Analysis and control of PDE systems - Transfer case

Transfer phenomena are physical processes that describe how different physical quantities move and propagate through space and time. These quantities can be heat, mass, momentum, electric charge, electric potential, concentration, and many others. Transfer phenomena are therefore studied in many areas of physics, such as fluid mechanics, thermodynamics, materials physics, particle physics and chemistry. As a result, they are essential elements of engineering. For example, heat transfer is an essential transport phenomenon in the design of cooling and heating systems. The diffusion of molecules across a membrane is also a transport phenomenon crucial to the functioning of living cells, and therefore plays a key role in bioengineering and medicine.

Understanding, controlling and observing systems involving transport phenomena are essential for solving practical problems in many fields. In general, the quantities transported involve quantities that vary as a function of spatial position and time. Models are therefore generally given by partial differential equations (PDEs). Tools adapted to this type of system are required. Automatics for systems described by PDEs has received particular attention in recent years. Numerous techniques have been developed in this field. In this article, we give an overview of the different methods for designing control laws for transport phenomena described by PDEs. The article is structured as follows. First, we present the usual methods for obtaining a representative mathematical model of these phenomena. In the second section, we present the various analytical elements required to study the behavior of such a system. The third section addresses the problem of control for systems of partial differential equations, before finally concluding.