Geodesic
On polyhedral estimates for reachable sets of differential systems with bilinear uncertainty
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 195-210 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Systems of nonlinear ordinary differential equations describing the dynamics of internal parallelotope-valued estimates for reachable sets of differential systems with bilinear uncertainty are derived. The existence and uniqueness of solutions are proved at least on some subinterval of the time interval under consideration and, in the case of single-valued additive terms in the right-hand sides of the original equations, on the whole interval. Two types of differential inclusions that guarantee the extension of estimates to the whole interval are obtained. The second of these inclusions produces nondegenerate parallelepiped-valued estimates. Some conditions are specified under which the proposed estimates are most efficient. Differential equations for external parallelepiped-valued estimates of reachable sets were obtained earlier. Here, for unification, they are given in the form of equations describing the dynamics of the centers and matrices of parallelotopes. Results of numerical simulation are presented.
Keywords: reachable sets, bilinear uncertainty, polyhedral estimates, parallelepipeds, interval analysis.
Mots-clés : parallelotopes 
@article{TIMM_2012_18_4_a17,
     author = {E. K. Kostousova},
     title = {On polyhedral estimates for reachable sets of differential systems with bilinear uncertainty},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {195--210},
     year = {2012},
     volume = {18},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a17/}
}
TY  - JOUR
AU  - E. K. Kostousova
TI  - On polyhedral estimates for reachable sets of differential systems with bilinear uncertainty
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2012
SP  - 195
EP  - 210
VL  - 18
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a17/
LA  - ru
ID  - TIMM_2012_18_4_a17
ER  - 
%0 Journal Article
%A E. K. Kostousova
%T On polyhedral estimates for reachable sets of differential systems with bilinear uncertainty
%J Trudy Instituta matematiki i mehaniki
%D 2012
%P 195-210
%V 18
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a17/
%G ru
%F TIMM_2012_18_4_a17
E. K. Kostousova. On polyhedral estimates for reachable sets of differential systems with bilinear uncertainty. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 195-210. http://geodesic.mathdoc.fr/item/TIMM_2012_18_4_a17/

[1] Aschepkov L. T., Lifantova S. V., “Vypuklost mnozhestva dostizhimosti bilineinoi upravlyaemoi sistemy”, Izv. RAN. Tekhn. kibernetika, 1994, no. 3, 24–28

[2] Gusev M. I., “Otsenki mnozhestv dostizhimosti mnogomernykh upravlyaemykh sistem s nelineinymi perekrestnymi svyazyami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15, no. 4, 2009, 82–94

[3] Demidovich B. P., Sbornik zadach i uprazhnenii po matematicheskomu analizu, Nauka, M., 1972, 544 pp.

[4] Demyanov V. F., Malozemov V. N., Vvedenie v minimaks, Nauka, M., 1972, 368 pp. | MR

[5] Zaslavskii B. G., Poluektov R. A., Upravlenie ekologicheskimi sistemami, Nauka, M., 1988, 296 pp. | MR | Zbl

[6] Kinëv A. N., Rokityanskii D. Ya., Chernousko F. L., “Ellipsoidalnye otsenki fazovogo sostoyaniya lineinykh sistem s parametricheskimi vozmuscheniyami i neopredelennoi matritsei nablyudenii”, Izv. RAN. Teoriya i sistemy upravleniya, 2002, no. 1, 5–13 | MR

[7] Kornoushenko E. K., “Intervalnye pokoordinatnye otsenki dlya mnozhestva dostizhimykh sostoyanii lineinoi statsionarnoi sistemy. I”, Avtomatika i telemekhanika, 1980, no. 5, 12–22 ; “II”, Автоматика и телемеханика, 1980, No 12, 10–17 ; “III”, Автоматика и телемеханика, 1982, No 10, 47–52 ; “IV”, Автоматика и телемеханика, 1983, No 2, 81–87 | MR | Zbl | MR | Zbl | MR | Zbl | MR | Zbl

[8] Kostousova E. K., “O vneshnikh poliedralnykh otsenkakh dlya mnozhestv dostizhimosti sistem s bilineinoi neopredelennostyu”, Prikl. matematika i mekhanika, 66:4 (2002), 559–571 | MR | Zbl

