Section
|
Mathematics
|
Title
|
The stability of completely controllable systems
|
Author(-s)
|
Narmanov A.Ya.a,
Abdishukurova G.M.a
|
Affiliations
|
National University of Uzbekistana
|
Abstract
|
The subject of this paper is the stability of completely controllable systems defined on a smooth manifold. It is known that the controllability sets of symmetric systems generate singular foliations. In the case when the controllability sets have the same dimension, a regular foliation arises. Thus, the possibility of applying the methods of foliation theory to control theory problems arises. This paper presents some of the authors' results on the possibility of applying the theorems on the stability of leaves to the problems on the stability of completely controllable systems and on the geometry of attainability sets. Smoothness throughout the work will mean smoothness of class $C^{\infty}$.
|
Keywords
|
control systems, controllability sets, orbit of vector fields, singular foliation
|
UDC
|
517.936, 517.925.53
|
MSC
|
37C10, 57R27
|
DOI
|
10.35634/vm220106
|
Received
|
17 December 2021
|
Language
|
English
|
Citation
|
Narmanov A.Ya., Abdishukurova G.M. The stability of completely controllable systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 81-93.
|
References
|
- Narmanov A.Ya. Controllability sets of control systems that are fibers of a foliation of codimension one, Differentsial'nye Uravneniya, 1983, vol. 19, no. 9, pp. 1627-1630 (in Russian). http://mi.mathnet.ru/eng/de4960
- Narmanov A.Ya. A stability theorem for noncompact leaves of a foliation of codimension one, Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya, 1983, no. 19 (4), pp. 100-102. https://zbmath.org/?q=an:0564.57020
- Narmanov A.Ya. Dependence of the control set on the target point, Differentsial'nye Uravneniya, 1985, vol. 21, no. 9, pp. 1504-1508 (in Russian). http://mi.mathnet.ru/eng/de5629
- Narmanov A.Ya. On the dependence of the controllability set on the target point, Differential Equations, 1995, vol. 31, no. 4, pp. 555-558.
- Narmanov A.Ya. On the transversal structure of the controllability sets of symmetric control systems, Differential Equations, 1996, vol. 32, no. 6, pp. 786-789.
- Narmanov A.Ya., Saitova S.S. On the geometry of the reachability set of vector fields, Differential Equations, 2017, vol. 53, no. 3, pp. 311-316. https://doi.org/10.1134/S001226611703003X
- Narmanov A.Ya. Stability of completely controllable systems, Differential Equations, 2000, vol. 36, no. 10, pp. 1475-1483. https://doi.org/10.1007/BF02757386
- Stefan P. Accessibility and foliations with singularities, Bulletin of the American Mathematical Society, 1974, vol. 80, no. 6, pp. 1142-1145. https://doi.org/10.1090/S0002-9904-1974-13648-7
- Sussmann H.J. Orbits of families of vector fields and integrability of systems with singularities, Bulletin of the American Mathematical Society, 1973, vol. 79, no. 1, pp. 197-199. https://doi.org/10.1090/S0002-9904-1973-13152-0
- Levitt N., Sussmann H.J. On controllability by means of two vector fields, SIAM Journal on Control, 1975, vol. 13, no. 6, pp. 1271-1281. https://doi.org/10.1137/0313079
- Azamov A., Narmanov A.Ya. On the limit sets of orbits of systems of vector fields, Differential Equations, 2004, vol. 40, no. 2, pp. 271-275. https://doi.org/10.1023/B:DIEQ.0000033716.96100.06
- Narmanov A.Ya., Saitova S.S. On the geometry of orbits of Killing vector fields, Differential Equations, 2014, vol. 50, no. 12, pp. 1584-1591. https://doi.org/10.1134/S0012266114120027
- Agrachev A., Sarychev A. Control in the spaces of ensembles of points, SIAM Journal on Control and Optimization, 2020, vol. 58, no. 3, pp. 1579-1596. https://doi.org/10.1137/19M1273049
- Petrov N.N. Matrix resolving functions in a linear problem of group pursuit with multiple capture, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 2, pp. 185-196 (in Russian). https://doi.org/10.21538/0134-4889-2021-27-2-185-196
- Gauthier J.P., Bornard G. An openness condition for the controllability of nonlinear systems, SIAM Journal on Control and Optimization, 1982, vol. 20, no. 6, pp. 808-814. https://doi.org/10.1137/0320058
- Sachkov Yu.L. Coadjoint orbits and time-optimal problems for step-2 free nilpotent Lie groups, Mathematical Notes, 2020, vol. 108, no. 6, pp. 867-876. https://doi.org/10.1134/S0001434620110280
- Tamura I. Topology of foliations: an introduction, AMS, 1992.
- Schweitzer P.A. Some problems in foliation theory and related areas, Differential Topology, Foliations and Gelfand-Fuks Cohomology: Proceedings of the Symposium held at the Pontifica Universidade Católica do Rio de Janeiro, 5-24 January, 1976, Berlin-Heidelberg: Springer, 1978, pp. 240-252. https://doi.org/10.1007/BFb0063516
- Inaba T. On stability of proper leaves of codimension one foliations, Journal of the Mathematical Society of Japan, 1977, vol. 29, no. 4, pp. 771-778. https://doi.org/10.2969/jmsj/02940771
- Hermann R. On the differential geometry of foliations, Annals of Mathematics, 1960, vol. 72, no. 3, pp. 445-457. https://doi.org/10.2307/1970226
- Molino P. Riemannian foliations, Boston: Birkhäuser, 1988. https://doi.org/10.1007/978-1-4684-8670-4
- Reinhart B.L. Foliated manifolds with bundle-like metrics, Annals of Mathematics, 1959, vol. 69, no. 1, pp. 119-132. https://doi.org/10.2307/1970097
- Tondeur Ph. Foliations on Riemannian manifolds, New York: Springer, 1988. https://doi.org/10.1007/978-1-4613-8780-0
- Abdishukurova G.M., Narmanov A.Ya. Diffeomorphisms of foliated manifolds, Methods of Functional Analysis and Topology, 2021, vol. 27, no. 1, pp. 1-9.
- Narmanov A.Ya., Zoyidov A.N. On the group of diffeomorphisms of foliated manifolds, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 49-58. https://doi.org/10.35634/vm200104
- Narmanov A., Qosimov O. On the geometry of the set of orbits of Killing vector fields on Euclidean space, Journal of Geometry and Symmetry in Physics, 2020, vol. 55, pp. 39-49. https://doi.org/10.7546/jgsp-55-2020-39-49
- Petrov N.N. Control systems like convex vectogram systems. Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya, 1974, no. 1-1, pp. 63-69. https://zbmath.org/?q=an:0275.93007
- Petrov N.N. Controllability of autonomous systems, Differentsial'nye Uravneniya, 1968, vol. 4, no. 4, pp. 606-617 (in Russian). http://mi.mathnet.ru/eng/de328
|
Full text
|
|