zbMATH — the first resource for mathematics

Local invariants of smooth control systems. (English) Zbl 0681.49018
Summary: Methods are presented for locally studying smooth nonlinear control systems on the manifold \({\mathbb{M}}\). The technique of chronological calculus [see the first two authors, Math. USSR, Sb. 35, 727-785 (1979); translation from Mat. Sb., n. Ser. 107(149), 467-532 (1978; Zbl 0408.34044); J. Sov. Math. 17, 1650-1675 (1981); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 11, 135-176 (1980; Zbl 0473.58021)] is intensively exploited. The concept of chronological connection is introduced and is used when obtaining the invariant expressions in the form of Lie bracket polynomials for high-order variations of a nonlinear control system.
The theorem on adduction of a family of smooth vector fields to the canonical form proved in Section 4 is then applied to the construction of a nilpotent polynomial approximation for a control system. Finally, the relation between the attainable sets of an original system and an approximating one is established; it implies some conclusions on the local controllability of these systems.

49K15 Optimality conditions for problems involving ordinary differential equations
93B03 Attainable sets, reachability
93C10 Nonlinear systems in control theory
Full Text: DOI
[1] Agrachev, A. A. and Gamkrelidze, R. V.: The exponential representation of flows and chronological calculus,Math. sbornik 107 (149) (1978), 467-532; English transl. inMath. USSR Sbornik,35 (1979), 727-785. · Zbl 0408.34044
[2] Agrachev, A. A. and Gamkrelidze, R. V.: The chronological algebras and the nonstationary vector fields,Itogi Nauki: Problemy geometrii, Vol. 11, VINITI, Moscow, 1980, pp. 135-176; English transl. inJ. Soviet Math. · Zbl 0473.58021
[3] Agrachev, A. A., Vakhrameev, S. A., and Gamkrelidze, R. V.: The differential-geometric and group-theoretic methods in optimal control theory,Itogi Nauki: Problemy geometrii, Vol 14, VINITI, Moscow, 1983, pp. 3-56; English transl. inJ. Soviet Math. · Zbl 0542.93045
[4] Sarychev, A. V.: Stability of mappings of Hilbert space and equivalence of control systems,Matem sbornik,113 (155) (1980), 146-160; English transl. inMath. USSR Sbornik. · Zbl 0468.93039
[5] Arnol’d, V. I.:Mathematical Methods, in Classical Mechanics, Nauka, Moscow, 1974; English transl., Springer-Verlag, Berlin and New York, 1978.
[6] Sussmann, H. J.: A Lie-Volterra expansion for nonlinear systems, inLect. Notes Contr. and Inf. Sci. No. 58, 1984, pp. 822-828. · Zbl 0538.93032
[7] Serre, J. P.:Lie Algebra and Lie Groups, Benjamin, New York, 1965. · Zbl 0132.27803
[8] Hermes, H.: Control systems which generate decomposable Lie Algebras,J. Diff. Equations 44 (1982), 166-187. · Zbl 0496.49021 · doi:10.1016/0022-0396(82)90012-2
[9] Sussmann, H. J.: Lie brackets and local controllability: a sufficient condition for scalar-input systems,SIAM J. Control Optim. 21, (1983), 685-713. · Zbl 0523.49026 · doi:10.1137/0321042
[10] Stefani, G.: Polynomial approximations to control systems and local controllability,Proc. 24th IEEE Conf. Decision and Control, Ft. Lauderdale, Flo., Dec. 11-13, 1985, Vol. 1. IEEE, New York, 1985, pp. 33-38.
[11] Stefani, G.: On the local controllability of a scalar-input control systems, Theory and applied nonlinear control systems, Selected Paper7th Int. Symp. Math. Theory Networks and Systems, Stockholm, 10-14 June 1985?. Elsevier, North Holland, Amsterdam, 1986, pp. 167-179.
[12] Stefani, G.: Local properties of nonlinear control systems, Sci. Pap. Inst. Techn. Cybern. Techn. Univ. Wroclow, 1985, No. 29, pp. 219-226.
[13] Bianchini, R. M. and Stefani, G.: Sufficient conditions of local controllability,Proc. 25th Conf. on Decision and Control, Athens, Greece, 1986, Vol. 2, IEEE, New York, 1986, pp. 967-970.
[14] Agrachev, A. A. and Sarychev, A. V.: Filtration of Lie algebra of vector fields and nilpotent approximation of control systems,Dokl. Akad. Nauk USSR 295 (1987), 771-781. English transl. in Soviet Math. Dokl.,36 (1988), 104-108. · Zbl 0850.93106
[15] Vershik, A. M. and Gershkovitch, V. Ja: Nonholonomic dynamic systems. Geometry of distributions and variational problems,Fundament. napravlenia 16 (1987), 5-85; English transl. inEncyclopaedia of Mathematical Sciences, Vol. 16, Springer-Verlag (to appear).
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.