Uniform stabilization of beams by means of a pointwise feedback. (Q1848331)

The author considers the well-known simplified Euler-Bernoulli system with a feedback pointwise control. The basic equation is: \(\partial^2_t u+ \partial^4_x u+ F(\partial_t u)= 0\) in \(\Omega\times (0,\infty)\), with built in boundary conditions, and specified initial conditions: \(u(x,0)= u^0(x)\), \(\partial_t u(x,0)= u^1(x)\). \(F\) is the feedback function. The author notes that a very large number of publications have already discussed topics closely related to this project. The feedback applied at an interior point \(\xi\) is defined by the author, but essentially it is the Dirac delta and its special derivative (different limits from left or right of the point \(\xi\)). The author shows that to prove existence and uniqueness, it suffices to look at the infinitesimal generator of the \(C^0\) semigroup of linear contractions in the energy space. An interesting theorem is proved about the set of nodal and trinodal points of the eigenvalue problem: \(\partial^4_x\phi_n= \lambda_n\phi_n\) with zero boundary conditions. The author shows that this set \(N\) is countable, which implies that strong stabilization occurs almost everywhere in \(\Omega\). In his arguments the author skillfully uses the multiplier technique. This article is well written and provides proofs of theorems which have important implications in designing stabilizing controls for beams.