Stabilisation uniforme d'une équation des poutres d'Euler-Bernoulli. (Uniform stabilization of an Euler-Bernoulli beam equation).
The aim of this very important paper is the study of the equation for Euler-Bernoulli's beam governed by the linear combination of a velocity and the velocity of a rotation applied to one extremity of the system. The author verifies that the problem is well-posed and it has stability with respect to some conditions imposed on the feedback coefficients. Moreover, the author estimates the optimal decrease of energy using the method of Shkalikov. Here, for the transversal displacement at the point \([x,t]\) the following system is considered: \[ \begin{cases} y_{tt}+y_{xxxx} =0\text{ in }(0,1) & t>0\\ y(0,t)=y_x(0,t)=0 & t>0\\ -y_{xx}(1,t)=f_0(t) & t>0\\ y_{xxx}(1,t)=f_1(t) & t>0\\ y(\cdot,0)=y_0(\cdot),y_t(\cdot,0)=y_1(\cdot)& \text{in }(0,1)\end{cases}\tag{P} \] with \(f_0(t)=-2\varepsilon\mu_{1,1}y_t(1,t)+\mu_{1,2}y_{xt}(1,t),f_1(t)=\mu_{2,1}y_{xt}(1,t)-2\varepsilon \mu_{2,2}y_{xt}(t)\), where \(\varepsilon=\pm 1\), \(\mu_{i,j}\geq 0\), \(1\leq i,j\leq 2\) are positive constants such that \(\mu_{1,2}\mu_{2,1}>(\mu_{1,1}+\mu_{2,2})^2\). Main result: For any initial data \([y_0,y_1]^T\in H = V\times L_2(0,1)\) \((V = \{y\in H^2(0,1), y(0) = y_x(0)= 0\})\), the energy \(E(y(t))= (1/2)\int^1_0(| y_t(x,t)|^2+| y_{xx}(x,t)|^2)dx\) of (P) is a Lyapunov function and \(E(y(t))\) converges asymptotically to zero in the Hilbert space \(H\) as \(t\to \infty\). Moreover, for fixed \(M>1\), there exists a positive constant \(\omega >0\) such that for any initial data in \(H\), one has: \(E(y(t)) \leq ME(0)\exp(-\omega t)\) for any \(t>0\). Furthermore, if \(S = \{(\Phi_n,\Psi_n), n\in \mathbb{Z}\}\) is a system of eigenvectors of an associated operator, \(A\), suitably normalized, then \(S\) forms a Riesz basis for the energy space \(H\). Precise proofs of the main results are presented.
scientific article; zbMATH DE number 2068415