Construction of the optimal result function and dispersing lines in time-optimal problems with a nonconvex target set

P. D. Lebedev   ; A. A. Uspenskii

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 22 (2016) no. 2, pp. 188-198

Résumé

Algorithms for constructing the optimal result function are proposed for a planar time-optimal problem with a circular velocity vectogram and a nonconvex target set with smooth boundary. The algorithms work with the case where the solution of the problem has a complicated (segmented) structure of the singular set. Differentiable dependences are detected for smooth segments of the singular set, which makes it possible to consider and construct these segments as arcs of integral curves. An example of the time-optimal problem is considered, for which the optimal result function and its singular set are calculated numerically. A visualization of the results is implemented.

Détail

Keywords: time-optimal problem, dispersing line, nonconvex set, optimal trajectory, differential equation.

Bibliographie

[1] Aizeks R., Differentsialnye igry, per. s angl. V.I. Arkina, E.N. Simakovoi, ed. M.I. Zelikin, Mir, M., 1967, 479 pp. | MR

[2] Lebedev P.D., Uspenskii A.A., Ushakov V.N., “Postroenie minimaksnogo resheniya uravnenii tipa eikonala”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:2 (2008), 182–191 | MR | Zbl

[3] Lebedev P.D., Uspenskii A.A., “Analiticheskoe i chislennoe konstruirovanie funktsii optimalnogo rezultata dlya odnogo klassa zadach bystrodeistviya”, Prikl. matematika i informatika, tr. fak. VMK Mosk. un-ta, 27, 2007, 65–79

[4] Slyusarev G.G., Geometricheskaya optika, Izd-vo AN SSSR, M., 1946, 332 pp. | MR

[5] Kazakov A.L., Lebedev P.D., “Algoritmy postroeniya optimalnykh upakovok dlya kompaktnykh mnozhestv na ploskosti”, Vychislitelnye metody i programmirovanie, 16:3 (2015), 307–317

[6] Kryazhimskii A.V., Osipov Yu.S., “O modelirovanii upravleniya v dinamicheskoi sisteme”, Izv. AN SSSR. Tekhnicheskaya kibernetika, 1983, no. 2, 51–60 | MR

[7] Savelov A.A., Ploskie krivye. Sistematika, svoistva, primeneniya, Librokom, M., 2010, 294 pp.

[8] Uspenskii A.A., “Neobkhodimye usloviya suschestvovaniya psevdovershin kraevogo mnozhestva v zadache Dirikhle dlya uravneniya eikonala”, Tr. In-ta matematiki i mekhaniki, 21:1 (2015), 250–263 | MR

[9] Rashevskii P.K., Kurs differentsialnoi geometrii, Editorial: URSS, M., 2003, 432 pp.

[10] Krasovskii N.N., “Igrovye zadachi dinamiki I”, Izv. AN SSSR. Tekhnicheskaya kibernetika, 1969, no. 5, 3–12 | MR

[11] Ushakov V.N., Uspenskii A.A., Malev A.G., “Otsenka defekta stabilnosti mnozhestva pozitsionnogo pogloscheniya, podvergnutogo diskriminantnym preobrazovaniyam”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17:2 (2011), 209–224

[12] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoi optimizatsii, In-t kompyuternykh tekhnologii, M.; Izhevsk, 2003, 336 pp.

[13] Kruzhkov S.N., “Obobschennye resheniya uravnenii Gamiltona - Yakobi tipa eikonala”, Mat. sb., 98:3 (1974), 450–493

[14] Pshenichnyi K., Vypuklyi analiz i ekstremalnye zadachi, Nauka, M., 1980, 319 pp. | MR

[15] Lebedev P.D., Uspenskii A.A., “Algoritmy postroeniya singulyarnykh mnozhestv dlya odnogo klassa zadach bystrodeistviya”, Vestnik Udm. un-ta. Ser. Matematika, mekhanika, kompyuternye nauki, 2010, no. 3, 30–41, Izhevsk

[16] Sedykh V.D., “On the topology of symmetry sets of smooth submanifolds in $\mathbb{R}^k$”, Adv. Stud. Pure Math.: Singularity Theory and Its Applications, 43 (2006), 401–419 | MR | Zbl

[17] Arnold V.I., Osobennosti kaustik i volnovykh frontov, FAZIS, M., 1996, 334 pp. | MR

[18] Mestetskii L.M., Nepreryvnaya morfologiya binarnykh izobrazhenii: figury, skelety, tsirkulyary, Fizmatlit, M., 2009, 288 pp.

[19] Lebedev P.D., Uspenskii A.A., “Postroenie funktsii optimalnogo rezultata v zadache bystrodeistviya na osnove mnozhestva simmetrii”, Avtomatika i telemekhanika, 2009, no. 7, 50–57 | MR | Zbl

[20] Patsko V.S., “Poverkhnosti pereklyucheniya v lineinykh differentsialnykh igrakh s fiksirovannym momentom okonchaniya”, Prikl. matematika i mekhanika, 68:4 (2004), 653–666 | MR | Zbl

[21] Lebedev P.D., Uspenskii A.A., “Geometriya i asimptotika volnovykh frontov”, Izv. vuzov. Matematika, 2008, no. 3 (550), 27–37 | MR | Zbl

[22] Uspenskii A.A., Lebedev P.D., “O mnozhestve predelnykh znachenii lokalnykh diffeomorfizmov pri evolyutsii volnovykh frontov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16:1 (2010), 171–186 | MR

[23] Ushakov V.N., Uspenskii A.A., Lebedev P.D., “Geometriya singulyarnykh krivykh dlya odnogo klassa zadach bystrodeistviya”, Vestn. S.-Peterb. un-ta Ser. 10, 2013, no. 3, 157–167

[24] Ketkov Yu.L., Ketkov A.Yu., Shults M.M., MATLAB7: programmirovanie, chislennye metody, BKhV-Peterburg, SPb., 2005, 752 pp.

[25] Demyanov V.F., Rubinov A.M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990, 432 pp. | MR

[26] Brus Dzh., Dzhiblin P., Krivye i osobennosti. Geometricheskoe vvedenie v teoriyu osobennostei, per. s angl. I.G. Scherbak, ed. V.I. Arnold, Mir, M., 1988, 262 pp. | MR