Dynamics of extensible beams with nonlinear non-compact energy-level damping

Abstract

This work is motivated by experimental studies (NASA Langley Research Center) of nonlinear damping mechanisms present in flight structures. It has been observed that the structures exhibit significant nonlinear damping effects which are functions of the energy of the system. The present work is devoted to the study of long-time dynamics to a class of extensible beams/plates featuring nonlocal nonlinear energy damping of hyperbolic nature. Such models arise frequently in aeroelasticity when modeling flight structures, see NASA-AirForce reports (Balakrishnan in A theory of nonlinear damping in flexible structures. Stabilization of flexible structures, 1988; Balakrishnan and Taylor in Proceedings Damping 89, Flight Dynamics Lab and Air Force Wright Aeronautical Labs, WPAFB, 1989). The main mathematical challenge in this context is twofold: (1) nonlinear and potentially degenerate energy damping coefficient, (2) energy damping at a critical level where the usual compactness arguments (critical to the theory of attractors) do not apply. Our investigation sheds some light on a long-time behavior of such class of problems, providing new results in the area of existence of attractors and their properties within this hyperbolic-like framework. This should be contrasted with widely studied parabolic-like models involving structural damping which is known to be poorly understood. The goal is achieved by developing new methodology which allows to circumvent the difficulties related to the lack of compactness and non-locality of the nonlinear damping. The results are achieved through a rigorous analysis that reveals an interplay between extensibility, non-locality, and nonlinear energy damping of critical exponent.

Appendix: a short review on long time behavior of evolution operators

In order to keep this work self-contained, we find convenient to recall several definitions characterizing a long time behavior of dynamic evolutions such as $S_{\epsilon }(t)$. This can be found in many references, including [1, 8, 9, 11, 21, 25].

1.1 Definitions

Let (X, S(t)) be a dynamical system, where X is a Banach space.

Definition A.1

A global attractor for (X, S(t)) is a compact set ${\mathfrak {A}}\subset X$ that is fully invariant and uniformly attracting, that is, $S(t){\mathfrak {A}}={\mathfrak {A}}$ for all $t\ge 0$ and for every bounded subset $B\subset X$

$$\begin{aligned} {\text{ dist }}_X(S(t)B,{\mathfrak {A}}) = \sup _{x\in S(t)B}\inf _{y\in {\mathfrak {A}}}d(x,y) \rightarrow 0 \quad \text{ as } \quad t \rightarrow \infty . \end{aligned}$$

Definition A.2

A bounded set $D \subset X$ is an absorbing set for S(t) if for any bounded set $B\subset X$, there exists $t_B\ge 0$ such that

$$\begin{aligned} S(t)B\subset D, \quad \forall \, t \ge t_B, \end{aligned}$$

which defines (X, S(t)) as a dissipative dynamical system.

Definition A.3

(X, S(t)) is said to be (ultimate) dissipative iff it posesses a bounded absorbing set B. If X is a Banach space, then a value $R>0$ is said to be a radius of dissipativity of (X, S(t)) iff $B\subset \{x\in X:\,\Vert x\Vert _X\le R\}$.

Definition A.4

We say that S(t) is asymptotically smooth in X, if for any bounded positive invariant set $B \subset X$, there exists a compact set $K \subset {\overline{B}}$, such that

$$\begin{aligned} {\text{ dist }}_X(S(t)B,K) = 0 \quad \text{ as } \quad t \rightarrow \infty . \end{aligned}$$

Definition A.5

The fractal dimension of a compact set $K\subset X$ is defined by

$$\begin{aligned} \text{ dim }K_f^X=\lim \sup _{\varepsilon \rightarrow 0}\frac{\ln (n(K,\varepsilon ))}{\ln (1/\varepsilon )}, \end{aligned}$$

where $n(X,\varepsilon )$ is the minimal number of closed balls of the radius $\varepsilon$ which cover the set K.

