Abstract
A continuous multi-dimensional behavior is defined to be the solution space of a linear system of constant-coefficient partial differential equations. After revisiting the concepts of controllability and extendibility for systems of this type, the following main result is obtained: behaviors that are both controllable and extendable are characterized by a concatenation property that naturally generalizes the notion of strong controllability, originally introduced for two-dimensional discrete behaviors by Willems and the second author.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
L. Ehrenpreis (1970) Fourier analysis in several complex variables Wiley New York Occurrence Handle0195.10401
D. Eisenbud (1995) Commutative algebra with a view toward algebraic geometry Springer Berlin Occurrence Handle0819.13001
M. Fliess J. Lévine P. Martin P. Rouchon (1995) ArticleTitleFlatness and defect of non-linear systems International Journal Control 61 1327–1361 Occurrence Handle0838.93022
Malgrange B. (1964). Systèmes différentiels à coefficients constants. Séminaire Bourbaki 15 (1962/1963) (248), 11 p.
M Morf B.C. Lévy S.-Y. Kung (1977) ArticleTitleNew results in 2-D systems theory, part I Proceedings IEEE 65 861–872 Occurrence Handle10.1109/PROC.1977.10582
Oberst, U. (1990). Multidimensional constant linear systems. Acta Applicandae Mathematicae 20, 1–175 (1990).
V.P. Palamodov (1970) Linear differential operators with constant coefficients Springer Berlin Occurrence Handle0191.43401
H. Pillai S. Shankar (1999) ArticleTitleA behavioral approach to control of distributed systems SIAM Journal Control Optimization 37 388–408 Occurrence Handle10.1137/S0363012997321784 Occurrence Handle1655859 Occurrence Handle0919.93020
J.-F. Pommaret A. Quadrat (1999) ArticleTitleAlgebraic analysis of linear multidimensional control systems IMA Journal Mathematical Control Information 16 275–297 Occurrence Handle10.1093/imamci/16.3.275 Occurrence Handle1706658 Occurrence Handle01389547
Quadrat, A. Controllability and removable singularities: extension of the solutions of linear multidimensional control systems. Preprint.
P. Rocha J.C. Willems (1991a) ArticleTitleControllability of 2-D systems IEEE Transactions Automatic Control 36 413–423 Occurrence Handle10.1109/9.75100 Occurrence Handle1097094 Occurrence Handle0753.93008
P. Rocha J.C. Willems (1991b) ArticleTitleOn the representation of 2-D systems Kybernetika 27 225–230 Occurrence Handle1116836 Occurrence Handle0746.93016
P. Rocha J. Wood (1997) ArticleTitleA new perspective on controllability properties for dynamical systems Applied Mathematics Computer Science 7 869–879 Occurrence Handle1622473 Occurrence Handle0908.93011
P. Rocha E. Zerz (2005) ArticleTitleStrong controllability and extendibility of discrete multidimensional behaviors Systems and Control Letters 54 375–380 Occurrence Handle10.1016/j.sysconle.2004.08.015 Occurrence Handle2123710 Occurrence Handle1129.93339
H.L. Trentelman (2005) ArticleTitleOn flat systems behaviors and observable image representations Systems and Control Letters 51 51–55 Occurrence Handle10.1016/S0167-6911(03)00206-8 Occurrence Handle2026261
J. Wood E. Rogers D.H. Owens (1998) ArticleTitleA formal theory of matrix primeness Mathematics Control Signals Systems 11 40–78 Occurrence Handle1620017 Occurrence Handle0898.93008
E. Zerz (2000) Topics in multidimensional linear systems theory Springer Berlin Occurrence Handle1002.93002
E. Zerz (2001) ArticleTitleExtension modules in behavioral linear systems theory Multidimensional Systems and Signal Processing 12 309–327 Occurrence Handle10.1023/A:1011905623429 Occurrence Handle1031.93061 Occurrence Handle1868812
E. Zerz (2003) ArticleTitleOn the extension of solutions of linear constant-coefficient PDE Systems and Control Letters 50 127–133 Occurrence Handle10.1016/S0167-6911(03)00148-8 Occurrence Handle2012302 Occurrence Handle05055556
E. Zerz (2004) ArticleTitleMultidimensional behaviors: an algebraic approach to control theory for PDE International Journal Control 77 812–820 Occurrence Handle10.1080/00207170410001726822 Occurrence Handle1136.93320 Occurrence Handle2082175
Author information
Authors and Affiliations
Corresponding author
Additional information
Published online: April 2006
Rights and permissions
About this article
Cite this article
Zerz, E., Rocha, P. Controllability and Extendibility of Continuous Multi-dimensional Behaviors. Multidim Syst Sign Process 17, 97–106 (2006). https://doi.org/10.1007/s11045-005-6231-8
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11045-005-6231-8