Behaviors of \(n\)-D systems (Q2702495)

This paper is difficult to review, simply because it is already so condensed by the authors that it is hard to condense it much more.NEWLINENEWLINENEWLINEThey comment that in the most commonly accepted space formulation the input and output from the system are described in terms of the state, that is by a system of differential equations. In the lumped parameter representation this is for a system of ordinary differential equations, controllability and observability are the important concepts. In a more detailed view, one studies the evolution of the system, using semigroups determined by their infinitesimal generators. Great advances have been made in cases where the time evolution describes a class of physical processes. As the authors observe, the transition from very convenient description in terms of lumped parameters to the distributed parameters, that is from systems of ordinary differential equations to partial differential equations, produced several serious computational difficulties. The authors offer comments on the means of escaping many of these problems. Instead of regarding the response of the system primarily described by the input-output formalism, they discuss the behavioral approach, whereby a system is described by considering all possible trajectories. For shift invariant, distributed systems the behavior is a submodule (that is a commutative group with operators forming a ring) of some function space of generalized functions (or distributions). This submodule is a kernel of a matrix of constant coefficients partial differential operators of the (Noetherian) ring of differential operators, results in computational efficiency. The concepts of controllability and observability generalize perfectly to geometric descriptions, with algebraic characterization, whereby they become torsion modules, etc. \dots, all finitely generated. The authors use the now classical concepts of the theory of distributions: spaces \(D\) and \(D'\), \(S\) and \(S'\) as \(A\)-modules, with the structure defined by differentiation. Specifically, they let \(W\) be a submodule of \(D'\). They comment that many phenomena in engineering or physics are modelled by equations of the form: \(P(\partial)w= M(\partial)\ell\), where the manifest variable \(w\) is thus related to a latent variable \(\ell\).NEWLINENEWLINENEWLINEThe behavior of a system is defined as \(B= \{w\in w^k\mid P(\partial)w\in \text{Im}(M(\partial)\ell)\}\). The kernel of the morphism depends on the rows of the matrix \(P(\partial)\), while the image depends on the columns of \(P\). The fundamental theorem of Ehrenpreis-Malgrange-Palamodov basically states that every image of this map is also a kernel. It becomes clear that sequences of behaviors now can be rewritten as exact homological sequences.NEWLINENEWLINENEWLINEThe authors prove several propositions. The examples of application include the classical wave equation and also systems with storage functions, which may involve hidden variables.NEWLINENEWLINENEWLINEThe paper contains a broad outline of the ``behavior approach'', which already has shown possibilities of uncovering some basic, and otherwise not generally known properties of controlled systems, and in fact of quite arbitrary dynamical distributed parameters systems.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00062].