Parametric optimization of nonlinear Goursat–Darboux systems with phase constraints

V. S. Gavrilov   ; M. I. Sumin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 44 (2004) no. 6, pp. 1002-1022

Bibliographie

[1] Novozhenov M. M., Plotnikov V. I., “Obobschennoe pravilo mnozhitelei Lagranzha dlya raspredelennykh sistem s fazovymi ogranicheniyami”, Differents. ur-niya, 18:4 (1982), 584–592 | MR | Zbl

[2] Mackenroth U., “On some elliptic optimal control problems with state constraints”, Optimizat., 17 (1986), 595–607 | DOI | MR | Zbl

[3] Abergel F., Teman R., “Optimality conditions for some non-qualified problems of distributed control”, SIAM J. Control Optimizat., 27:1 (1989), 1–12 | DOI | MR | Zbl

[4] Matveev A. S., “K abstraktnoi teorii optimalnogo upravleniya sistemami s raspredelennymi parametrami”, Sibirskii matem. zhurnal, 29:1 (1988), 94–107

[5] Bergounioux M., “A penalization method for optimal control of elliptic problems with state constraints”, SIAM J. Control Optimizat., 30:2 (1992), 305–323 | DOI | MR | Zbl

[6] Casas E., “Boundary control of semilinear elliptic equations with pointwise state constraints”, SIAM J. Control Optimizat., 31:4 (1993), 993–1006 | DOI | MR | Zbl

[7] Li X., Yong J., Optimal control theory for insfinite dimensional systems, Birkhäuser Verlag, Basel, 1995 | MR | Zbl

[8] Bonnans J. F., Casas E., “An extension of Pontryagin's principle for state-constrained opitmal contorl of semilinear elliptic equations and variational inequalities”, SIAM J. Control Optimizat., 33:1 (1995), 274–298 | DOI | MR | Zbl

[9] Casas E., “Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations”, SIAM J. Control Optimizat., 35:4 (1997), 1297–1327 | DOI | MR | Zbl

[10] Raymond J.-P., Zidani H., “Pontryagin's principle for state-constrained control problems governed by parabolic equations with unbounded controls”, SIAM J. Control Optimizat., 36:6 (1998), 1853–1879 | DOI | MR | Zbl

[11] Casas E., Raymond J.-P., Zidani H., “Pontryagin's principle for local solutions of control problems with mixed control-state constraints”, SIAM J. Control Optimizat., 39:4 (2000), 1182–1208 | DOI | MR

[12] Sumin M. I., “Optimal control of semilinear elliptic equation with state constraint: Maximum principle for minimizing sequence, regularity, normality, sensitivity”, Control and Cybernetics, 29:2 (2000), 449–472 | MR | Zbl

[13] Sumin M. I., “Suboptimalnoe upravlenie polulineinymi ellipticheskimi uravneniyami s fazovymi ogranicheniyami. I: Printsip maksimuma dlya minimiziruyuschikh posledovatelnostei, normalnost”, Izv. vuzov. Matematika, 2000, no. 6, 33–44 | MR | Zbl

[14] Sumin M. I., “Suboptimalnoe upravlenie polulineinymi ellipticheskimi uravneniyami s fazovymi ogranicheniyami. II: Chuvstvitelnost, tipichnost regulyarnogo printsipa maksimuma”, Izv. vuzov. Matematika, 2000, no. 8, 52–63 | MR | Zbl

[15] Sumin M. I., Matematicheskaya teoriya suboptimalnogo upravleniya raspredelennymi sistemami, Dis. ...dokt. fiz.-matem. nauk, Nizhegorodskii gos. un-t, N. Novgorod, 2000

[16] Sumin M. I., “Suboptimalnoe upravlenie polulineinym ellipticheskim uravneniem s fazovym ogranicheniem i granichnym upravleniem”, Diferents. ur-niya, 37:2 (2001), 260–275 | MR | Zbl

[17] O. V. Vasilev (red.), Usloviya ekstremuma i konstruktivnye metody resheniya v zadachakh optimizatsii giperbolicheskikh sistem, Nauka, Novosibirsk, 1993 | MR | Zbl

[18] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977 | MR

[19] Gavrilov B. C., Sumin M. I., “Optimalnoe upravlenie sistemami Gursa–Darbu s potochechnymi fazovymi ogranicheniyami. II: Minimiziruyuschie posledovatelnosti, parametricheskaya optimizatsiya”, Matem. modelirovanie i optimalnoe upravlenie, Vestn. NNGU, 1, no. 26, N. Novgorod, 2003, 126–137

