Topological properties of observability for a system of parabolic type

Abstract

The purpose of the present paper is to demonstrate topological properties of observable regions in a distributed parameter system. A parabolic partial differential equation with constant coefficients is considered. According to Sakawa's definition, observability is defined to be the possibility of the unique determination of the initial value by point measurements, or by spatially averaged measurements. Furthermore, n-mode observability is defined to be the possibility of the unique determination of the coefficients corresponding to the first n eigenvalues, based on the expansion of the solution by eigenfunctions. Then it is proved that n-mode observability is generic, that is, open and dense, whereas observability is shown to be dense in the whole space of measurements. In case of point measurements, it is shown that observability is valid almost everywhere with respect to the Lebesque measure. Moreover genericity of n-mode controllability and the related properties of controllability will be shown for the dual systems with controls.