Definition A.6

Let ${\mathcal {N}}$ be the set of stationary points of the dynamical system (X, S(t)):

$$\begin{aligned} {\mathcal {N}}=\{v\in X:S(t)v=v\;\;\text{ for } \text{ all }\;\;t\ge 0\}. \end{aligned}$$

We define the unstable manifold $\textrm{M}^u({\mathcal {N}})$ emanating from set ${\mathcal {N}}$ as a set of all $y\in X$ such that there exists a full trajectory $\Upsilon =\{u(t):t\in {\mathbb {R}}\}$ with the properties

$$\begin{aligned} u(0)=y\quad \text{ and }\quad \lim _{t\rightarrow -\infty }\text{ dist}_X(u(t),{\mathcal {N}})=0. \end{aligned}$$

Definition A.7

The dynamical system (X, S(t)) is said to be gradient if there exists a strict Lyapunov function for (X, S(t)) on the whole phase space X.

Definition A.8

Let X, Y be two reflexive Banach spaces with X compactly embedded in Y and set $H=X\times Y$. Consider the dynamical system (H, S(t)) given by an evolution operator

$$\begin{aligned} S(t)z = (u(t),u_{t}(t),), \qquad z=(u_0, u_1) \in H , \end{aligned}$$

(A.1)

where the functions u and $\theta$ possess the regularity

$$\begin{aligned} u \in C({\mathbb {R}}^{+}; X)\cap C^{1}({\mathbb {R}}^{+}; Y), \quad \end{aligned}$$

(A.2)

Then one says that (H, S(t)) is quasi-stable on a set $B \subset H$ if there exist a compact seminorm $n_X$ on X and nonnegative scalar functions a(t) and c(t) locally bounded in $[0,\infty )$, and $b(t)\in L^1({\mathbb {R}}^{+})$ with $\lim _{t\rightarrow \infty }b(t)=0,$ such that

$$\begin{aligned} \Vert S(t)z^1 - S(t)z^2 \Vert _{H}^2 \le a(t)\Vert z^1 - z^2 \Vert _{H}^2, \end{aligned}$$

(A.3)

and

$$\begin{aligned} \Vert S(t)z^1 - S(t)z^2 \Vert _{H}^2 \le b(t)\Vert z^1 - z^2 \Vert _{H}^2 + c(t) \sup _{0<s<t}\left[ n_X(u^1(s)-u^2(s)) \right] ^2 , \end{aligned}$$

(A.4)

for any $z^1,z^2 \in B$. The inequality (A.4) is often called stabilizability inequality.

Quasi-stable systems enjoy many interesting properties that include finite dimension and smoothness, cf. [8, 9].

1.2 Abstract results

Finally, we provide several abstract theorems pertaining to long time-behavior of dynamical systems, which have been used in the process of proofs related to Sects. 3–4.

It is well known that the properties of dissipativity and asymptotic smoothness are critical for proving existence of global attractors. In fact, the following result is well-known [8, 9].

Theorem A.1

(Theorem 2.3, [8]) Let S(t) be a dissipative semigroup defined on a metric space H. Then S(t) has a compact global attractor in H if and only if it is asymptotically smooth in H.

The following result establishes a convenient criteria for asymptotic smoothness of a dynamical system.

Theorem A.2

(Theorem 7.1.11, [9]) Let (X, S(t)) be a dynamical system on a complete metric space X endowed with a metric d. Assume that for any bounded positively invariant set B in X and for any $\varepsilon > 0$ there exists $T=T_{\varepsilon ,B}$ such that

$$\begin{aligned} d(S(T)y_1,S(T)y_2) < \varepsilon +\Psi _{\epsilon ,B,T}(y_1,y_2),\quad y_i\in B, \end{aligned}$$

(A.5)

where $\Psi _{\varepsilon ,B,T}(y_1,y_2)$ is a functional defined on $B\times B$ such that

$$\begin{aligned} \liminf _{m\rightarrow \infty }\liminf _{n\rightarrow \infty }\Psi _{\varepsilon ,B,T}(y_n,y_m)=0 \end{aligned}$$

(A.6)

for every sequence ${y_n}$ from B. Then (X, S(t)) is an asymptotically smooth dynamical system.

The following result also guarantees that quasi-stable systems are also asymptotically smooth.

Proposition A.3

(Proposition 7.9.5, [9]) Let Assumptions (A.1) and (A.2) be in force. Assume that the dynamical system (H, S(t)) is quasi-stable on every bounded forward invariant set ${\mathcal {B}}$ in H. The, (H, S(t)) is asymptotically smooth.

The following result gives the characterization of the attractor for gradient systems.

Theorem A.4

(Theorem 7.5.6, [9]) Let a dynamical system (X, S(t)) possess a compact global attractor ${\mathfrak {A}}$. Assume that there exists a strict Lyapunov function on ${\mathfrak {A}}$. Then ${\mathfrak {A}}=\textrm{M}^u({\mathcal {N}})$. Moreover, the global attractor ${\mathfrak {A}}$ consists of full trajectories $\Upsilon =\{u(t):t\in {\mathbb {R}}\}$ with the properties

$$\begin{aligned} \lim _{t\rightarrow -\infty }\text{ dist}_{X}(u(t),{\mathcal {N}})=0\quad \text{ and }\quad \lim _{t\rightarrow +\infty }\text{ dist}_{X}(u(t),{\mathcal {N}})=0. \end{aligned}$$

The next two results show that quasi-stable systems enjoy nice properties that include both finite-dimensional and smoothness.

Theorem A.5

(Theorem 7.9.6, [9]) Assume that the dynamical system (H, S(t)) possess a compact global attractor ${\mathfrak {A}}$ and is quasi-stable on ${\mathfrak {A}}$. Then the atractor ${\mathfrak {A}}$ of has a finite fractal dimension $dim^H_f{\mathfrak {A}}$.

Theorem A.6

(Theorem 7.9.8, [9]) Assume that the dynamical system (H, S(t)) possess a compact global attractor ${\mathfrak {A}}$ and is quasi-stable on ${\mathfrak {A}}$. Moreover, we assume that (A.4) holds with the function c(t) possessing the property $c_{\infty }=\sup _{t\in {\mathbb {R}}^+}c(t)<\infty$. Then any full trajectory $\{(u(t);u_t(t);\theta (t)):t\in {\mathbb {R}}\}$ that belongs to the global attractor enjoys the following regularity properties,

$$\begin{aligned} u_t\in L^{\infty }({\mathbb {R}};X)\cap C({\mathbb {R}};Y),\quad u_{tt}\in L^{\infty }({\mathbb {R}};Y),\quad \theta \in L^{\infty }({\mathbb {R}};Z). \end{aligned}$$

Moreover, there exists $R>0$ such that

$$\begin{aligned} |u_t(t)|_X^2+|u_{tt}(t)|_Y^2+|\theta _t(t)|_Z^2\le R^2,\quad t\in {\mathbb {R}}, \end{aligned}$$

where R depends on the constant $c_{\infty }$, on the semigroup $\eta _X$ in Definition A.8, also on the embedding properties of X into Y.

Finally, the following abstract result deals with upper-semicontinuity of attractors, see for instance the books by Robinson [21] and Chueshov [7].

Theorem A.7

(Theorem 10.16, [21]; Proposition 2.3.30, [7]) Assume that for each $\epsilon \in [0,\epsilon _0)$, $\epsilon _0>0$, the semigroup $S_{\epsilon }(t)$ have a global attractor ${\mathfrak {A}}_{\epsilon }\subset H$ such that:

• (i) the attractors are uniformly bounded, i.e.: there exists a bounded set $B_0\subset H$ such that ${\mathfrak {A}}_{\epsilon }\subset B_0$ for all $\epsilon \in [0,\epsilon _0)$;

• (ii) there exists $t_0 > 0$ such that the semigroup $S_{\epsilon }(t) x$ converge to $S_{0}(t) x$ as $\epsilon \rightarrow 0$ for every $t \ge t_0$ uniformly with respect to $x\in B_0$, i.e.:

  $$\begin{aligned} \sup _{x\in B_0 }|S_{\epsilon }(t)x-S_{0}(t)x|\rightarrow 0\quad \text{ as }\quad \epsilon \rightarrow 0^+. \end{aligned}$$

Then, the Hausdorff semidistance

$$\begin{aligned} {\text{ dist }}_H({\mathfrak {A}}_{\epsilon },{\mathfrak {A}}_0)\rightarrow 0\quad \text{ as }\quad \epsilon \rightarrow 0^+. \end{aligned}$$