Reconstruction of boundary disturbances: the case of Neumann boundary conditions

V. I. Maksimov

Trudy Instituta matematiki i mehaniki, Dynamical systems and control problems, Tome 11 (2005) no. 1, pp. 160-176

Résumé

A problem of dynamical reconstruction of boundary disturbances in a nonlinear parabolic equation is considered. In the case when a disturbance is concentrated in the Neumann boundary conditions, two solving algorithms stable with respect to informational noises and computational errors are described.

Bibliographie

[1] Alifanov O. M., Artyukhin E. A., Rumyantsev S. V., Ekstremalnye metody resheniya nekorrektnykh zadach, Nauka, M., 1988 | MR | Zbl

[2] Lavrentev M. M., Romanov V. G., Shishatskii S. P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka. SO, Novosibirsk, 1980 | MR

[3] Prilepko A. I., Solovev V. V., “O razreshimosti obratnykh kraevykh zadach opredeleniya koeffitsienta pri mladshei proizvodnoi v parabolicheskom uravnenii”, Differents. uravneniya, 23:1 (1987), 136–143 | MR | Zbl

[4] Gusev M. I., Kurzhanskii A. B., “Obratnye zadachi dinamiki upravlyaemykh sistem”, Mekhan. i nauchno-tekhn. progress. T. 1. Obschaya i prikladnaya mekhanika, Nauka, M., 1987, 187–195 | MR

[5] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978 | MR

[6] Kryazhimskii A. V., Osipov Yu. S., “O modelirovanii upravleniya v dinamicheskoi sisteme”, Izv. AN SSSR. Tekhn. kibernetika, 1983, no. 2, 51–60 | MR

[7] Osipov Yu. S., Kryazhimskii A. V., Inverse problems for ordinary differential equations: dynamical solutions, Gordon and Breach, London, 1995 | MR | Zbl

[8] Osipov Yu. S., Kryazhimskii A. V., Maksimov V. I., Zadachi dinamicheskoi regulyarizatsii dlya sistem s raspredelennymi parametrami, Izd-vo Instituta matematiki i mekhaniki UrO RAN, Sverdlovsk, 1991

[9] Maksimov V. I., Zadachi dinamicheskogo vosstanovleniya vkhodov beskonechnomernykh sistem, Izd-vo Instituta matematiki i mekhaniki UrO RAN, Ekaterinburg, 2000

[10] Osipov Yu. S., Kryazhimskii A. V., Maksimov V. I., “Obratnye zadachi dinamiki dlya parabolicheskikh sistem”, Differents. uravneniya, 36:5 (2000), 579–597 | MR | Zbl

[11] Krasovskii N. N., Subbotin A. I., Pozitsionnye differentsialnye igry, Nauka, M., 1974 | MR | Zbl

[12] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1978 | Zbl

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

[14] Maksimov V. I., Pandolfi L., J. Optimization Theory and Applications, 103:2 (1999), 401–420 | DOI | MR | Zbl

[15] Osipov Yu. S., Pandolfi L., Maksimov V. I., “Problems of dynamic reconstruction and robust boundary control: the case of Dirichlet boundary conditions”, J. Inverse and Ill-Posed Problems, 9:2 (2001), 149–162 | MR | Zbl

[16] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR

[17] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

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

[19] Lasiecka I., Triggiani R., “Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems”, Appl. Math. and Optim., 23:2 (1991), 109–154. | DOI | MR | Zbl

[20] Lasiecka I., “Boundary control of parabolic systems: regularity of optimal solutions”, Appl. Math. and Optim., 4:4 (1978), 301–328 | MR

[21] Lasiecka I., “Unified theory for abstract parabolic boundary problemsva semigroup approach”, Appl. Math. and Optim., 6:4 (1980), 287–334 | DOI | MR

[22] Lasiecka I., Triggiani R., Differential and algebraic Riccati equations with applications to bound-ary/point control problems: continuous theory and approximation theory, Lecture Notes in Control and Information Sciences, 164, Springer-Verlag, Berlin, 1991 | MR | Zbl

[23] Tröltzsch F., “On convergence of semidiscrete Ritz–Galerkin schemes applied to the boundary control of parabolic equations with non-linear boundary condition”, Z. Angew. Math. Mech., 72:7 (1992), 291–301 | DOI | MR | Zbl

[24] Fattorini H., “Ordinary differential equations in linear topological spaces. I and II”, J. Different. Equations, 5 (1968), 72–105 ; 6 (1969), 50–70 | DOI | MR | DOI | MR | Zbl

[25] Baiokki K., Kapelo A., Variatsionnye i kvazivariatsionnye neravenstva, Nauka, M., 1988 | MR

[26] Barbu V., “Boundary control problems with convex cost criterion”, SIAM J. Control Optim., 18:2 (1980), 227–243 | DOI | MR | Zbl

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

[28] Barbu V., Optimal control of variational inequalities, Pitman, London, 1984 | MR | Zbl

[29] Osipov Yu. S., Okhezin S. P., “K teorii differentsialnykh igr v parabolicheskikh sistemakh”, DAN SSSR, 226:6 (1976), 1267–1270 | MR | Zbl