Controllability on Infinite-Dimensional Manifolds: A Chow–Rashevsky Theorem

Acta Applicandae Mathematicae

Volumn 134, Issue 1, 2014, Pages 229-246

  Khajeh Salehani, Mahdi ^a   Markina, Irina ^a  

^a UNIVERSITY OF BERGEN   (Norway)

Author keywords

Controllability; Convenient locally convex spaces; Geometric control; Infinite dimensional manifolds

Indexed keywords

CONTROL SYSTEM ANALYSIS; GEOMETRY;

GEOMETRIC CONTROL; GEOMETRIC CONTROL PROBLEMS; GEOMETRIC CONTROL THEORY; GLOBAL CONNECTIVITY; INFINITE DIMENSIONAL; INFINITE-DIMENSIONAL MANIFOLDS; LOCALLY CONVEX SPACES; SUB-RIEMANNIAN GEOMETRY;

CONTROLLABILITY;

EID: 84911950740     PISSN: 01678019     EISSN: 15729036     Source Type: Journal    
DOI: 10.1007/s10440-014-9880-5     Document Type: Article

References (62)

1. Agrachev, A.A.: Is it possible to recognize local controllability in a finite number of differentiations? In: Open Problems in Mathematical Systems and Control Theory. Comm. Control Engrg. Ser., pp. 15–18. Springer, London (1999)
2. Agrachev, A.A., Caponigro, M.: Controllability on the group of diffeomorphisms. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 26(6), 2503–2509 (2009)
3. Agrachev, A.A., Gamkrelidze, R.V.: Exponential representation of flows and a chronological enumeration. Mat. Sb. 107(149), 467–532 (1978) (in Russian), 639 pp.
4. Agrachev, A.A., Gamkrelidze, R.V.: Local controllability and semigroups of diffeomorphisms. Acta Appl. Math. 32(1), 1–57 (1993)
5. Agrachev, A.A., Gamkrelidze, R.V.: Local controllability for families of diffeomorphisms. Syst. Control Lett. 20(1), 67–76 (1993)
6. Agrachev, A.A., Sachkov, Y.L.: Control theory from the geometric viewpoint. In: Control Theory and Optimization II. Encyclopaedia of Mathematical Sciences, vol. 87. Springer, Berlin (2004)
7. Bianchini, R.-M., Kawski, M.: Needle variations that cannot be summed. SIAM J. Control Optim. 42(1), 218–238 (2003)
8. Bianchini, R.-M., Stefani, G.: Controllability along a trajectory: a variational approach. SIAM J. Control Optim. 31(4), 900–927 (1993)
9. Bloch, A.M.: Nonholonomic mechanics and control. In: Interdisciplinary Applied Mathematics, vol. 24. Springer, New York (2003). With the collaboration of J. Baillieul, P. Crouch and J. Marsden. With scientific input from P.S. Krishnaprasad, R.M. Murray and D. Zenkov
10. Boltyanski, V.G., Gamkrelidze, R.V., Pontryagin, L.S.: On the theory of optimal processes. Dokl. Akad. Nauk SSSR 110, 7–10 (1956) (in Russian)
11. Brockett, R.W.: Systems theory on group manifolds and coset spaces. SIAM J. Control 10, 265–284 (1972)
12. Brunovsky, P.: Local controllability of odd systems. Banach Cent. Publ. 1, 39–44 (1976)
13. Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems: Modeling, Analysis, and Design for Simple Mechanical Control Systems. Texts in Applied Mathematics, vol. 49. Springer, New York (2005)
14. Carathéodory, C.: Untersuchungen über die grundlagen der thermodynamik. Math. Ann. 67, 355–386 (1909)
15. Chow, W.-L.: Über systeme von linearen partiellen differentialgleichungen erster ordnung. Math. Ann. 117, 98–105 (1939)
16. Crouch, P.E., Byrnes, C.I.: Symmetries and local controllability. In: Algebraic and Geometric Methods in Nonlinear Control Theory. Math. Appl., vol. 29, pp. 55–75. Reidel, Dordrecht (1986)
17. Dudnikov, P.I., Samborski, S.N.: Criterion of controllability for systems in Banach space (generalization of Chow’s theorem). Ukr. Mat. Zh. 32(5), 649–653 (1980) (in Russian)
18. Eells, J.: A setting for global analysis. Bull. Am. Math. Soc. 72, 571–807 (1966)
19. Eells, J., Elworthy, K.D.: Open embeddings of certain Banach manifolds. Ann. Math. 91, 465–485 (1970)
20. Elliot, D.L.: A consequence of controllability. J. Differ. Equ. 10, 364–370 (1971)
21. Frobenius, G.: Über die unzerlegbaren diskreten, Bewegungsgruppen. Sitzungsber. Akad. Wiss. Berlin 29, 654–665 (1911)
22. Grong, E., Markina, I., Vasil’ev, A.: Sub-Riemannian geometry on infinite-dimensional manifolds. Preprint (2012). arXiv:1201.2251
23. Grong, E., Markina, I., Vasil’ev, A.: Sub-Riemannian structures corresponding to Kählerian metrics on the universal Teichmueller space and curve. Preprint (2012). arXiv:1202.6135
24. Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull., New Ser., Am. Math. Soc. 7(1), 65–222 (1982)
25. Heintze, E., Liu, X.: Homogeneity of infinite-dimensional isoparametric submanifolds. Ann. Math. 149(1), 149–181 (1999)
26. Hermann, R.: On the accessibility problem in control theory. In: LaSalle, J.P., Lefschetz, S. (eds.) Int. Symp. on Nonlinear Differential Equations, pp. 325–332. Academic Press, New York (1963)
