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 w _tt + w _xxxx + q(t, x)w = 0 with two types of boundary conditions w = w _xx = 0 or w = w _x = 0, and q(t, x) being a suitable potential. The author shows that the sharp observability constant is of order $\exp \left( {C\left\| q \right\|_\infty ^{\tfrac{2} {7}} } \right)$\exp \left( {C\left\| q \right\|_\infty ^{\tfrac{2} {7}} } \right) 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.