Sequential differentiation in nonsmooth infinite-dimensional extremal problems

S. Ya. Serovaǐskiǐ

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2008), pp. 48-62.

Bibliographie

[1] Lions Zh.–L., Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972, 414 pp. | MR

[2] Sea Zh., Optimizatsiya, teoriya i algoritmy, Mir, M., 1973, 244 pp. | MR

[3] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 480 pp. | MR | Zbl

[4] Vasilev F. P., Metody resheniya ekstremalnykh zadach. Zadachi minimizatsii v funktsionalnykh prostranstvakh, regulyarizatsiya, approksimatsiya, Nauka, M., 1981, 400 pp. | MR

[5] Oben Zh.-P., Ekland I., Prikladnoi nelineinyi analiz, Mir, M., 1988, 512 pp. | MR

[6] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977, 342 pp. | MR | Zbl

[7] Serovajsky S. Ya., “Functional gradient calculation and extended differentiation of operators”, J. of Inverse and Ill-Posed Problems, 13:4 (2005), 383–396 | DOI | MR | Zbl

[8] Serovaiskii S. Ya., “Differentsirovanie obratnoi funktsii v prostranstvakh bez normy”, Funktsion. analiz i ego pril., 27:4 (1993), 84–87 | MR | Zbl

[9] Ekland I., Temam R., Vypuklyi analiz i variatsionnye problemy, Mir, M., 1979, 400 pp. | MR

[10] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 280 pp. | Zbl

[11] Serovaiskii S. Ya., “Optimalnoe upravlenie dlya uravnenii ellipticheskogo tipa s negladkoi nelineinostyu”, Differents. uravneniya, 39:10 (2003), 1420–1424 | MR

[12] Serovaiskii S. Ya., “Priblizhennoe reshenie optimizatsionnykh zadach dlya singulyarnykh beskonechnomernykh sistem”, Sib. matem. zhurn., 2003, no. 3, 660–673 | MR

[13] Serovaiskii S. Ya., “Priblizhennoe reshenie zadachi optimalnogo upravleniya dlya singulyarnogo uravneniya ellipticheskogo tipa s negladkoi nelineinostyu”, Izv. vuzov. Matematika, 2004, no. 1, 80–86 | MR

[14] Serovaiskii S. Ya., “Sekventsialnye proizvodnye operatorov i ikh prilozheniya v negladkikh zadachakh optimalnogo upravleniya”, Izv. vuzov. Matematika, 2006, no. 12, 75–87 | MR

[15] Antosik P., Mikusinskii Ya., Sikorskii R., Teoriya obobschennykh funktsii. Sekventsialnyi podkhod, Mir, M., 1976, 311 pp.

[16] Wolfersdorf L., “Optimal control for processing governed by mildly nonlinear equations of parabolic type”, Z. Angew. Math. Mech., 56 (1976), 531–538 | DOI | MR | Zbl

[17] Ahmed K. L., “Optimization control of systems governed by a class of nonlinear evolutional equations in a reflexive Banach space”, J. Optim. Theory Appl., 28 (1978), 57–81 | DOI

[18] Matveev A. S., Yakubovich V. A., “Optimalnoe upravlenie nekotorymi sistemami s raspredelennymi parametrami”, Sib. matem. zhurn., 19:5 (1978), 1109–1140 | MR | Zbl

[19] Noussair E. S., Nababan S., Teo K. L., “On the existence of optimal control for quasilinear parabolic partial differential equations”, J. Optim. Theory and Appl., 34:1 (1981), 291–315 | DOI | MR

[20] Barbu V., “Necessary conditions for control problems governed by nonlinear partial differential equations”, Nonlinear Partial Differential Equations and Their Applications, College de France Sem. (Paris, 1979/80), Res. Notes in Math., 60, Pitman, London, 1982, 19–47 | MR

[21] Friedman A., “Nonlinear optimal control problems for parabolic equations”, SIAM J. Optim., 22:5 (1984), 805–816 | DOI | MR | Zbl

[22] Yong J., “Existence theory of optimal controls for distributed parameter systems”, Kodai Math. J., 15 (1992), 193–220 | DOI | MR | Zbl

[23] Ledzewicz U., “On distributed parameter control systems in the abnormal case in the case of nonoperators equality constraints”, J. Appl. Math. Stochast. Anal., 6:2 (1993), 137–151 | DOI | MR | Zbl

[24] Wang G., Liu C., “Maximum principle for state-constrained control of semilinear parabolic differential equation”, J. Optim. Theory Appl., 115:1 (2002), 183–209 | DOI | MR | Zbl

[25] Burbaki N., Teoriya mnozhestv, Nauka, M., 1965, 455 pp.

[26] Burbaki N., Obschaya topologiya. Osnovnye struktury, Nauka, M., 1968, 272 pp. | MR

[27] Dedonne Zh., Osnovy sovremennogo analiza, Mir, M., 1964, 430 pp.

[28] Zolezzi T., “A characterizations of well-posed optimal control systems”, SIAM J. Control and Optim., 19:5 (1981), 604–616 | DOI | MR | Zbl

[29] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978, 336 pp. | MR

[30] Krein S. G., Funktsionalnyi analiz, Nauka, M., 1972, 544 pp. | Zbl

[31] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972, 588 pp.

[32] Chernousko F. L., Kolmanovskii V. B., “Vychislitelnye i priblizhennye metody optimalnogo upravleniya”, Itogi nauki i tekhn. Matem. analiz, 14, VINITI, M., 1977, 101–166

[33] Serovajsky S. Ya., “Optimal control for singular equation with nonsmooth nonlinearity”, Journal of Inverse and Ill-Posed Problems, 14:6 (2006), 621–631 | DOI | MR