Geodesic
On the numerical solution of a minmax control problem with a positional functional
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 3, pp. 58-75 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We consider a minmax feedback control problem for a linear dynamic system with a positional quality criterion, which is the norm of the set of deviations of the motion from given target points at given times. The problem is formalized as a positional differential game. A numerical method is given for finding an approximate value of the game and constructing an optimal (minmax and maxmin) control law. The method is based on the recursive construction of upper convex (concave) hulls of auxiliary program functions. In addition, we use the “pixel” approximation of the domains of convexified functions and the approximate construction of the upper convex hull of a function as the lower envelope of a finite set of support hyperplanes of its subgraph.
Keywords: optimal control, differential games, numerical methods. 
@article{TIMM_2014_20_3_a3,
     author = {M. I. Gomoyunov and D. V. Kornev and N. Yu. Lukoyanov},
     title = {On the numerical solution of a~minmax control problem with a~positional functional},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {58--75},
     year = {2014},
     volume = {20},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2014_20_3_a3/}
}
TY  - JOUR
AU  - M. I. Gomoyunov
AU  - D. V. Kornev
AU  - N. Yu. Lukoyanov
TI  - On the numerical solution of a minmax control problem with a positional functional
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2014
SP  - 58
EP  - 75
VL  - 20
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2014_20_3_a3/
LA  - ru
ID  - TIMM_2014_20_3_a3
ER  - 
%0 Journal Article
%A M. I. Gomoyunov
%A D. V. Kornev
%A N. Yu. Lukoyanov
%T On the numerical solution of a minmax control problem with a positional functional
%J Trudy Instituta matematiki i mehaniki
%D 2014
%P 58-75
%V 20
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2014_20_3_a3/
%G ru
%F TIMM_2014_20_3_a3
M. I. Gomoyunov; D. V. Kornev; N. Yu. Lukoyanov. On the numerical solution of a minmax control problem with a positional functional. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 3, pp. 58-75. http://geodesic.mathdoc.fr/item/TIMM_2014_20_3_a3/

[1] Krasovskii N. N., Upravlenie dinamicheskoi sistemoi, Nauka, M., 1985, 516 pp. | MR

[2] Krasovskii N. N., Reshetova T. N., “O programmnom sinteze garantirovannogo upravleniya”, Problemy upravleniya i teoriya informatsii, 17:6 (1988), 333–343 | MR

[3] Krasovskii A. N., Krasovskii N. N., Control under Lack of Information, Birkhäuser, Berlin etc., 1995, 322 pp. | MR

[4] Krasovskii N. N., Lukoyanov N. Yu., “Zadacha konfliktnogo upravleniya s nasledstvennoi informatsiei”, Prikladnaya matematika i mekhanika, 60:6 (1996), 885–900 | MR

[5] Isakova E. A., Logunova G. V., Patsko V. S., “Postroenie stabilnykh mostov v lineinoi differentsialnoi igre s fiksirovannym momentom okonchaniya”, Algoritmy i programmy resheniya lineinykh differentsialnykh igr, UNTs AN SSSR, Sverdlovsk, 1984, 127–158

[6] Tarasev A. M., Uspenskii A. A., Ushakov V. N., “Konechno-raznostnyi metod postroeniya funktsii optimalnogo garantirovannogo rezultata”, Gagarinskie nauchnye chteniya po kosmonavtike i aviatsii 1991, Sb. tr., Nauka, M., 1992, 166–172

[7] Grigorenko N. L., Kiselev Yu. N., Lagunova N. V., Silin D. B. [i dr.], “Metody resheniya differentsialnykh igr”, Matematicheskoe modelirovanie, v. 1, Izd-vo MGU, M., 1993, 296–316

[8] Cardaliaguet P., Quincampoix M., Saint-Pierre P., “Set-valued numerical analysis for optimal control and differential games”, Stochastic and Differential Games: Theory and Numerical Methods, Annals of the Int. Soc. of Dynamic Games, 4, eds. Martino Bardi, T. E. S. Raghavan, T. Parthasarathy, Birkhäuser, Boston, 1999, 177–247 | MR | Zbl

[9] Bardi M., Falcone M., Soravia P., “Numerical methods for pursuit-evasion games via viscosity solutions”, Stochastic and Differential Games: Theory and Numerical Methods, Ann. Internat. Soc. Dynam. Games, 4, Birkhauser, Boston, 1999, 105–175 | MR | Zbl

