Geodesic
The necessary optimality conditions for a~nonlinear stationary system whose state functional is not differentiable with respect to the control
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2010), pp. 32-46.

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We consider a control system described by a nonlinear elliptic equation. Its control-state mapping is extended differentiable but not Gateaux differentiable for large values of the domain dimension and the nonlinearity index. We obtain a necessary optimality condition for various state functionals.
Keywords: optimal control, nonlinear elliptic equation, extended differentiability, optimality conditions. 
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S. Ya. Serovaiskii. The necessary optimality conditions for a~nonlinear stationary system whose state functional is not differentiable with respect to the control. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 6 (2010), pp. 32-46. http://geodesic.mathdoc.fr/item/IVM_2010_6_a3/

[1] Serovaiskii S., “Calculation of functional gradients and extended differentiation of operators”, J. of Inverse and Ill-Posed Problems, 13:4 (2005), 383–396 | DOI | MR

[2] Serovaiskii S. Ya., Optimizatsiya i differentsirovanie. T. 1. Minimizatsiya funktsionalov. Statsionarnye sistemy, Print-S, Almaty, 2006, 390 pp. | MR | Zbl

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

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

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

[6] Neustadt L. W., “An abstract variational theory with application to a broad class of optimization problems”, SIAM J. Control., 4:3 (1966), 505–527 | DOI | MR | Zbl

[7] Neustadt L. W., “An abstract variational theory with application to a broad class of optimization problems. II”, SIAM J. Control., 5:1 (1967), 90–137 | DOI | MR | Zbl

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

[9] Matveev A. S., Yakubovich V. A., Abstraktnaya teoriya optimalnogo upravleniya, Izd-vo SPb un-ta, SPb, 1994, 361 pp. | MR | Zbl

[10] Dmitruk A. V., Milyutin A. A., Osmolovskii N. P., “Teorema Lyusternika i teoriya ekstremuma”, UMN, 35:6 (1980), 11–46 | MR | Zbl

[11] Lions Zh.-L., Upravlenie singulyarnymi raspredelennymi sistemami, Nauka, M., 1987, 368 pp. | MR

[12] Raitum U. E., Zadachi optimalnogo upravleniya dlya ellipticheskikh uravnenii, Zinatne, Riga, 1989, 280 pp. | MR | Zbl

[13] Alibert J.-J., Raymond J.-P., “Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls”, Numer. Funct. Anal. Optim., 18:3–4 (1997), 235–250 | DOI | MR | Zbl

[14] Arada N., Raymond J.-P., “State-constrained relaxed problems for semilinear elliptic equations”, J. Math. Anal. Appl., 223:1 (1998), 248–271 | DOI | MR | Zbl

[15] Fursikov A. V., Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Nauchnaya kniga, Novosibirsk, 1999, 352 pp. | Zbl

[16] Voisei M. D., “Mathematical programming problems governed by nonlinear elliptic PDEs”, SIAM J. Optim., 18:4 (2007), 1231–1249 | DOI | MR | Zbl

[17] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972, 596 pp. | MR | Zbl

[18] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1977, 744 pp. | MR | Zbl

[19] Oben Zh.-P., Ekland I., Prikladnoi nelineinyi analiz, Mir, M., 1988, 510 pp. | MR

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

[21] Arutyunov A. V., “Neobkhodimye usloviya vtorogo poryadka v zadachakh optimalnogo upravleniya”, Dokl. RAN, 371:1 (2000), 10–13 | MR | Zbl

[22] Averbukh V. I., Smolyanov O. G., “Teoriya differentsirovaniya v lineinykh topologicheskikh prostranstvakh”, UMN, 22:6 (1967), 201–260 | MR | Zbl

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

[24] Lions Zh.-L., Madzhenes E., Neodnorodnye granichnye zadachi i ikh prilozheniya, Mir, M., 1971, 372 pp. | Zbl