Monitoring distributed parameters systems

What is traditionally referred to as automation is a term that encompasses all techniques used to act on a dynamic system for which x(t), the state of the system at time t, is a vector of ${ℝ}^n$ , and therefore finite in size. In most cases, these systems are governed by differential equations, which may be linear or nonlinear (see  ) in this document database (ref.  ).

The purpose of this article is to discuss the control or regulation (the terms are equivalent) of systems governed by partial differential equations. The essential difference lies in the fact that, at each moment t, the state of the system, now denoted y(t), is a function of a space variable x (we will therefore also denote it y(x, t)); we can therefore consider y(t) as an element of a functional space that is not finite-dimensional, hence the terminology of infinite-dimensional dynamic systems. The term distributed system, which seems to have become established in the literature, comes from the fact that y(t) is a state "distributed" over the domain Ω of space. ${ℝ}^n$ , n = 1, 2, 3, in which the phenomena modeled by the partial differential equation occur. There are therefore very close links with automation, which will be emphasized throughout the presentation of the problems and methods.

A distinctive feature of this subject is that the study of stationary (time-independent) systems is highly relevant, and it is through this type of situation that the subject can be approached.

Finally, it should be noted that many problems that, at first glance, do not appear to be optimal control problems can naturally be reduced to them: this is the case, for example, with parameter identification and shape optimization problems.

Readers will find brief reminders in the appendix, in paragraph 8