Geodesic
Differential equations of ellipsoidal estimates for reachable sets of a nonlinear dynamical control system
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 1, pp. 223-232
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The problem of estimating trajectory tubes of a nonlinear control system with uncertainty in initial data is considered. It is assumed that the dynamical system has a special structure, in which nonlinear terms are quadratic in phase coordinates and the values of the uncertain initial states and admissible controls are subject to ellipsoidal constraints. Differential equations are found that describe the dynamics of the ellipsoidal estimates of reachable sets of the nonlinear dynamical system under consideration. To estimate reachable sets of the nonlinear differential inclusion corresponding to the control system, we use results from the theory of ellipsoidal estimation and the theory of evolution equations for multivalued states of dynamical systems under uncertainty.
Keywords: reachable set, trajectory tubes, set-valued estimates, differential inclusions, ellipsoidal estimation, control systems, dynamical systems. 
@article{TIMM_2010_16_1_a16,
     author = {T. F. Filippova},
     title = {Differential equations of ellipsoidal estimates for reachable sets of a~nonlinear dynamical control system},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {223--232},
     year = {2010},
     volume = {16},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2010_16_1_a16/}
}
TY  - JOUR
AU  - T. F. Filippova
TI  - Differential equations of ellipsoidal estimates for reachable sets of a nonlinear dynamical control system
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2010
SP  - 223
EP  - 232
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TIMM_2010_16_1_a16/
LA  - ru
ID  - TIMM_2010_16_1_a16
ER  - 
%0 Journal Article
%A T. F. Filippova
%T Differential equations of ellipsoidal estimates for reachable sets of a nonlinear dynamical control system
%J Trudy Instituta matematiki i mehaniki
%D 2010
%P 223-232
%V 16
%N 1
%U http://geodesic.mathdoc.fr/item/TIMM_2010_16_1_a16/
%G ru
%F TIMM_2010_16_1_a16
T. F. Filippova. Differential equations of ellipsoidal estimates for reachable sets of a nonlinear dynamical control system. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 1, pp. 223-232. http://geodesic.mathdoc.fr/item/TIMM_2010_16_1_a16/

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

[2] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidiffepentsialnoe ischislenie, Nauka, M., 1990, 431 pp. | MR

[3] Krasovskii N. N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp. | MR

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

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

[6] Filippova T. F., “Postroenie mnogoznachnykh otsenok mnozhestv dostizhimosti nekotorykh nelineinykh dinamicheskikh sistem s impulsnym upravleniem”, Tr. In-ta matematiki i mekhaniki UrO RAN, 15, no. 4, 2009, 262–269

[7] Chernousko F. L., Otsenivanie fazovogo sostoyaniya dinamicheskikh sistem, Nauka, M., 1988, 320 pp. | MR

[8] Aubin J. P., Frankowska H., Set-valued analysis, Birkhauser, Boston, 1990, 461 pp. | MR | Zbl

[9] Dontchev A. L., Farkhi E. M., “Error estimates for discretized differential inclusions”, Computing, 41:4 (1989), 349–358 | DOI | MR | Zbl

[10] Dontchev A. L., Lempio F., “Difference methods for differential inclusions: a survey”, SIAM Rev., 34:2 (1992), 263–294 | DOI | MR | Zbl

[11] Filippova T. F., “Sensitivity problems for impulsive differential inclusions”, Proc. of the 6th WSEAS International conference on applied mathematics, Corfu Island, 2004, 1–6

[12] Filippova T. F., “Set-valued solutions to impulsive differential inclusions”, Math. Comput. Model. Dyn. Syst., 11:2 (2005), 149–158 | MR | Zbl

[13] Filippova T. F., “State estimation in control problems under uncertainty and nonlinearity”, Proc. of the 6th Vienna International conference on mathematical modelling, MATHMOD–2009, Vienna, 2009, 1–7

[14] Filippova T. F., Berezina E. V., “On state estimation approaches for uncertain dynamical systems with quadratic nonlinearity: theory and computer simulations”, Lecture Notes in Computer Science, 4818, Springer, Berlin, 2008, 326–333 | MR | Zbl

[15] Krasovskii N. N., Subbotin A. I., Positional differential games, Springer-Verlag, Berlin, 1988, 517 pp. | Zbl

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

[17] Kurzhanski A. B., Filippova T. F., “On the theory of trajectory tubes – a mathematical formalism for uncertain dynamics, viability and control”, Advances in nonlinear dynamics and control: a report from Russia, Progress in Systems and Control Theory, 17, ed. A. B. Kurzhanski, Birkhäuser, Boston etc., 1993, 122–188 | MR

[18] Panasyuk A. I., “Equations of attainable set dynamics. Part 1: Integral funnel equations”, J. Optimiz. Theory Appl., 62:2 (1990), 349–366 | DOI | MR

[19] Veliov V. M., “Second order discrete approximations to strongly convex differential inclusions”, Systems and Control Letters, 13:3 (1989), 263–269 | DOI | MR | Zbl

[20] Veliov V., “Second-order discrete approximation to linear differential inclusions”, SIAM J. Numer. Anal., 29:2 (1992), 439–451 | DOI | MR | Zbl

[21] Wolenski P. R., “The exponential formula for the reachable set of a Lipschitz differential inclusion”, SIAM J. Contr. and Optimiz., 28:5 (1990), 1148–1161 | DOI | MR | Zbl