Distributions and the Lie algebras their bases can generate
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- by Henry Hermes PDF
- Proc. Amer. Math. Soc. 106 (1989), 555-565 Request permission
Abstract:
The problem is to determine when a smooth, $k$-dimensional distribution ${D^k}$ defined on an $n$-manifold ${M^n}$, locally admits a vector field basis which generates a nilpotent, solvable or even finite-dimensional Lie algebra. We show that for every $2 \leq k \leq n - 1$ there exists a (nonregular at $p \in {M^n}$) distribution ${D^k}$ on ${M^n}$ which does not locally (near $p$) admit a vector field basis generating a solvable Lie algebra. From classical results on the equivalence problem, it is shown that for $1 \leq k \leq 4$ and ${D^k}$ regular at $p \in {M^4}$, ${D^k}$ admits a local vector field basis generating a nilpotent Lie algebra.References
- Henry Hermes, Albert Lundell, and Dennis Sullivan, Nilpotent bases for distributions and control systems, J. Differential Equations 55 (1984), no. 3, 385–400. MR 766130, DOI 10.1016/0022-0396(84)90076-7
- Tadashi Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan 18 (1966), 398–404. MR 199865, DOI 10.2969/jmsj/01840398
- Héctor J. Sussmann, An extension of a theorem of Nagano on transitive Lie algebras, Proc. Amer. Math. Soc. 45 (1974), 349–356. MR 356116, DOI 10.1090/S0002-9939-1974-0356116-6
- H. Hermes, Distributions having bases which generate finite-dimensional Lie algebras, Systems Control Lett. 8 (1987), no. 4, 375–380. MR 884888, DOI 10.1016/0167-6911(87)90105-8
- Kuo-tsai Chen, Decompostion of differential equations, Math. Ann. 146 (1962), 263–278. MR 140621, DOI 10.1007/BF01470955
- Henry Hermes, Local controllability and sufficient conditions in singular problems. II, SIAM J. Control Optim. 14 (1976), no. 6, 1049–1062. MR 420707, DOI 10.1137/0314065
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
- V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1974. MR 0376938
- Elie Cartan, Œuvres complètes. Partie II. Vol. 1. Algèbre, formes différentielles, systèmes différentiels. Vol. 2. Groupes infinis, systèmes différentiels, théories d’équivalence, Gauthier-Villars, Paris, 1953 (French). MR 0058523 E. Goursat, Lecons sur le Probleme de Pfaff, Hermann Pub., Paris, 1922.
- Henry Hermes, Involutive subdistributions and canonical forms for distributions and control systems, Theory and applications of nonlinear control systems (Stockholm, 1985) North-Holland, Amsterdam, 1986, pp. 123–135. MR 935372
- Robert B. Gardner, Differential geometric methods interfacing control theory, Differential geometric control theory (Houghton, Mich., 1982) Progr. Math., vol. 27, Birkhäuser, Boston, Mass., 1983, pp. 117–180. MR 708501
- Robert Hermann, The theory of equivalence of Pfaffian systems and input systems under feedback, Math. Systems Theory 15 (1981/82), no. 4, 343–356. MR 683051, DOI 10.1007/BF01786990
- Robert B. Gardner, Invariants of Pfaffian systems, Trans. Amer. Math. Soc. 126 (1967), 514–533. MR 211352, DOI 10.1090/S0002-9947-1967-0211352-5
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 106 (1989), 555-565
- MSC: Primary 58A30; Secondary 17B30, 53A55, 93B27
- DOI: https://doi.org/10.1090/S0002-9939-1989-0969317-1
- MathSciNet review: 969317