The Structure of Optimal Synthesis in the Vicinity of Singular Manifolds for Problems Affine with Respect to Control

M. I. Zelikin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and dynamical systems, Tome 236 (2002), pp. 174-196.

Résumé

A classification of phase portraits of optimal synthesis in the vicinity of universal singular manifolds is considered for systems of constant rank that are affine with respect to the control. Both the phase state and the control are assumed to be multidimensional. The classification is based on the order of singular extremals and on the involutiveness (or noninvolutiveness) of the velocity indicatrix. It is shown that the synthesis of optimal trajectories is a fibered space over the base $W$ formed by singular optimal trajectories; the fibers consist of nonsingular optimal trajectories. For a multidimensional control, the singular manifold $W$ is stratified. In the involutive case, the fibers are one-dimensional. In the noninvolutive case, the fibers are multidimensional and contain trajectories with switching at increasing frequency (chattering trajectories); the dimension of the fibers and the structure of the field of trajectories inside the fibers depend on the order of the singular extremals. Application of the theory developed to classical problems of the mechanics of controlled systems and to the evaluation of exact constants in Kolmogorov-type inequalities for derivatives is described.

Bibliographie

[1] Baitman M. M., Sintez optimalnykh traektorii na ploskosti, Zinatne, Riga, 1971 | MR | Zbl

[2] Bushaw D. W., Differential equations with a discontinuous forcing term, Princeton Univ., Princeton, NJ, 1952

[3] Boltyanskii V. G., “Dostatochnye usloviya optimalnosti i obosnovanie metoda dinamicheskogo programmirovaniya”, Izv. AN SSSR. Ser. mat., 28:3 (1964), 481–514 | MR

[4] Brunovsky P., “Existence of regular synthesis for general control problems”, J. Diff. Equat., 38:3 (1980), 317–343 | DOI | MR | Zbl

[5] Borisov V. F., “O chisle predelnykh tsiklov faktor-sistemy $n$-mernoi zadachi Fullera”, Mat. sb., 187:12 (1996), 3–20 | MR | Zbl

[6] Zelikin M. I., “On the singular arcs”, Probl. Contr. Inform. Theory, 14:2 (1985), 75–88 | MR | Zbl

[7] Zelikin M. I., “Sintez optimalnykh traektorii na prostranstvakh predstavlenii grupp Li”, Mat. sb., 132:4 (1987), 541–555 | MR

[8] Zelikin M., Borisov V., Theory of chattering control: With applications to astronautics, robotics, economics, and engineering, Birkhäuser, Boston etc., 1994, 233 pp. | MR | Zbl

[9] Zelikin M. I., Borisov V. F., “Rezhimy uchaschayuschikhsya pereklyuchenii v zadachakh optimalnogo upravleniya”, Tr. MIAN, 197, 1991, 85–166 | MR

[10] Zelikina L. F., “Mnogomernyi sintez i teoremy o magistrali v zadachakh optimalnogo upravleniya”, Veroyatnostnye problemy upravleniya v ekonomike, M., 1977, 33–114 | MR

[11] Zelikina L. F., “K voprosu o regulyarnom sinteze”, DAN SSSR, 267:3 (1982), 532–535 | MR | Zbl

[12] Manita L. A., “Povedenie ekstremalei v okrestnosti osobykh rezhimov i negladkie funktsii Lyapunova v zadachakh optimalnogo upravleniya”, Fund. i prikl. matematika, 2:2 (1996), 449–485 | MR

[13] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961

[14] Tom R., Lokalnye topologicheskie svoistva differentsiruemykh otobrazhenii Osobennosti differentsiruemykh otobrazhenii, Mir, M., 1968 | MR

[15] Feldbaum A. A., Osnovy teorii optimalnykh avtomaticheskikh sistem, 2-e izd., Nauka, M., 1966 | MR

[16] Filippov A. F., “Differentsialnye uravneniya s razryvnoi pravoi chastyu”, Mat. sb., 51:1 (1960), 99–128 | Zbl

[17] Agrachev A., Gauthier J.-P., Kupka I., “Generic singularities of sub-Riemannian metrics on $\mathbb{R}^3$”, C. R. Acad. Sci. Paris. Sér. 1, 322:4 (1996), 377–384 | MR | Zbl

[18] Kelley H. J., Kopp R. E., Moyer H. G., “Singular extremals”, Topics in optimization, ed. G. Leitmann, Acad. Press, New York, 1967, 63–101 | MR

[19] Krener A. J., Schättler H., “The structure of small-time reachable sets in low dimensions”, SIAM J. Contr. and Optim., 27:1 (1989), 120–147 | DOI | MR | Zbl

[20] Kupka I., “The ubiquity of Fuller's phenomenon”, Nonlinear controllability and optimal control, Monogr. Textbooks Pure and Appl. Math., 133, ed. H. J. Sussmann, Dekker, New York, 1990, 313–350 | MR | Zbl

[21] Lewis R. M., “Definitions of order and junction conditions in singular optimal control problems”, SIAM J. Contr. and Optim., 18:1 (1980), 21–32 | DOI | MR | Zbl

[22] Lobry C., “Contrôlabilité des systèmes non linéaires”, SIAM J. Contr. and Optim., 8 (1970), 573–605 | DOI | MR | Zbl

[23] Schättler H., “On the local structure of time-optimal bang-bang trajectories in $\mathbb{R}^3$”, SIAM J. Contr. and Optim., 26:1 (1988), 186–204 | DOI | MR | Zbl

[24] Sussmann H. J., “The structure of time-optimal trajectories for single input systems in the plane: the general real-analytic case”, SIAM J. Contr. and Optim., 25:4 (1987), 868–904 | DOI | MR | Zbl

[25] Landau E., “Einige Ungleichungen für zweimal differenzierbare Functionen”, Proc. London Math. Soc., 2:13 (1913), 43–49 | Zbl

[26] Hadamard J., “Sur le module maximum d'une function et de ses derivees”, C. R. Acad. Sci. Paris, 41 (1914), 68–72

[27] Kolmogorov A. N., “O neravenstvakh mezhdu verkhnimi granyami posledovatelnykh proizvodnykh proizvolnoi funktsii na beskonechnom intervale”, Uchen. zap. MGU, no. 30, 1939, 3–16

[28] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[29] Tikhomirov V. M., Nekotorye voprosy teorii priblizhenii, MGU, M., 1976 | MR

[30] Magaril-Ilyaev G. G., Tikhomirov V. M., “O neravenstvakh dlya proizvodnykh kolmogorovskogo tipa”, Mat. sb., 188:12 (1997), 73–106 | MR

[31] DeVore R. A., Lorentz G. G., Constructive approximation, Grundl. Math. Wissensch., 303, Springer-Verl., Berlin, 1993 | MR | Zbl

[32] Magaril-Ilyaev G. G., “O neravenstvakh Kolmogorova na polupryamoi”, Vestn. MGU. Matematika. Mekhanika, 1976, no. 5, 33–41

[33] Zelikin M. I., “Sintez optimalnykh traektorii, opredelyayuschii sloenie Riba”, Tr. MIAN, 233, 2001, 89–94 | MR | Zbl

[34] Zelikin M. I., Zelikina L. F., “Asimptotika ukloneniya funktsionala ot optimalnogo znacheniya pri zamene chetteringa suboptimalnym rezhimom”, UMN, 54:3 (1999), 163–164 | MR | Zbl

[35] Zelikin M. I., Zelikina L. F., “O tochnykh konstantakh v neravenstvakh tipa Kolmogorova”, Tr. MIAN, 227, 1999, 137–145 | MR | Zbl