×

zbMATH — the first resource for mathematics

On local linearization of control systems. (English) Zbl 1203.93044
Summary: We consider the problem of topological linearization of smooth (\(C^{\infty }\) or \(C^{\omega }\)) control systems, i.e., of their local equivalence to a linear controllable system via point-wise transformations on the state and the control (static feedback transformations) that are topological but not necessarily differentiable. We prove that the local topological linearization implies the local smooth linearization, at generic points. At arbitrary points, it implies the local conjugation to a linear system via a homeomorphism that induces a smooth diffeomorphism on the state variables, and, except at “strongly” singular points, this homeomorphism can be chosen to be a smooth mapping (the inverse map needs not be smooth). Deciding whether the same is true at “strongly” singular points is tantamount to solve an intriguing open question in differential topology.
MSC:
93B18 Linearizations
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
37C10 Dynamics induced by flows and semiflows
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] V. I. Arnold, Equations différentielles ordinaires. MIR (1974).
[2] V. I. Arnold, Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Mir, Moscow (1980).
[3] L. Baratchart, M. Chyba, and J.-B. Pomet, A Grobman–Hartman theorem for control systems. J. Dynam. Differ. Equations 19 (2007), 75–107 (2007). http://dx.doi.org/10.1007/s10884-006-9014-5 . · Zbl 1126.34029 · doi:10.1007/s10884-006-9014-5
[4] P. Brunovský, A classification of linear controllable systems. Kybernetika 6 (1970), 176–188. · doi:10.1007/BF00273963
[5] S. Celikovský, Topological equivalence and topological linearization of controlled dynamical systems. Kybernetika 31 (1995), No. 2, 141–150. · Zbl 0863.93013
[6] E. A. Coddington and N. Levinson, Theory of ordinary differential equations. McGraw-Hill, New York–Toronto–London (1955). · Zbl 0064.33002
[7] F. Colonius and W. Kliemann, Some aspects of control systems as dynamical systems. J. Dynam. Differ. Equations 5 (1993), 469–494. · Zbl 0784.34050 · doi:10.1007/BF01053532
[8] P. Hartman, Ordinary differential equations. Birkhäuser (1982). · Zbl 0476.34002
[9] L. R. Hunt, R. Su, and G. Meyer, Design for multi-input nonlinear systems. In: R. Brockett (ed.), Differential geometric control theory. Birkhäuser (1983), pp. 258–298.
[10] A. Isidori, Nonlinear control systems. Springer-Verlag (1995). · Zbl 0878.93001
[11] B. Jakubczyk, Equivalence and invariants of nonlinear control systems. In: Nonlinear controllability and optimal control. Dekker, New York (1990), pp. 177–218. · Zbl 0712.93027
[12] B. Jakubczyk and W. Respondek. On linearization of control systems. Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 517–522. · Zbl 0489.93023
[13] A. Juditsky, H. Hjalmarsson, A. Benveniste, B. Delyon, L. Ljung, J. Sjöberg, and Q. Zhang, Nonlinear black-box models in system identification: mathematical foundations. Automatica 31 (1995), No. 12, 1725–1750. · Zbl 0846.93019 · doi:10.1016/0005-1098(95)00119-1
[14] V. Jurdjevic, Geometric control theory. Cambridge Stud. Adv. Math. 51, Cambridge Univ. Press (1997). · Zbl 0940.93005
[15] V. Jurdjevic and H. J. Sussmann, Controllability of nonlinear systems. J. Differ. Equations 12 (1972), 95–116. · Zbl 0237.93027 · doi:10.1016/0022-0396(72)90035-6
[16] C. Lobry, Contrôlabilité des systèmes non linéaires. SIAM J. Control 8 (1970), 573–605. · Zbl 0207.15201 · doi:10.1137/0308042
[17] J. R. Munkres, Elements of algebraic topology. Addison-Wesley (1984). · Zbl 0673.55001
[18] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras. J. Math. Soc. Jpn. 18 (1966), 398–404. · Zbl 0147.23502 · doi:10.2969/jmsj/01840398
[19] H. Nijmeijer and A. J. van der Schaft, Nonlinear dynamical control systems. Springer-Verlag, New York (1990). · Zbl 0701.93001
[20] R. Roussarie, Modèles locaux de champs et de formes. Astérisque 30 (1975).
[21] W. Rudin, Analyse réelle et complexe. Masson, Paris (1975). · Zbl 0333.28001
[22] J. Sjöberg, Q. Zhang, L. Ljung, A. Benveniste, B. Delyon, P.Y. Glorennec, H. Hjalmarsson, and A. Juditsky, Nonlinear black-box modeling in system identification: a unified overview. Automatica 31 (1995), No. 12, 1691–1724. · Zbl 0846.93018 · doi:10.1016/0005-1098(95)00120-8
[23] E. D. Sontag, Mathematical control theory. Springer-Verlag, New York (1998). · Zbl 0945.93001
[24] M. Spivak, A comprehensive introduction to differential geometry. Publish or Perish, Houston (1979). · Zbl 0439.53001
[25] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 317 (1973), 171–188. · Zbl 0274.58002 · doi:10.1090/S0002-9947-1973-0321133-2
[26] K. Tchoń, The only stable normal forms of affine systems under feedback are linear. Syst. Control Lett. 8 (1987), 359–365. · Zbl 0641.93034 · doi:10.1016/0167-6911(87)90103-4
[27] A. J. van der Schaft, Linearization and input-output decoupling for general nonlinear systems. Syst. Control Lett. 5 (1984), 27–33. · Zbl 0557.93039 · doi:10.1016/0167-6911(84)90005-7
[28] S. van Strien, Smooth linearization of hyperbolic fixed points without resonance conditions. J. Differ. Equations 85 (1990), No. 1, 66–90. · Zbl 0726.58039 · doi:10.1016/0022-0396(90)90089-8
[29] J. C. Willems, Topological classification and structural stability of linear systems. J. Differ. Equations 35 (1980), 306–318. · Zbl 0416.93053 · doi:10.1016/0022-0396(80)90031-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.