# zbMATH — the first resource for mathematics

A behavioral approach to control of distributed systems. (English) Zbl 0919.93020
This paper develops a theory of control for distributed systems (i.e. those defined by systems of constant coefficient partial differential operators) via the behavioral approach of Willems. The study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates. As in the lumped case, behaviors of distributed ARMA systems can be reduced to AR behaviors. This paper first studies the notion of AR controllable distributed systems following the corresponding definition for lumped systems due to Willems. It shows that, as in the lumped case, the class of controllable AR systems is precisely the class of MA systems. It then shows that controllable 2-D distributed systems are necessarily given by free modules, whereas this is not the case for $$n$$-D distributed systems, $$n \geq 3$$. This therefore points out an important difference between these two cases. This paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems.
Control is the process of restricting a behavior to a specific desirable autonomous subbehavior. A notion of stability generalizing bounded input-bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems. This paper then solves this problem for a class of distributed behaviors.

##### MSC:
 93B25 Algebraic methods 93C20 Control/observation systems governed by partial differential equations 35B37 PDE in connection with control problems (MSC2000) 35E20 General theory of PDEs and systems of PDEs with constant coefficients 93C35 Multivariable systems, multidimensional control systems 93B55 Pole and zero placement problems 93D25 Input-output approaches in control theory
Full Text: