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Quadratic mappings in geometric control theory. (English) Zbl 1267.93035
J. Sov. Math. 51, No. 6, 2667-2734 (1990); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 20, 111–206 (1988).
Summary: The article is dedicated to local investigation of mappings of type “entry-exit” of smooth controlled systems. The homological theory of quadratic mappings and the geometry of the Lagrange Grassmannian are used for the study of sets of the level and the form of mappings of type “entry-exit,” including for the obtaining of the necessary and sufficient conditions for local optimality.

93B27 Geometric methods
49K15 Optimality conditions for problems involving ordinary differential equations
Full Text: DOI
[1] A. A. Agrachev, ?The topology of quadratic mappings and the Hessians of smooth mappings,?Itogi Nauki Tekh. Ser. Algebra, Topologiya, Geometriya 26, 85?124 (1988).
[2] A. A. Agrachev, S. A. Vakhrameev, and R. V. Gamkrelidze, ?Differential geometric and group theoretic methods in optimal control theory,?Itogi Nauki Tekh. Ser. Probl. Geometr. 14, 3?56 (1983). · Zbl 0542.93045
[3] A. A. Agrachev and R. V. Gamkrelidze, ?The principle of second-order optimality for time-optional problems,?Mat. Sb. 100(142), 610?643 (1976).
[4] A. A. Agrachev and R. V. Gamkrelidze, ?Exponential representation of flows and a chronological enumeration,?Mat. Sb. 107(149), 467?532 (1978). · Zbl 0408.34044
[5] A. A. Agrachev and R. V. Gamkrelidze, ?Chronological algebras and nonstationary vector fields,?Itogi Nauki Tekh., Ser. Probl. Geometr. 11, 135?176 (1980).
[6] A. A. Agrachev and R. V. Gamkrelidze, ?The index of extremality and quasiextremal controls,?Dokl. Akad. Nauk SSSR 284, No. 4, 777?781 (1985). · Zbl 0587.49018
[7] A. A. Agrachev and R. V. Gamkrelidze, ?The Morse index and the Maslov index for extremals of controlled systems,?Dokl. Akad. Nauk SSSR 287, No. 3, 521?524 (1986). · Zbl 0614.58016
[8] A. A. Agrachev and R. V. Gamkrelidze, ?Local invariants of smooth controlled systems,? Manuscript deposited in VINITI, October 4, 1986, No. 7020-V Dep.
[9] V. I. Arnol’d,Mathematical Methods of Classical Mechanics [in Russian], Nauka, Moscow (1974).
[10] R. Gabasov and F. M. Kirillova,Singular Optimal Controls [in Russian], Nauka, Moscow (1973).
[11] R. V. Gamkrelidze, ?First-order necessary conditions and the axiomatics of extremal problems,?Trudy Mat. Inst. Steklov. 112, Pt. 1, 152?180 (1971).
[12] V. Guillemin and S. Sternberg,Geometric Asymptotics, Mathematical Surveys No. 14, Am. Math. Soc., Providence, RI (1977).
[13] G. Lion and M. Vergne,The Weil Representation, Maslov Index, and Theta Series, Progress in Mathematics, No. 6, Birkhuser, Boston (1980). · Zbl 0444.22005
[14] L. S. Pontryagin, B. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko,The Mathematical Theory of Optimal Processes [in Russian], Fizmatgiz, Moscow (1961).
[15] A. V. Sarychev, ?Index of second variation of a controlled system,?Mat. Sb. 113(155), 464?486 (1980).
[16] R. V. Gamkrelidze, ?Exponential representations of solutions of ordinary differential equations,? in:Proc. Equadif. IV, Lecture Notes in Mathematics 703, Springer, Berlin (1979), pp. 118?129.
[17] M. R. Hastens, ?Applications of the theory of quadratic forms in Hilbert space to the calculus of variations,?Pacific J. Math. 1, No. 5, 525?582 (1951). · Zbl 0045.20806
[18] A. J. Krener, ?The high-order maximal principle and its applications to singular extremals,?SIAM J. Control. Optim. 15, No. 2, 256?293 (1977). · Zbl 0354.49008
[19] M. Morse,The Calculus of Variations in the Large, Am. Math. Soc., Providence, RI (1934). · Zbl 0011.02802
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