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P. Vettori, S. Zampieri (2006)
Stability and stabilizability of delay-differential systems
L. Ehrenpreis (1970)
Fourier analysis in several complex variables
A. Quadrat (2003)
The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part II: Internal StabilizationSIAM J. Control. Optim., 42
REFERENCES
(2014)
Convolution behaviors and topological algebra
A. Quadrat (2003)
The fractional representation approach to synthesis problems: an algebraic analysis viewpointSiam Journal on Control and Optimization
M. Fliess, H. Mounier (1995)
Interpretation and Comparison of Various Types of Delay System Controllabilities 1IFAC Proceedings Volumes, 28
J. Delsarte (1935)
Les fonctions moyenne-périodiquesJ. Math. Pures Appl., 14
Y. Yamamoto (1988)
Pseudo-rational input/output maps and their realizations: a fractional representation approach to inSiam Journal on Control and Optimization
H. Mounier (1998)
Algebraic interpretations of the spectral controllability of a linear delay system, 10
H. Matsumura, M. Reid (1989)
Commutative Ring Theory
L. Schwartz (1966)
Théorie des distributions
C. Berenstein, B. Taylor (1979)
A new look at interpolation theory for entire functions of one variableAdvances in Mathematics, 33
L. Schwartz (1947)
Theorie Generale des Fonctions Moyenne-PeriodiquesAnnals of Mathematics, 48
M. Fliess, H. Mounier (1995)
Proceedings IFAC Conference System Structure and Control
P. Vettori, S. Zampieri (2000)
Controllability of systems described by convolutional or delay-differential equationsSIAM J. Control Optim., 39
H. Bourlès, U. Oberst (2009)
Duality for Differential-Difference Systems over Lie GroupsSIAM J. Control. Optim., 48
S. Eijndhoven, L. Habets (2003)
Equivalence of Convolution Systems in a Behavioral FrameworkMathematics of Control, Signals and Systems, 16
H. Gluesing-Luerssen, P. Vettori, S. Zampieri (2000)
The algebraic structure of DD systems : a behavioral perspective
P. Vettori, S. Zampieri (2001)
Some results on systems described by convolutional equationsIEEE Trans. Autom. Control., 46
P. Vettori, S. Zampieri (2002)
Module theoretic approach to controllability of convolutional systemsLinear Algebra and its Applications
H. Bourlès, U. Oberst (2014)
Proc. MTNS 2014
A. Quadrat (2003)
The Fractional Representation Approach to Synthesis Problems: An Algebraic Analysis Viewpoint Part I: (Weakly) Doubly Coprime FactorizationsSIAM J. Control. Optim., 42
Y. Yamamoto (1988)
Pseudo-rational input/output maps and their realizations: A fractional representation approach to infinite-dimensional systemsSIAM J. Control Optim., 26
L. Ehrenpreis (1956)
Solutions of Some Problems of Division: Part III. Division in the Spaces, D , H, Q A , OAmerican Journal of Mathematics, 78
L. Ehrenpreis (1960)
Solutions of some problems of division IVAm. J. Math., 82
If H has an invertible common denominator f ( 0 (cid:54) = f ∈ E (cid:48) , fH ∈ E (cid:48) p × m ) then Q ∈ E (cid:48) k × m and hence B is a behavior by item 2.
