New results on controllability and stability for degenerate Euler-Bernoulli type equations

Abstract

In this paper we study the controllability and the stability for a degenerate beam equation in divergence form via the energy method. The equation is clamped at the left end and controlled by applying a shearing force or a damping at the right end. 1. Introduction. Starting from a paper of G. Leugering [21], we consider the following viscoelastic beam of length 1:which is clamped at the left end, x = 0, and controlled by applying a shearing force or a damping at the right end, x = 1. In particular, at x = 1, we consider the following conditions y(t, 1) = 0 and y x (t, 1) = f (t),(2) where f ∈ L 2 loc [0, +∞), or βy(t, 1) − (ay xx ) x (t, 1) + y t (t, 1) = 0, γy x (t, 1) + (ay xx )(t, 1) + y tx (t, 1) = 0, (3)