Sharp observability inequalities for the 1-D plate equation with a potential

This paper deals with the problem of sharp observability inequality for the 1-D plate equation wtt + wxxxx + q(t, x)w = 0 with two types of boundary conditions w = wxx = 0 or w = wx = 0, and q(t, x) being a suitable potential. The author shows that the sharp observability constant is of order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exp \left( {C\left\| q \right\|_\infty ^{\tfrac{2} {7}} } \right)$$\end{document} for ‖q‖∞ ≥ 1. The main tools to derive the desired observability inequalities are the global Carleman inequalities, based on a new point wise inequality for the fourth order plate operator.