N. Bourbaki (1987)
Topological Vector Spaces
O. Forster (1964)
Primärzerlegung in Steinschen AlgebrenMathematische Annalen, 154
L. Ehrenpreis (1960)
Solution of Some Problems of Division. Part IV. Invertible and Elliptic OperatorsAmerican Journal of Mathematics, 82
C. Berenstein, B. Taylor (1980)
Mean-periodic functionsInternational Journal of Mathematics and Mathematical Sciences, 3
L. Habets (1997)
System Equivalence for AR-Systems over Rings—with an Application to Delay-Differential SystemsMathematics of Control, Signals and Systems, 12
R. Bellman, K. Cooke (1967)
Differential-Difference Equations
P. Vettori, S. Zampieri (2000)
Controllability of systems described by convolutional or delay-differential equationsProceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228), 1
U. Oberst (1995)
Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficientsApplicable Algebra in Engineering, Communication and Computing, 6
L. Hörmander (2005)
Analysis of Linear Partial Differential Operators II
P. Rocha, J. Willems (1997)
Behavioral Controllability of Delay-Differential SystemsSiam Journal on Control and Optimization, 35
F. Parreau, Y. Weit (1989)
Schwartz's theorem on mean periodic vector-valued functionsBulletin de la Société Mathématique de France, 117
L. Hörmander (1990)
The analysis of linear partial differential operators
Y. Yamamoto, J.C. Willems (2008)
Proceedings of the 47th Conference on Decision and Control
Computations and constructions : In contrast to the theory of finitely presented modules over the Bézout domain O (cid:84) C ( s )[ σ, σ − 1 ] from above [16]
P. Vettori, S. Zampieri (2006)
Constructive Algebra and Systems Theory
H. Glüsing-Lüerssen (1997)
A Behavioral Approach To Delay-Differential SystemsSiam Journal on Control and Optimization, 35
H. Gluesing-Luerssen (2001)
Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach
D. Brethé, J.J. Loiseau (1996)
Proc. 4th IEEE Mediterranean Symp. Control and Autom.
J. Kahane (1954)
Sur quelques problèmes d'unicité et de prolongement relatifs aux fonctions approchables par des sommes d'exponentiellesAnnales de l'Institut Fourier, 5
Y. Yamamoto, J. Willems (2008)
Behavioral controllability and coprimeness for a class of infinite-dimensional systems2008 47th IEEE Conference on Decision and Control
L. Schwartz (1947)
Théorie générale des fonctions moyenne-périodiquesAnn. Math., 48
H. Bourlès, U. Oberst (2012)
Elimination, fundamental principle and duality for analytic linear systems of partial differential-difference equations with constant coefficientsMathematics of Control, Signals, and Systems, 24
J. Loiseau (2000)
Algebraic tools for the control and stabilization of time-delay systemsVaccine
We investigate one-dimensional ‘generalized convolution behaviors’ (gen. beh.) that comprise differential and delay-differential behaviors in particular. We thus continue work of, for instance, Brethé, van Eijndhoven, Fliess, Gluesing-Luerssen, Habets, Loiseau, Mounier, Rocha, Vettori, Willems, Yamamoto, Zampieri of the last twenty-five years. The signal space for these behaviors is the space E of smooth complex-valued functions on the real line. The ring of operators is the commutative integral domain E′ of distributions with compact support with its convolution product that acts on E by a variant of the convolution product and makes it an E′-module. Both E and E′ carry their standard topologies. Closed E′-submodules of finite powers of E were introduced and studied by Schwartz already in 1947 under the name ‘invariant varieties’ and are called gen. beh. here. A gen. beh. is called a behavior if it can be described by finitely many convolution equations. The ring E′ is not noetherian and therefore the standard algebraic arguments from one-dimensional differential systems theory have to be completed by methods of topological algebra. Standard constructions like elimination or taking (closed) images of behaviors may lead to gen. beh. and therefore the consideration of the latter is mandatory. It is not known whether all gen. beh. are indeed behaviors, but we show that many of them are, in particular all autonomous ones. The E′-module E is neither injective nor a cogenerator and, in particular, does not admit elimination in Willems’ sense. But the signal submodule PE of all polynomial-exponential signals is injective for finitely generated modules and thus admits elimination. This is a useful replacement and approximation of the injectivity of E since the polynomial-exponential part of any gen. beh. is dense in it. We also describe a useful replacement of the cogenerator property and thus establish a strong relation between convolution equations and their solution spaces. Input/output structures of gen. beh. exist and are used to prove that also many nonautonomous generalized behaviors are indeed behaviors. The E′-torsion elements of E, i.e., the smooth functions which satisfy at least one nonzero convolution equation, are called ‘mean-periodic functions’ and were studied by many outstanding analysts. Their results are significant for gen. beh.
Acta Applicandae Mathematicae – Springer Journals
Published: Feb 5, 2015
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