Discrete regularization of optimal control problems on ill-posed monotone variational inequalities

O. A. Liskovets

Izvestiya. Mathematics , Tome 37 (1991) no. 2, pp. 321-335.

Résumé

The author considers the problem of minimizing an abstract functional which depends on the control and on the solution of an ill-posed variational inequality (v.i.) with a bounded monotone exact operator with the $\mu$-property (these properties are not needed for approximate operators, nor is the $\mu$-property in the pseudomonotone case). Solvability of the problem is shown. For its regularization the original v.i. is regularized by means of v.i. with a small step. A number of generalizations are indicated.

Bibliographie

[1] Mikhlin S. G., Chislennaya realizatsiya variatsionnykh metodov, Nauka, M., 1966 | MR

[2] Dyuvo G., Lions Zh.-L., Neravenstva v mekhanike i fizike, Nauka, M., 1980 | MR

[3] Glovinski R., Lions Zh.-L., Tremoler R., Chislennoe issledovanie variatsionnykh neravenstv, Mir, M., 1979 | MR

[4] Kinderlerer D., Stampakkya G., Vvedenie v variatsionnye neravenstva i ikh prilozheniya, Mir, M., 1983 | MR

[5] Vaiberg M. M., Variatsionnyi metod i metod monotonnykh operatorov, Nauka, M., 1972

[6] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[7] Gaevskii X., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

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

[9] Litvinov V. G., Optimizatsiya v ellipticheskikh granichnykh zadachakh s prilozheniyami k mekhanike, Nauka, M., 1987 | MR

[10] Mignot F., “Controle dans les inéquations variationnelles”, C.R. Acad. Sci. Ser. A., 280:4 (1975), 197–200 | MR | Zbl

[11] Mignot F., Puel J. P., “Optimal control in some variational inequalities”, SIAM J. Control Optimiz., 22:3 (1984), 466–476 | DOI | MR | Zbl

[12] Hlaváček I., Bock I., Lovišek J., “Optimal control of a variational inequality with applications to structural analysis, I”, Appl. Math. and Optimiz., 11:2 (1984), 111–143 | DOI | MR | Zbl

[13] Saquez C., Bermudez A., “Optimal control of variational inequalities”, Proc. 23rd IEEE Conf. Decis. and Control, v. 1, Las Vegas, 1984, 249–251

[14] Friedman A., “Optimal control for variational inequalities”, SIAM J. Control Optimiz., 24:3 (1986), 439–451 | DOI | MR | Zbl

[15] Bermudez A., Saguez C., “Optimal control of a Signorini problem”, SIAM J. Control. Optimiz., 25:3 (1987), 576–582 | DOI | MR | Zbl

[16] Bock I., Lovišek J., “Optimal control problems for variational inequalities with controls in coefficients and in unilateral constraints”, Apl. Mat., 32:4 (1987), 301–314 | MR | Zbl

[17] Melnik V. S., “Metod monotonnykh operatorov v teorii optimalnykh sistem s ogranicheniyami”, DAN USSR. Ser. A, 1984, no. 7, 71–73 | MR

[18] Alber Ya. I., “O reshenii lineinykh uravnenii s monotonnymi operatorami v banakhovom prostranstve”, Sib. matem. zhurn., 16:1 (1975), 3–11 | MR

[19] Alber Ya. I., Ryazantseva I. P., “Variatsionnye neravenstva s razryvnymi monotonnymi otobrazheniyami”, DAN SSSR, 262:6 (1982), 1289–1293 | MR

[20] Bakushinskii A. B., Polyak B. T., “O reshenii variatsionnykh neravenstv”, DAN SSSR, 219:5 (1974), 1038–1041

[21] Liskovets O. A., “Regulyarizatsiya zadach s razryvnymi monotonnymi nemonotonno vozmuschaemymi operatorami”, DAN SSSR, 272:1 (1983), 30–34 | MR | Zbl

[22] Liskovets O. A., “Diskretnaya regulyarizatsiya zadach s proizvolno vozmuschaemymi monotonnymi operatorami”, DAN SSSR, 289:5 (1986), 1056–1059 | MR

[23] Liskovets O. A., “Regulyarizatsiya zadach s monotonnymi operatorami pri diskretnoi approksimatsii prostranstv i operatorov”, ZhVM i MF, 27:1 (1987), 3–15 | MR | Zbl

[24] Stummel F., “Diskrete Konvergenz linearer Operatoren, I”, Math. Ann., 190:1 (1970), 45–92 | DOI | MR | Zbl

[25] Grigorieff R. D., “Über die diskrete Approximation nichtlinearer Gleichungen erster Art”, Math. Nachr., 69 (1975), 253–272 | DOI | MR | Zbl

[26] Anselone P. M., Ansorge R., “A unified framework for the discretization of nonlinear operator equations”, Numer. Funct. Anal. Optimiz., 4:1 (1981), 61–99 | DOI | MR | Zbl

[27] Vainikko G. M., “Regulyarnaya skhodimost operatorov i priblizhennoe reshenie uravnenii”, Itogi nauki i tekhniki. Matematicheskii analiz, 16, VINITI, M., 1978, 5–53

[28] Pankov A. A., “Diskretnye approksimatsii vypuklykh mnozhestv i skhodimost reshenii variatsionnykh neravenstv”, Math. Nachr., 91 (1979), 7–22 | DOI | MR | Zbl

[29] Mosco U., “Convergence of convex sets and of solutions of variational inequalities”, Advances Math., 3:4 (1969), 510–585 | DOI | MR | Zbl

[30] Liskovets O. A., Variatsionnye metody resheniya neustoichivykh zadach, Nauka i tekhnika, Minsk, 1981 | MR | Zbl

[31] Lapin A. V., “Reshenie variatsionnykh neravenstv s nelineinymi polukoertsitivnymi operatorami”, Vychislitelnye protsessy i sistemy, no. 4, Nauka, M., 1986, 219–264 | MR