[20] Plotnikov V. I., Sumin M. I., “O postroenii minimiziruyuschikh posledovatelnostei v zadachakh upravleniya sistemami s raspredelennymi parametrami”, Zh. vychisl. matem. i matem. fiz., 22:1 (1982), 49–56 | MR | Zbl

[21] Gamkrelidze R. V., Osnovy optimalnogo upravleniya, Izd-vo Tbilisskogo un-ta, Tbilisi, 1977 | MR | Zbl

[22] Danilova O. A., Matveev A. S., “Netraditsionnye usloviya suschestvovaniya optimalnogo upravleniya dlya sistemy Gursa–Darbu”, Izv. AN SSSR. Ser. matem., 62:5 (1998), 79–102 | MR | Zbl

[23] Sumin M. I., “Suboptimalnoe upravlenie sistemami s raspredelennnymi parametrami: minimiziruyuschie posledovatelnosti, funktsiya znachenii”, Zh. vychisl. matem. i matem. fiz., 37:1 (1997), 23–41 | MR | Zbl

[24] Sumin M. I., “Suboptimalnoe upravlenie sistemami s raspredelennymi parametrami: svoistva normalnosti, subgradientnyi dvoistvennyi metod”, Zh. vychisl. matem. i matem. fiz., 37:2 (1997), 162–178 | MR | Zbl

[25] Alekseev V. M., Tikhomirov V. M., Fomin S. V., Optimalnoe upravlenie, Nauka, M., 1979 | MR

[26] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1998 | MR

[27] Borwein J. M., Strojwas H. M., “Proximal analysis and boundaries of closed sets in Banach space. Part I: Theory”, Canad. J. Math., 38:2 (1986), 431–452 ; “Part II: Applications”, 39:2 (1987), 428–472 | DOI | MR | Zbl | DOI | MR | Zbl

[28] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988 | MR | Zbl

[29] Mordukhovich B. S., Shao Y., “Nonsmooth sequential analysis in asplund spaces”, Trans. Amer. Math. Soc., 346:4 (1996), 1235–1280 | DOI | MR

[30] Plotnikov V. I., Sumin B. I., “Optimizatsiya ob'ektov s raspredelennymi parametrami, opisyvaemykh sistemami Gursa–Darbu”, Zh. vychisl. matem. i matem. fiz., 12:1 (1972), 61–77 | MR | Zbl

[31] Tikhomirov V. M., “Vypuklyi analiz”, Itogi nauki i tekhn. Ser. Sovrem. probl. matem. Fundamentalnye napravleniya, 14, VINITI, 1987, 5–101 | MR

[32] Gavrilov B. C., Sumin M. I., “Optimalnoe upravlenie sistemami Gursa–Darbu s potochechnymi fazovymi ogranicheniyami. I: Printsip maksimuma Pontryagina”, Matem. modelirovanie i optimalnoe upravlenie, Vestn. NNGU, 1, no. 25, N. Novgorod, 2002, 175–183

[33] Plotnikov V. I., Sumin M. I., “Problemy ustoichivosti nelineinykh sistem Gursa–Darbu”, Different ur-niya, 8:5 (1972), 845–856 | MR | Zbl

[34] Plotnikov V. I., Sumin M. I., “Optimalnoe upravlenie ob'ektami s raspredelennymi parametrami, opisyvaemymi negladkimi sistemami Gursa–Darbu s ogranicheniyami tipa neravenstva”, Differents. ur-niya, 20:5 (1984), 851–860 | MR | Zbl

[35] Sumin M. I., “Dostatochnye usloviya optimalnosti v negladkikh zadachakh optimalnogo upravleniya raspredelennymi sistemami”, Differents. ur-niya, 22:2 (1986), 326–337 | MR | Zbl

[36] Sumin M. I., “O minimiziruyuschikh posledovatelnostyakh v zadachakh optimalnogo upravleniya pri ogranichennykh fazovykh koordinatakh”, Differents. ur-niya, 22:10 (1986), 1719–1731 | MR

[37] Sumin M. I., “Optimalnoe upravlenie ob'ektami, opisyvaemymi kvazilineinymi ellipticheskimi uravneniyami”, Differents. ur-niya, 25:8 (1989), 1406–1416 | MR

[38] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981 | MR

[39] Bertsekas D., Uslovnaya optimizatsiya i metody mnozhitelei Lagranzha, Radio i svyaz, M., 1987 | MR | Zbl

[40] Ward A. L., “Differentiability of vector monotone functions”, Proc. London Math. Soc., 32:2 (1935), 339–362 | DOI

[41] Saks S., Teoriya integrala, Izd-vo inostr. lit., M., 1949