[10] Polovinkin E. S., Ivanov G. E., Balashov M. V., Konstantinov R. V. [i dr.], “Algoritmy chislennogo resheniya lineinykh differentsialnykh igr”, Mat. sb., 192:10 (2001), 95–122 | DOI | MR | Zbl

[11] Mikhalev D. K., Ushakov V. N., “O dvukh algoritmakh priblizhennogo postroeniya mnozhestva pozitsionnogo pogloscheniya v igrovoi zadache sblizheniya”, Avtomatika i telemekhanika, 2007, no. 11, 178–194 | MR | Zbl

[12] Botkin N. D., Hoffmann K.-H., Turova V. L., “Stable numeric schemes for solving Hamilton–Jacobi–Bellman–Isaacs equation”, SIAM J. Scientific Computing, 33:2 (2011), 992–1007 | DOI | MR | Zbl

[13] Ivanov G. E., Kazeev V. A., “Minimaksnyi algoritm postroeniya optimalnoi strategii upravleniya v differentsialnoi igre s lipshitsevoi platoi”, Zhurn. vychisl. matematiki i mat. fiziki, 51:4 (2011), 594–619 | MR | Zbl

[14] Dvurechenskii P. E., Ivanov G. E., “Algoritmy vychisleniya operatorov Minkovskogo i ikh primenenie v differentsialnykh igrakh”, Zhurn. vychisl. matematiki i mat. fiziki, 54:2 (2014), 224–255 | DOI

[15] Krasovskii N. N., Tretyakov V. E., “Stokhasticheskii programmnyi sintez dlya pozitsionnoi differentsialnoi igry”, Dokl. AN SSSR, 259:1 (1981), 24–27 | MR

[16] Fleming W. H., “The convergence problem for differential games”, J. Math. Anal. Appl., 3:1 (1961), 102–116 | DOI | MR | Zbl

[17] Pontryagin L. S., “O lineinykh differentsialnykh igrakh. 1”, Dokl. AN SSSR, 174:6 (1967), 1278–1280 | Zbl

[18] Pshenichnyi B. N., “Struktura differentsialnykh igr”, Dokl. AN SSSR, 184:2 (1969), 285–287

[19] Friedman A., “On the definition of differential games and the existence of value and of saddle points”, J. Diff. Eq., 7 (1970), 69–91 | DOI | MR | Zbl

[20] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR | Zbl

[21] Krasovskii A. N., “Postroenie smeshannykh strategii na osnove stokhasticheskikh programm”, Prikl. matematika i mekhanika, 51:2 (1987), 186–192 | MR

[22] Lokshin M. D., “O differentsialnykh igrakh s integralnymi ogranicheniyami na upravlyayuschie vozdeistviya”, Differents. uravneniya, 28:1 (1992), 1952–1961 | MR | Zbl

[23] Lukoyanov N. Yu., “O zadache konfliktnogo upravleniya pri smeshannykh ogranicheniyakh na upravlyayuschie vozdeistviya”, Differents. uravneniya, 31:9 (1995), 1473–1482 | MR | Zbl

[24] Lukoyanov N. Yu., “K voprosu vychisleniya tseny differentsialnoi igry dlya pozitsionnogo funktsionala”, Prikl. matematika i mekhanika, 62:2 (1998), 188–198 | MR | Zbl

[25] Gomoyunov M. I., Lukoyanov N. Yu., “Ob ustoichivosti odnoi protsedury resheniya zadachi upravleniya na minimaks pozitsionnogo funktsionala”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20, no. 1, 2014, 68–82

[26] Kornev D. V., “O chislennom reshenii pozitsionnykh differentsialnykh igr s neterminalnoi platoi”, Avtomatika i telemekhanika, 2012, no. 11, 60–75 | MR | Zbl

[27] Gomoyunov M. I., Kornev D. V., “K voprosu vychisleniya tseny differentsialnoi igry v klasse kontrstrategii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19, no. 1, 2013, 59–68

[28] Kornev D. V., Lukoyanov N. Yu., “O chislennom reshenii differentsialnykh igr s neterminalnoi platoi v klassakh smeshannykh strategii”, Vestn. Udmurt. un-ta. Matematika. Mekhanika. Kompyuternye nauki, 2013, no. 3, 34–48

[29] Balashov M. V., “O $P$-svoistve vypuklykh kompaktov”, Mat. zametki, 71:3 (2002), 323–333 | DOI | MR | Zbl

[30] Rokafellar R., Vypuklyi analiz, Mir, M., 1973, 469 pp.