of the form (6.12) and (6.14) respectively.

We introduce the quantities

From Lemma 6.5, we have that if $\Omega_j\Phi_0 < 1$, $C_j = \Omega_j\varphi < 1$ for any $j\leq M$ and

where $\Phi_0$ and $\varphi$ are defined in Proposition 3.2 and in Lemma 6.3, then

where $L$ is obtained in the proof of Lemma 6.5. Fixed $\delta$, we recall that $S$ and $L$ are constants depending on $\chi_1, \dots \chi_n$ and $\chi_{v, 1}, \dots \chi_{v, n}$. Hence, since from Lemma 6.4 we have that

from (6.24) we obtain that

where $C$ is a constant depending on $A^2$, $\mathfrak{g}$ and $\omega$, that is the thesis.

7.   The intermediate piers model

In this section we show how the analysis performed in this paper can be useful in order to get some more information about the stability of real world structures such as suspension bridges.

While in the first part of the paper (Theorem 2.3 and Theorem 2.4) we study the general case given by (2.5), in the second part (Theorem 2.5) we focus in particular on the case when $\theta = 0$ and

In particular, taking $\mathcal{H} = L^2(I)$ with $I = [-\pi, \pi]$, $A = -\partial_{xx}$ and $\mathcal{D}(A) = \{v\in {H}^2(I)\cap {H}^1_0(I):v(-\pi) = v(\pi) = v(-a\pi) = v(b\pi) = 0\}$ for $a, b\in (0, 1)$, the results of Section 6 apply to the system

This choice of the forcing term comes from the fact that, in engineering literature (see ^[35]), the load due to the vortex shedding of the wind along the structure of the bridge is usually modeled in this way with $g(x)\equiv g_\infty \in \mathbb{R}$. The coefficient $g_\infty$ depends on the wind speed and on the geometry of the structure and $\omega$ is the frequency at which vortex shedding occurs. More precisely, we have that in engineering applications $g(x, t) = W^2\sin(\omega t),$ where $W$ is the scalar velocity of the wind blowing on the deck of the bridge and $\omega$ can be expressed in terms of the structural constants of the bridge and the aerodynamic parameters of the air. We refer to the European Eurocode ^[23] (see also ^[8]) for a more detailed discussion.

The peculiar expression of the forcing term allows us to improve the estimate on the asymptotic $\mathcal{H}^2-$norm of the solution of (7.1) that one is able to obtain with no other information on $g$ than the value of $\limsup_{t\to \infty}{\lVert{g(t)}\rVert}$. A comparison between the general estimate on $\limsup_{t\to \infty}{\lVert{u}\rVert}_2$ (see Proposition 3.2) obtained by using the methods of [8,Lemma 22] and the one obtained by using the antiperiodicity of the forcing term (see Lemma 6.4) is given in Figure 2. The data considered are $a = b = 14/25$, $\delta = 1.5$, and $\omega = 20$. The maximum value of $g_\infty$ considered represents the largest value of $g_\infty$ such that Lemma 6.4 can be applied.

The improvement in the estimates on the asymptotic $\mathcal{H}^2-$norm is obtained by using also ultimate bounds of the asymptotic amplitude of each mode. We represent in Figure 3 a comparison between these estimates, obtained in Lemma 6.4, and a numerical estimate on the asymptotic amplitude of each of the first $20$ modes. Fixed $\delta = 1.5$ and $g_\infty = 1.5$, we considered the cases when $\omega = 5$ (left) and $\omega = 10$ (right). We considered different positions of the piers, namely we chose $a = b = 14/25$ (up) and $(a, b) = (0.51, 0.67)$ (down). Each of these choices respect the hypothesis of Lemma 6.4. We remark that the mode with largest amplitude is such that $\sqrt{\alpha_j}/\omega \approx 1$.

The estimates on each single mode of $u$ allow us to study more precisely how the asymptotic $\mathcal{H}^2-$norm of $u$ varies as the position of the piers vary, i.e., as $a$ and $b$ varies (see Lemma 6.4). Since most suspension bridges have symmetrical piers with $a = b\in [1/2, 2/3]$, we restrict ourselves to the case where $(a, b)\in [1/2, 2/3]\times [1/2, 2/3]$. We represent in Figure 4 the estimate on the asymptotic $\mathcal{H}^2-$norm given by Lemma 6.4 in function of $a$ and $b$, with $\delta = 1.5$, $g_\infty = 1.5$ and $\omega = 10$ fixed. We remark that this figure does not give any information about the stability of the bridge as $a$ and $b$ vary. In fact, the stability of a bridge is more endangered by the concentration of the energy on a single mode than by the generalized oscillation of the structure.

In order to study the distribution of the $\mathcal{H}^2-$norm among the modes, we introduce the concept of family of asymptotic $\eta-$prevailing modes.

Definition 7.1. Let $0 < \eta < 1$. We say that a weak solution of (2.5) has a family $S = \{j_1, \dots j_n\}$ of asymptotic $\eta-$prevailing modes if

In Figure 5 we plot the number of $\eta-$prevailing modes for $\eta = 0.1$. The value of the parameters is the same as in Figure 4, namely $\delta = 1.5$, $g_\infty = 1.5$ and $\omega = 10$. We can observe that the asymptotic $\mathcal{H}^2-$norm concentrates on few modes as $a = b$. Moreover, we notice how the energy turns out to be more dispersed among the modes when $a\neq b$.

In conclusion, we are able to assert that under suitable smallness conditions on the asymptotic amplitude of the forcing term and on the nonlinearity, we are able to perform a rather accurate modal analysis for the nonlinear nonlocal beam equations considered. In particular, Figure 5, allows us to conclude that the more stable configurations are achieved when $a\neq b$. This suggests that, according to the model considered, asymmetric suspension bridges are more stable than suspension bridges where the piers are symmetric with respect to the center of the deck.

Acknowledgements

The author would like to express his sincere gratitude to an anonymous referee for the useful comments, remarks and recommendations which definitely helped to improve the readability and the quality of the paper.

Conflict of interest

The author declares no conflict of interest.