Control of Systems Governed by Partial Differential Equations

In many applications, such as diffusion and structural vibrations, the physical quantity of interest depends on both position and time. Some examples are shown in Figures 1-3. These systems are modelled by partial differential equations (PDE’s) and the solution evolves on an infinite-dimensional Hilbert space. For this reason, these systems are often called infinite-dimensional systems. In contrast, the state of a system modelled by an ordinary differential equation evolves on a finite-dimensional system, such as R, and these systems are called finite-dimensional. Since the solution of the PDE reflects the distribution in space of a physical quantity such as the temperature of a rod or the deflection of a beam, these systems are often also called distributedparameter systems (DPS). Systems modelled by delay differential equations also have a solution that evolves on an infinite-dimensional space. Thus, although the physical situations are quite different, the theory and controller design approach is quite similar to that of systems modelled by partial differential equations. However, delay differential equations will not be discussed directly in this article. The purpose of controller design for infinite-dimensional systems is similar to that for finite-dimensional systems. Every controlled system must of course be stable. Beyond that, the goals are to improve the response in some well-defined manner, such as by solving a linear-quadratic optimal con-