Boundary stabilization of a linear elastodynamic systems with variable coefficients (Q2761586)

The paper under review follows a sequence of papers, by A. Guesmia, Lasiecka and Triggiani, which study a stabilizing control of elastic objects. Here they authors tackle the boundary stabilization of a well-studied classical elastodynamical system. (The word system has been abused, but it is hard to find another word which describes so many objects.) This system is the one proposed by Lagrange: \(u''- \text{div}(\sigma(u))= 0\) in \(\Omega\times\mathbb{R}_+\), with \(u= 0\) on \(\Gamma_0\times\mathbb{R}_+\), \(\sigma(u)+ A(u)+ Bu'= 0\) on \(\Gamma_1\times \mathbb{R}_+\), with the initial conditions assigned to \(u\), which belong to the space \(H^1\) on the boundary, and \(L^2\) in \(\Omega\). The authors observe the well-known fact that this system is well posed. They look at the present state of research and propose to offer a constructive proof of a boundary stabilization result for such a system (with variable coefficients). Twenty pages later they complete the proof, which proceeds at a leisurely pace. They discuss the metric and the \(C^\infty\) diffeomorphism, identify the eigenvalues of orthogonal projections with principal curvatures of the boundary, and look closely at the \(g_{ij}\) components of their metric after having introduced a functional which they prove to be a norm. Korn's inequality (comment: more precisely one of Korn's inequalities) is used in the proof. They use Komornik's multiplier technique to show that the energy functional regarded as a function of time is exponentially decreasing. Two cases had to be considered separately: 1) when the measure of \(\Gamma_0\) is equal to zero, and 2) when it is positive.NEWLINENEWLINENEWLINEThe paper is well written, the arguments are easy to follow.