Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side

M. I. Zelikin ; L. V. Lokutsievskii ; R. Hildebrand

Contemporary Mathematics. Fundamental Directions, Optimal control, Tome 56 (2015), pp. 5-128.

Résumé

In this paper, we consider linear-quadratic deterministic optimal control problems where the controls take values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in a finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely, the chaotic behavior of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighborhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us a possibility of calculating the entropy and the Hausdorff dimension of the nonwandering set, which appears to have a Cantor-like structure as in Smale's horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behavior is generic for piecewise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.

Bibliographie

[1] Agrachev A. A., Gamkrelidze R. V., “Printsip optimalnosti vtorogo poryadka dlya zadachi bystrodeistviya”, Mat. sb., 100(142):4(8) (1976), 610–643 | MR | Zbl

[2] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Editorial URSS, M., 1989 | MR

[3] Dmitruk A. V., “Kvadratichnye usloviya pontryaginskogo minimuma v zadache optimalnogo upravleniya, lineinoi po upravleniyu. I. Teorema o rasshifrovke”, Izv. AN SSSR. Ser. mat., 50:2 (1986), 284–312 | MR | Zbl

[4] Dmitruk A. V., “Kvadratichnye dostatochnye usloviya minimalnosti anormalnykh subrimanovykh geodezicheskikh”, Itogi nauki i tekhn. Ser. Sovr. mat. prilozh. Temat. obz., 65, 1999, 5–89 | MR | Zbl

[5] Zarisskii O., Samyuel P., Kommutativnaya algebra, v. 1, Izd-vo inostr. lit., M., 1963

[6] Zelikin M. I., Borisov V. F., “Osobye optimalnye rezhimy v zadachakh matematicheskoi ekonomiki”, Sovr. mat. prilozh., 11, 2003, 3–161 | MR | Zbl

[7] Zelikin M. I., Kiselev D. D., Lokutsievskii L. V., “Optimalnoe upravlenie i teoriya Galua”, Mat. sb., 204:11 (2013), 83–98 | DOI | MR | Zbl

[8] Zelikin M. I., Lokutsievskii L. V., Khildebrand R., “Geometriya okrestnostei osobykh ekstremalei v zadachakh s mnogomernym upravleniem”, Tr. MIAN, 277, 2012, 74–90 | MR | Zbl

[9] Zelikin M. I., Lokutsievskii L. V., Khildebrand R., “Stokhasticheskaya dinamika algebr Li skobok Puassona v okrestnosti tochki negladkosti gamiltoniana”, Dokl. RAN, 450:1 (2013), 1–6

[10] Zelikin M. I., Melnikov N. B., Khildebrand R., “Topologicheskaya struktura fazovogo portreta tipichnogo sloya optimalnogo sinteza dlya zadach s nakopleniem pereklyuchenii”, Tr. MIAN, 233, 2001, 125–152 | MR | Zbl

[11] Zelikina L. F., Zelikin M. I., Khlyustov K. V., “Osobye stratifitsirovannye mnogoobraziya dlya involyutivnykh upravlyaemykh sistem”, Diff. uravn., 37:9 (2001), 1161–1167 | MR | Zbl

[12] Katok A. B., Khasselblat B., Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Faktorial, M., 1999

[13] Lokutsievskii L. V., “Gamiltonovost potoka osobykh traektorii”, Mat. sb., 205:3 (2014), 133–160 | DOI | MR | Zbl

[14] Lokutsievskii L. V., “Osobye rezhimy v upravlyaemykh sistemakh s mnogomernym upravleniem iz mnogogrannika”, Izv. RAN. Ser. mat., 78:5 (2014), 167–190 | DOI | MR | Zbl

[15] Milyutin A. A., Ilyutovich A. E., Osmolovskii N. P., Chukanov S. V., Optimalnoe upravlenie v lineinykh sistemakh, Nauka, M., 1993 | MR

[16] Mischenko A. S., Fomenko A. T., “Obobschennyi metod Liuvillya integrirovaniya gamiltonovykh sistem”, Funkts. analiz i ego prilozh., 12:2 (1978), 46–56 | MR | Zbl

[17] Pontryagin L. S., Boltyanskii V. G., Gamkrelidze R. V., Mischenko E. F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1969

[18] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985 | MR

[19] Falconer K., Fractal Geometry. Mathematical Foundations and Applications, Wiley, Chichester, 2003 | MR | Zbl

[20] Fuller A. T., “Dimensional properties of optimal and sub-optimal nonlinear control systems”, J. Franklin Inst., 289 (1970), 379–393 | MR | Zbl

[21] Hildebrand R., Lokutsievskiy L. V., Zelikin M. I., “Generic fractal structure of finite parts of trajectories of piecewise smooth hamiltonian systems”, Russ. J. Math. Phys., 20:1 (2013), 25–32 | MR | Zbl

[22] Kelley H. J., Kopp R. E., Moyer H. G., “Singular extremals”, Topics in Optimization, Academic Press, N.Y., 1967, 63–101 | MR

[23] Krener A. J., “The high order maximum principle and its application to singular extremals”, SIAM J. Control Optim., 15:2 (1977), 256–293 | MR | Zbl

[24] Kupka I., “Fuller's phenomena”, Progr. Systems Control Theory, Birkhäuser, Boston, 1990, 129–142 | MR

[25] Lewis R. M., “Defenitions of order and junction condition in singular control problems”, SIAM J. Control Optim., 18:1 (1980), 21–32 | MR | Zbl

[26] Lokutsievskiy L. V., “Generic structure of the lagrangian manifold in chattering problems”, Sb. Math., 205:3 (2014), 432–458 | DOI | MR | Zbl

[27] Lokutsievskii L. V., Zelikin M. I., Hildebrand R., “Fractal structure of hyperbolic Lipschitzian dynamical systems”, Russ. J. Math. Phys., 19:1 (2012), 27–44 | MR

[28] Marchal C., “Chattering arcs and chattering controls”, J. Optim. Theory Appl., 11:5 (1973), 441–468 | MR | Zbl

[29] McDannel J. P., Powers W. F., “Necessary conditions for joining optimal singular and non-singular subarcs”, SIAM J. Control Optim., 9 (1971), 161–173 | MR

[30] Zelikin M. I., Borisov V. F., Theory of chattering control with applications to astronautics, robotics, economics, and engineering, Birkhäuser, Boston, 1994 | MR | Zbl