[9] Kostousova E. K., “O poliedralnykh otsenkakh mnozhestv dostizhimosti mnogoshagovykh sistem s bilineinoi neopredelennostyu”, Avtomatika i telemekhanika, 2011, no. 9, 49–60 | MR | Zbl

[10] Kostousova E. K., Kurzhanskii A. B., “Garantirovannye otsenki tochnosti vychislenii v zadachakh upravleniya i otsenivaniya”, Vychisl. tekhnologii, 2:1 (1997), 19–27 | MR

[11] Krasnoselskii M. A., Polozhitelnye resheniya operatornykh uravnenii, Nauka, M., 1962, 396 pp. | MR

[12] Kuntsevich V. M., Kurzhanskii A. B., “Oblasti dostizhimosti lineinykh i nekotorykh klassov nelineinykh diskretnykh sistem i upravleniya imi”, Probl. upravleniya i informatiki, 2010, no. 1, 5–21 | MR

[13] Kurzhanskii A. B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp. | MR | Zbl

[14] Lankaster P., Teoriya matrits, Nauka, M., 1978, 280 pp. | MR

[15] Lisin D. V., Filippova T. F., “Ob otsenivanii traektornykh trubok differentsialnykh vklyuchenii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 6, no. 2, 2000, 435–445 | MR | Zbl

[16] Nikolskii M. S., “Ob upravlyaemykh variantakh modeli L. Richardsona v politologii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17, no. 1, 2011, 121–128

[17] Pontryagin L. S., Obyknovennye differentsialnye uravneniya, Nauka, M., 1970, 332 pp. | MR | Zbl

[18] Filippov A. F., Differentsialnye uravneniya s razryvnoi pravoi chastyu, Nauka, M., 1985, 224 pp. | MR

[19] Chernousko F. L., “Ellipsoidalnaya approksimatsiya mnozhestv dostizhimosti lineinoi sistemy s neopredelennoi matritsei”, Prikl. matematika i mekhanika, 60:6 (1996), 940–950 | MR

[20] Sharyi S. P., Konechnomernyi intervalnyi analiz, Intervalnyi analiz i ego prilozheniya: [sait], URL: , 604 pp., (data obrascheniya 13.09.2012) http://www.nsc.ru/interval/index.php?j=Library/InteBooks/index

[21] Barmish B. R., Sankaran J., “The propagation of parametric uncertainty via polytopes”, IEEE Trans. Automat. Control, AC-24:2 (1979), 346–349 | DOI | MR | Zbl

[22] Digailova I. A., Kurzhanski A. B., “The joint model and state estimation problem under set-membership uncertainty: Proc. of the 15-th IFAC Congress, Barselona, 2002”: Kurzhanskii A. B., Izbr. tr., Izd-vo Mosk. un-ta, M., 2009, 182–193, 756 pp.

[23] Filippova T. F., “Differential equations of ellipsoidal state estimates in nonlinear control problems under uncertainty”, Discrete Contin. Dyn. Syst., 8th AIMS Conference, Suppl. Dynamical Systems, Differential Equations and Applications, 2011, 410–419

[24] Kostousova E. K., “Control synthesis via parallelotopes: optimization and parallel computations”, Optimiz. Methods Software, 14:4 (2001), 267–310 | DOI | MR | Zbl

[25] Kostousova E. K., “On polyhedral estimates for trajectory tubes of dynamical discrete-time systems with multiplicative uncertainty”, Discrete Contin. Dyn. Syst., 8th AIMS Conference, Suppl. Dynamical Systems, Differential Equations and Applications, 2011, 864–873

[26] Kurzhanski A. B., Vályi I., Ellipsoidal calculus for estimation and control, Birkhäuser, Boston, 1997, 321 pp. | MR | Zbl

[27] Nazin S. A., Polyak B. T., “Interval parameter estimation under model uncertainty”, Math. Comput. Model. Dyn. Syst., 11:2 (2005), 225–237 | DOI | MR | Zbl

[28] Polyak B. T., Nazin S. A., Durieu C., Walter E., “Ellipsoidal parameter or state estimation under model uncertainty”, Automatica J. IFAC, 40:7 (2004), 1171–1179 | DOI | MR | Zbl