Geodesic
Approximation of a singularly perturbed elliptic problem of optimal control
Sbornik. Mathematics, Tome 191 (2000) no. 10, pp. 1421-1431 
A. R. Danilin. Approximation of a singularly perturbed elliptic problem of optimal control. Sbornik. Mathematics, Tome 191 (2000) no. 10, pp. 1421-1431. http://geodesic.mathdoc.fr/item/SM_2000_191_10_a0/
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Voir la notice de l'article provenant de la source Math-Net.Ru

A problem of optimal control of solutions of an elliptic equation with a small parameter at higher derivatives is considered in a rectangle with two sides parallel to the characteristic of the limit equation. The limit problem is found and asymptotic estimates for solutions of a problem that approximates the original problem are obtained.

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

[2] Danilin A. R., “Asimptotika ogranichennykh upravlenii dlya singulyarnoi ellipticheskoi zadachi v oblasti s maloi polostyu”, Matem. sb., 189:11 (1998), 27–60 | MR | Zbl

[3] Kapustyan V. E., “Asimptotika ogranichennykh upravlenii v optimalnykh ellipticheskikh zadachakh”, Dokl. AN Ukrainy. Ser. matematika, estestvoznanie, tekhnicheskie nauki, 1992, no. 2, 70–74 | MR

[4] Ilin A. M., Lelikova E. F., “Asimptotika reshenii nekotorykh ellipticheskikh uravnenii v neogranichennykh oblastyakh”, Matem. sb., 119 (161):3 (1982), 307–324 | MR | Zbl

[5] Lelikova E. F., “Metod sraschivaniya asimptoticheskikh razlozhenii dlya uravneniya $\varepsilon\Delta u-au_z=f$ v parallelepipede”, Differents. uravneniya, 14:9 (1978), 1638–1648 | MR | Zbl

[6] Kalyakin L. A., “Asimptotika resheniya sistemy dvukh lineinykh uravnenii MGD s singulyarnym vozmuscheniem. I: Standartnaya zadacha v ellipticheskom sloe”, Differents. uravneniya, 18:10 (1982), 1724–1738 | MR | Zbl

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

[8] Banach S., Steinhaus H., “Sur le principle de la condensation de singularités”, Fund. Math., 9 (1927), 50–61 | Zbl

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

[10] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo LGU, L., 1950

[11] Ilin A. M., Soglasovanie asimtoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR