A behavioral approach to control of distributed systems

This paper develops a theory of control for distributed systems (i.e., thos e defined by systems of constant coefficient partial differential operators ) via the behavioral approach of Willems. The study here is algebraic in th e sense that it relates behaviors of distributed systems to submodules of f ree modules over the polynomial ring in several indeterminates. As in the l umped case, behaviors of distributed ARMA systems can be reduced to AR beha viors. This paper first studies the notion of AR controllable distributed s ystems following the corresponding definition for lumped systems due to Wil lems. It shows that, as in the lumped case, the class of controllable AR sy stems is precisely the class of MA systems. It then shows that controllable 2-D distributed systems are necessarily given by free submodules, whereas this is not the case for n-D distributed systems, n greater than or equal t o 3. This therefore points out an important difference between these two ca ses. 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 au tonomous subbehavior. A notion of stability generalizing bounded input{boun ded output stability of lumped systems is proposed and the pole placement p roblem is defined for distributed systems. This paper then solves this prob lem for a class of distributed behaviors.