Numerical analysis and simulation for Rayleigh beam equation with dynamical boundary controls

In this paper, the Rayleigh beam system with two dynamical boundary controls is treated. Theoretically, the well-posedness of the weak solution is obtained. Later, we discretize the system by using the Implicit Euler scheme in time and the P3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P^3$$\end{document} Hermite finite element in space. In addition, we show the decay of the discrete energy and we establish some a priori error estimates. Finally, some numerical simulations are presented.


Introduction
In [1], Wehbe considered the Rayleigh beam equation with two dynamical boundary controls and the energy decay was theoretically established. In [2], the authors consider a clamped Rayleigh beam equation subject to only one dynamical boundary feedback. First, they considered the Rayleigh beam equation subject to only one dynamical boundary control moment, and later, they considered the Rayleigh beam equation subject to only one dynamical boundary control force and established in both cases a theoretical energy decay.
In [3], Rao considered the exact controllability of a hybrid system consisting of an elastic beam, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by one partial differential equation and two ordinary differential equations. Using the HUM method, they proved that the hybrid system is exactly controllable in an arbitrarily short time in the usual energy space.
In [4], the authors established, using a multiplier method, the polynomial energy decay rate for the smooth solutions of Kirchhoff plates equations. Consequently, they obtained the strong stability in the absence of compactness of the infinitesimal operator.
In [5], Rincon and Copetti studied numerically a locally damped wave equation. Error estimates for the semi-discrete and fully discrete schemes in the energy norm were provided.
In [6], a dynamic contact problem between a viscoelastic beam and a deformable obstacle is considered. The classical Timoshenko beam model is used and the contact is modeled using the well-known normal compliance contact condition. Fully discrete approximations were introduced, A priori error estimates were proved to obtain the linear convergence of the algorithm under an additional regularity condition. A numerical analysis for Bresse system is done in [8] and [7].
The aim of this paper is to study numerically the equation of Rayleigh beam, which is clamped at one end and subjected to two dynamical boundary controls at the other end. At an instant time t, the position y(x, t) of a given point x of the beam is governed by the following system of equations: ⎧ ⎪ ⎨ ⎪ ⎩ y tt − γ y x xtt + y x x x x = 0, y(0, t) = y x (0, t) = 0, where γ > 0 is a physical constant, and ξ and η designate, respectively, the dynamical boundary force and moment controls applied at the free end of the beam. The dynamical controls ξ and η are given by the following integral system: where 0 < x < 1, t > 0 and the following initial conditions: Let y be a smooth solution of the system (1)- (2). The associated energy E(t) is defined by: A straightforward calculations yields: Hence, system (1)-(2) is dissipative in the sense that the energy E(t) is a decreasing function of the time.
The notion of a dynamical control has been studied for the first time by the automaticians in the finitedimensional case (see Francis [9]). While, in the infinite-dimensional case, the notion of dynamical controls was studied by Russell [10].
Our purpose is to use the finite-element method and to obtain error estimates for the approximation of (1). Moreover, a fully discrete implicit scheme is proposed and analyzed. An outline of the contents of this paper is as follows. In Sect. 2, we show the well-posedness using the Faedo-Galerkin method (see [11]), as well as some regularity results. In Sect. 3, a semi-discrete Galerkin approximation to the solution of (1.1) is analyzed, and in Sect. 4, a fully discrete scheme is considered. We use the P 3 Hermite functions in space and the backward Euler method in time. We show that the fully discrete energy decays and derive some stability and error estimates. Finally, the results of numerical experiments illustrating the theoretical results are presented in the last section.

Existence and uniqueness
Denote by |.| and (., .) the norm and scalar product in L 2 (0, 1), respectively, and introduce the energy space: 1), and then use Green's formula with the initial and boundary conditions to obtain the weak form of (1).

Semi-discrete approximation
Let 0 = x 0 < x 1 < · · · < x s+1 = 1 be a uniform partition of the interval I = (0, 1) into sub-intervals and satisfying the following estimate (see [12]): Moreover, for all ψ ∈ H 2 E (I ) ∩ H 3 (I ), we have: We admit the following lemma: The semi-discrete finite element to (5) and use lemma (3.1) again, we get: (12) and the Poincare inequality, we get: Define the semi-discrete energy by: To show that the semi-discrete energy decays, we choose W = 2y h t , as a test function in (14) Therefore, we have: Finally, integrating from 0 to t to get stability: Where we used lemma (3.2) and the regularity on the initial data, and hence: The next theorem provides us with an error bounds for the piecewise linear approximation (14).

Theorem 3.3 Under the assumption of Theorem
Proof Introducingŷ = y t andŷ h = y h t , hence the continuous (5) and the semi-discrete (14) problems can be written as: In particular, the first equality is true ∀W ∈ V h E . Take υ = W ∈ V h E and subtract the two equalities to get: Eŷ and e h = y h − π h E y, and we get: Take W =ê h ∈ V h E and use (13), to obtain: Using (12), we have: Combining these results with (15) to obtain: Integrating from 0 to t, we find: Use the above estimates with the fact that: The regularity on y 0 , y 1 , and y, gives: Applying Gronwall inequality, to obtain: Therefore: where we used the regularity on y.

Fully discrete approximation
In this section, we introduce a fully discrete finite-element method to (11). Given an integer N > 0, our numerical scheme can be stated as find y n , n = 2, ..., N and ξ n , η n , n = 1, ..., N , such that, ∀W ∈ V h E , we have: where t = T /N is the time step, y 0 = π h E y 0 , y 1 = y 0 + tπ h E y 1 , ξ 0 = ξ 0 , and η 0 = η 0 . Writing Hermite basis for V h E , which will be defined in Sect. 5.
We find that the method defined requires the linear system of (2s + 4) algebraic equations which can be written in the matrix form as: Here: And C n = (c n 1 , c n 2 , ..., c n 2s+2 , η n , ξ n ) T , is the vector to be determined at each time step. Since the matrix M is non-singular, the system has a unique solution.
In a similar manner to the continuous case, the decay of the energy associated with the fully discrete problem and stability estimates are proved. Letŷ define the energy of the fully discrete problem by: Taking, W =ŷ n+1 in (16) 1 , we get: x (1) = 0. Using (16) 2 and (16) 3 , we get: y n+1 x (1) =η n+1 + η n+1 , and y n+1 (1) Using the elementary equality: We get: Therefore: For the stability, we multiply (21) by 2 t and sum from i = 1 to n, to get: Use lemma (3.2) and the definitions of y 1 andŷ 1 , to obtain: To bound η 1 , we use (16) 2 for n = 0 , the second part of (13) with x s = 1, the definition of y 1 , and the fact that y 1 (1) ≤ |y 1x |.
Therefore, we have: In the same way, we use (16) 3 to get |ξ 1 | ≤ C. Therefore:

Finite-element method
In the finite-element method, the basis functions are usually polynomials of any degree, defined in each finite element. In this paper, we are going to use the Hermite finite element (see [12] Page 168).
To define a basis for V h E , we introduce the two reference functions: Define the following bases functions by: Note that, for all 1 ≤ i, j ≤ s + 1, we have: With the above, V h E becomes a subspace of H 2 E (I ) of dimension 2s + 2, and for all y h ∈ V h E , y h can be written in the form: let us rename the basis as {μ i } 2s+2 i=1 , where:
The graphs in Figs. 1, 2, 3, and 4 represent, respectively, the time evolution of the beam's position y n (x, t), moment control η n , force control ξ n , and the discrete energy E n . Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.