A BEHAVIORAL APPROACH TO CONTROL OF DISTRIBUTED SYSTEMS ∗ HARISH K. PILLAI † AND SHIVA SHANKAR † SIAM J. CONTROL OPTIM. c  1998 Society for Industrial and Applied Mathematics Vol. 37, No. 2, pp. 388–408 Abstract. 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 submodules, whereas this is not the case for n-D distributed systems, n ≥ 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. Key words. distributed systems, systems of partial differential equations, controllability, sta- bility, pole placement AMS subject classifications. 93C20, 93C35, 35B37, 35E20 PII. S0363012997321784 1. Introduction. In this paper we develop a theory of control of distributed systems patterned after the behavioral approach for lumped systems in Willems [10, 11]. Thus we study the control of behaviors of systems of linear, constant coefficient partial differential operators. In this paper, we demonstrate that, while behaviors of distributed systems are similar to the behaviors of lumped systems in some respects, there are nonetheless many points of departure between the two cases, especially in the techniques employed to arrive at the results. This is essentially due to the fact that lumped systems are defined over a principal ideal domain (PID), whereas distributed systems are not. In [10], Willems initiates his approach to the study of systems by first considering ARMA systems. (We adopt the terminology there to define various systems like ARMA, AR, and MA systems, which are again formally defined in section 2 below.) He establishes an “elimination theorem” for ARMA systems; i.e., he proves that every ARMA system is equivalent to an AR system. This follows from the fact that every submodule of a free module over the principal ideal domain R[ d dx ] is free. On the other hand, the elimination theorem for distributed systems requires the celebrated Ehrenpreis–Palamodov theorem (see Oberst [5, Corollary 38] for a constructive proof). Our study of distributed systems is algebraic in the sense that we set up a corre- spondence between smooth behaviors and submodules of free modules over polynomial rings (in several indeterminates). That this correspondence is one to one is the content of a hard theorem of Oberst [5] and is in fact a central result of his seminal paper. ∗ Received by the editors May 21, 1997; accepted for publication (in revised form) March 26, 1998; published electronically November 23, 1998. http://www.siam.org/journals/sicon/37-2/32178.html † Department of Electrical Engineering, Indian Institute of Technology, Powai, Bombay 400076, India (sshankar@ee.iitb.ernet.in). 388