Logarithmic spirals in optimal control problems with control in a disk

M. I. Ronzhina ; L. A. Manita

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXIV", Voronezh, May 3-9, 2023, Part 4, Tome 233 (2024), pp. 75-88.

Résumé

We study a neighborhood of singular second-order extremals in optimal control problems that are affine in a two-dimensional control in a disk. We study the stabilization problem for a linear system of second-order differential equations for which the origin is a singular second-order extremal. This problem can be considered as a perturbation of an analog of the Fuller problem with two-dimensional control in a disk. We prove that for this class of problems, optimal solutions keep their form of logarithmic spirals that arrive at a singular point in a finite time, while optimal controls make an infinite number of revolutions along the circle. Finally, we present a brief review of problems whose solutions have the form of such logarithmic spirals.

Détail

Keywords: two-dimensional control in a disk, singular extremal, blow-up of a singularity, logarithmic spiral, Hamiltonian system, Pontryagin's maximum principle

Bibliographie

[1] Zelikin M. I., Borisov V. F., “Rezhimy uchaschayuschikhsya pereklyuchenii v zadachakh optimalnogo upravleniya”, Tr. MIAN SSSR., 197 (1991), 85–166

[2] Zelikin M. I., Lokutsievskii L. V., Khildebrand R., “Tipichnost fraktalno-khaoticheskoi struktury integralnykh voronok v gamiltonovykh sistemakh s razryvnoi pravoi chastyu”, Sovr. mat. Fundam. napravl., 56 (2015), 5–128

[3] Zelikin M. I., Melnikov N. B., Khildebrand R., “Topologicheskaya struktura fazovogo portreta tipichnogo sloya optimalnogo sinteza dlya zadach s nakopleniem pereklyuchenii”, Tr. Mat. in-ta im. V. A. Steklova RAN., 233 (2001), 125–152 | Zbl

[4] Manita L. A., “Optimalnye rezhimy s uchaschayuschimisya pereklyucheniyami v zadachakh upravleniya manipulyatorami”, Prikl. mat. mekh., 64:1 (2000), 19–28 | MR | Zbl

[5] Manita L. A., “Optimalnyi osobyi rezhim i rezhim s uchaschayuschimisya pereklyucheniyami v zadache upravleniya kolebaniyami struny s zakreplennymi kontsami”, Prikl. mat. mekh., 74:5 (2010), 856–863

[6] Ronzhina M. I., “Optimalnye rezhimy s uchaschayuschimisya pereklyucheniyami v zadache upravleniya perevernutym dvukhzvennym mayatnikom”, Prikl. mat. mekh., 80:1 (2016), 24–33 | MR | Zbl

[7] Ronzhina M. I. , Manita L. A., Lokutsievskii L. V., “Resheniya gamiltonovoi sistemy s dvumernym upravleniem v okrestnosti osoboi ekstremali vtorogo poryadka”, Usp. mat. nauk., 76:5 (461) (2021), 201–202 | DOI | MR | Zbl

[8] Ronzhina M. I. , Manita L. A., Lokutsievskii L. V., “Okrestnost osobogo rezhima vtorogo poryadka v zadachakh s upravleniem iz kruga”, Tr. Mat. in-ta im. V. A. Steklova RAN., 315 (2021), 222–236 | DOI | MR | Zbl

[9] Chukanov S. V., Milyutin A. A., “Qualitative study of singularities for extremals of quadratic optimal control problem”, Russ. J. Math. Phys., 2:1 (1994), 31–48 | MR | Zbl

[10] Farkas M., Periodic Motions, Springer-Verlag, New York, 1994 | MR | Zbl

[11] Goh B. S., “Optimal singular rocket and aircraft trajectories”, Proc. 2008 Chinese Control and Decision Conference (Yantai, China, July 2-4, 2008), IEEE, 2008, 1531–1536

[12] Hartman P., Ordinary Differential Equations, Wiley, New York, 1964 | MR | Zbl

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

[14] Kupka I., “The ubiquity of Fuller's phenomenon”, Nonlinear Controllability and Optimal Control, Marcel Dekker, New York, 1990, 313–350 | MR | Zbl

[15] Lawden D. F., Optimal Trajectories for Space Navigation, Butterworths, London, 1963 | MR | Zbl

[16] Manita L. A., Ronzhina M. I., “Optimal spiral-like solutions near a singular extremal in a two-input control problem”, Discr. Cont. Dyn. Syst. B., 27:6 (2022), 3325–3343 | DOI | MR

[17] Robbins H. M., “A generalized Legendre–Clebsh condition for the singular cases of optimal control”, IBM J. Res. Develop., 11:4 (1967), 361–372 | DOI | MR | Zbl

[18] Ronzhina M. I., Manita L. A., “Singularity of optimal control for a Timoshenko beam”, J. Phys. Conf. Ser., 1740:012068 (2021)

[19] Ronzhina M. I., Manita L. A., “Spiral-like extremals near a singular surface in a rocket control problem”, Regul. Chaotic Dyn., 28:2 (2023), 148–161 | DOI | MR | Zbl

[20] Seywald, H., Kumar R. R., “Singular control in minimum time spacecraft reorientation”, J. Guid. Control Dyn., 16:4 (1993), 686–694 | DOI

[21] Shen H., Tsiotras P., “Time-optimal control of axi-symmetric rigid spacecraft using two controls”, J. Guid. Control Dyn., 22:5 (1999), 682–694 | DOI

[22] Witt E., “Die Unterringe der freien Lieschen Ringe”, Math. Z., 64 (1956), 195–216 | DOI | MR | Zbl

[23] Zelikin M. I., Borisov V. F., Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering, Birkhäuser, Boston, 1994 | MR | Zbl

[24] Zelikin M. I., Borisov V. F., “Optimal chattering feedback control”, J. Math. Sci., 114 (2003), 1227–1344 | DOI | MR

[25] Zelikin M. I., Manita L. A., “Optimal control for a Timoshenko beam”, C. R. Méc. Acad. Sci., 334:5 (2006), 292–297 | MR | Zbl

[26] Zhu J., Trélat E., Cerf M., “Minimum time control of the rocket attitude reorientation associated with orbit dynamics”, SIAM J. Control Optim., 54:1 (2016), 391–422 | DOI | MR | Zbl