ICM 2026 Invited Section Lecture - Gengsheng Wang

Geometric Characteristics of Observable RegionsObservability is one of the main research areas in control theory. Different forms of observability inequalities not only correspond to different control phenomena but also have their own mathematical meanings. For evolutionary partial differential equations (PDEs), looking into the geometric characteristics of their observable sets/regions is one of the key parts in studying the corresponding observability inequalities. However, research in this area is still far from well understood. In this paper, we first present four distinct observability inequalities within the framework of abstract evolution equations, together with their corresponding control problems. Then, for several specific evolution equations (such as wave equations, heat equations, Schrödinger equations, among others), we provide the definitions of observable sets/regions corresponding to the aforementioned inequalities. Subsequently, we review some existing results and main research methods concerning the geometric characteristics of these observable sets/regions. Finally, we outline several open problems related to the geometric characteristics of observable (and weakly observable, etc.) sets/regions.Uploaded 07-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices