Continuous dependence of linear differential systems on polynomial modules
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Continuous dependence of linear differential systems on polynomial modules Vakhtang Lomadze1 Received: 21 February 2020 / Accepted: 3 August 2020 / Published online: 9 August 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract Linear differential systems, in Willems’ behavioral system theory, are defined to be the solution sets to systems of linear constant coefficient PDEs, and they are naturally parameterized in a bijective way by means of polynomial modules. In this article, introducing appropriate topologies, this parametrization is made continuous in both directions. Moreover, the space of linear differential systems with a given complexity polynomial is embedded into a Grassmannian. Keywords Linear systems · Modules · Parametrization · Continuity · Jets · Grassmannian Mathematics Subject Classification 93B25 · 93C05 · 93C20
1 Introduction Let F be one of the fields R or C (equipped with the standard topology), U = C ∞ (Rn , F), s = (s1 , . . . , sn ) the sequence of indeterminates, ∂ = (∂1 , . . . , ∂n ) the sequence of partial differentiation operators, and q a fixed positive integer. Let F[s]•×q denote the set of polynomial matrices with coefficients in F that have q columns. A polynomial matrix R ∈ F[s]•×q translates into a partial differential equation with constant coefficients R(∂)w = 0, w ∈ U q the solution set of which we denote by Bh(R) and call the behavior of R. Following Willems, we define a linear differential system with signal number q to be a subset of U q that is representable as the behavior of some matrix in F[s]•×q . This is a very
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Vakhtang Lomadze [email protected] Ivane Javakhishvili Tbilisi State University, Tbilisi, Georgia
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Mathematics of Control, Signals, and Systems (2020) 32:385–409
special case of a (continuous) dynamical system with signal number q, by which, in the behavioral theory of Willems, one understands an arbitrary subset of U q . Let S(q) denote the set of all linear differential systems with signal number q. By the very definition, one has a canonical parametrization F[s]•×q → S(q) :
R → Bh(R).
(1)
In [8,9,16] Nieuwenhuis and Willems have initiated a fundamental study of the problem whether this parametrization can be made continuous. This requires, first of all, specifying appropriate topologies on F[s]•×q and S(q). The problem is very interesting from the applied point of view (as well as from the purely mathematical one). We quote from the introduction of [16]: “One of the important issues in the study of mathematical models is the continuity of their behavior as a function of the parameters describing the behavioral equations. Bifurcation theory and structural stability are examples of research areas which address such questions. However, also areas as system identification, robustness of control systems, and the performance of adaptive control schemes are other areas where (implicitly) continuity questions are raised. The question of continuity is indeed an important issue in automatic control theory in particular,
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