27. Hermann, R., Krener, A.J.: Nonlinear controllability and observability. IEEE Trans. Autom. Control AC-22(5), 728–740 (1977)
28. Hermes, H.: On local and global controllability. SIAM J. Control Optim. 12, 252–261 (1974)
29. Hermes, H.: Control systems which generate decomposable Lie algebras. J. Differ. Equ. 44(2), 166–187 (1982) (Special issue dedicated to J.P. LaSalle)
30. Hermes, H.: On local controllability. SIAM J. Control Optim. 20(2), 211–220 (1982)
31. Hermes, H.: Nilpotent and high-order approximations of vector field systems. SIAM Rev. 33(2), 238–264 (1991)
32. Hermes, H., Haynes, G.: On the nonlinear control problem with control appearing linearly. J. Soc. Ind. Appl. Math. 1, 85–108 (1963)
33. Isidori, A.: Nonlinear Control Systems, 3rd edn. Communications and Control Engineering Series Springer, Berlin (1995)
34. Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)
35. Jurdjevic, V.: Geometric Control Theory. Cambridge Studies in Advanced Mathematics, vol. 52. Cambridge University Press, Cambridge (1997)
36. Kalman, R.E.: Contributions to the theory of optimal control. Bol. Soc. Mat. Mexicana 2(5), 102–119 (1960)
37. Kalman, R.E.: Mathematical description of linear dynamical systems. SIAM J. Control 1, 152–192 (1963)
38. Kalman, R.E., Ho, Y.C., Narendra, K.S.: Controllability of linear dynamical systems. Contrib. Differ. Equ. 1, 186–213 (1962)
39. Kawski, M.: Control variations with an increasing number of switchings. Bull., New Ser., Am. Math. Soc. 18(2), 149–152 (1988)
40. Kawski, M.: Nilpotent Lie algebras of vector fields. J. Reine Angew. Math. 388, 1–17 (1988)
41. Kawski, M.: The complexity of deciding controllability. Syst. Control Lett. 15(1), 9–14 (1990)
42. Kawski, M.: In: Nonlinear Control and Combinatorics of Words—Geometry of Feedback and Optimal Control. Monogr. Textbooks Pure Appl. Math., vol. 207, pp. 305–346. Dekker, New York (1998)
43. Kawski, M., Sussmann, H.J.: Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory. In: Operators, Systems, and Linear Algebra. European Consort. Math. Indust., pp. 111–128. Teubner, Stuttgart (1997)
44. Krener, A.J.: A generalization of Chow’s theorem and the bang-bang theorem to nonlinear control systems. SIAM J. Control Optim. 12, 43–52 (1974)
45. Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. Am. Math. Soc., Providence (1997)
46. Lobry, C.: Contrôlabilité des systèmes non linéaires. SIAM J. Control Optim. 8, 573–605 (1970)
47. Lobry, C.: Controllability of nonlinear systems on compact manifolds. SIAM J. Control Optim. 1, 1–4 (1974)
48. Milnor, J.: Remarks on infinite-dimensional Lie groups. In: Relativity, groups and topology II, Les Houches, 1983, pp. 1007–1057. North-Holland, Amsterdam (1984)
49. Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs, vol. 91. Am. Math. Soc., Providence (2002)
50. Nijmeijer, H., van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, New York (1990)
51. Omori, H.: On Banach Lie groups acting on finite-dimensional manifolds. Tohoku Math. J. 30, 223–250 (1978)
52. Omori, H., de la Harpe, P.: About interactions between Banach Lie groups and finite-dimensional manifolds. J. Math. Kyoto Univ. 12, 543–570 (1972)
53. Palais, R.: Foundations of Global Nonlinear Analysis. Benjamin, New York (1968)
54. Rashevsky, P.K.: About connecting two points of complete nonholonomic space by admissible curve. Uch. Zapiski Ped. Inst. Libknexta 2, 83–94 (1938) (in Russian)
55. Sontag, E.D.: Controllability is harder to decide than accessibility. SIAM J. Control Optim. 26(5), 1106–1118 (1988)
56. Sussmann, H.J.: Orbits of families of vector fields and integrability of distributions. Trans. Am. Math. Soc. 180, 171–188 (1973)
57. Sussmann, H.J.: A sufficient condition for local controllability. SIAM J. Control Optim. 16(5), 790–802 (1978)
58. Sussmann, H.J.: Lie brackets and local controllability: a sufficient condition for scalar-input systems. SIAM J. Control Optim. 21(5), 686–713 (1983)
59. Sussmann, H.J.: A general theorem on local controllability. SIAM J. Control Optim. 25(1), 158–194 (1987)
60. Sussmann, H.J., Jurdjevic, V.: Controllability of nonlinear systems. J. Differ. Equ. 12, 95–116 (1972)
61. Vagner, V.V.: Differential geometry of nonholonomic manifolds. Tr. Semin. Vectorn. Tenzorn. Anal. 213, 269–314 (1935) (in Russian)
62. Yamamuro, S.: A Theory of Differentiation in Locally Convex Spaces. Mem. Amer. Math. Soc., vol. 17. Springer, Berlin (1979)