×

zbMATH — the first resource for mathematics

Generalized solvability and optimization of a parabolic system with a discontinuous solution. (English) Zbl 1113.35025
The paper considers optimal control problems governed by a linear parabolic equation with specificic jump conditions on the interface between two parts of the spatial reference domain. The author introduces a specific notion of a generalized solution for the state equation and gives some sufficient conditions for the existence of such solutions. These results are applied to cases with the right hand side of the equation as a control, among them the case with the pulse control, and some properties of differentiability of the cost functional with respect to the control and existence of an optimal control are established.

MSC:
35B37 PDE in connection with control problems (MSC2000)
49J20 Existence theories for optimal control problems involving partial differential equations
35K10 Second-order parabolic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Krutitskii, P.A., The mixed harmonic problem in a bounded cracked domain with Dirichlet condition on cracks, J. differential equations, 198, 2, 422-441, (2004) · Zbl 1086.35027
[2] Ferchichi, J.; Zolesio, J., Shape sensitivity for the laplace – beltrami operator with singularities, J. differential equations, 196, 2, 340-384, (2004) · Zbl 1039.49036
[3] Lyubimov, D.Yu.; Smirnov, L.G., Dirichlet and related problems for a composed space with annular slits at the interface in the axisymmetric case, Comput. math. math. phys., 44, 8, 1366-1373, (2004) · Zbl 1075.35090
[4] Albou, A.F.; Zubov, V.I.; Inyakin, V.A., Optimal control of the process of melting and solidification of a substance, Comput. math. math. phys., 44, 8, 1291-1305, (2004) · Zbl 1084.49032
[5] Albou, A.F.; Zubov, V.I., Optimal control of the process of solidification, Comput. math. math. phys., 44, 1, 34-44, (2004) · Zbl 1068.49029
[6] Krutitskii, P.A.; Sgibnev, A.I., Singularities of the solution gradient in the generalized jump problem for the Laplace equation outside a cut on the plane, Differential equations, 39, 9, 1225-1236, (2003) · Zbl 1161.35345
[7] Huang, J.; Zou, J., Some new a priori estimates for second-order elliptic and parabolic interface problems, J. differential equations, 184, 2, 570-586, (2002) · Zbl 1012.35013
[8] Bal, G., Transport through diffusive and nondiffusive regions, embedded objects, and clear layers, SIAM J. appl. math., 62, 5, 1677-1697, (2002) · Zbl 1020.45004
[9] Yang, J.; Kalliadasis, S.; Merkin, J.H.; Scott, S.K., Wave propagation in spatially distributed excitable media, SIAM J. appl. math., 63, 2, 485-509, (2002) · Zbl 1048.92002
[10] Piven’, V.F., Integral and integro – differential equations of the two-dimensional transmission problem for velocity fields on a time-dependent boundary, Differential equations, 38, 12, 1809-1814, (2002) · Zbl 1148.45300
[11] Tsurko, V.A., Finite-difference methods for approximate solutions of two-dimensional parabolic equations with discontinuous coefficients, Differential equations, 38, 7, 1070-1072, (2002) · Zbl 1023.65089
[12] Zakharov, E.V.; Safronov, S.I.; Tarasov, R.P., The problem of resonance scattering by a hole in an acoustically soft surface of revolution, Differential equations, 38, 9, 1239-1244, (2002) · Zbl 1290.35213
[13] Shishkin, G.I., Grid approximation of a singularly perturbed parabolic reaction – diffusion equation with a fast-moving source, Comput. math. math. phys., 42, 6, 788-801, (2002) · Zbl 1056.35015
[14] Krutitskii, P.A., The Neumann problem in a 2-D exterior domain with cuts and singularities at the tips, J. differential equations, 176, 1, 269-289, (2001) · Zbl 1013.35024
[15] Lederman, C.; Wolanski, N., Uniqueness in a two-phase free-boundary problem, Adv. differential equations, 6, 12, 1409-1442, (2001) · Zbl 1004.35123
[16] Komarenko, O.N., Self-adjoint operators generated by transmission problems with nonhomogeneous conjugation conditions, Ukrainian math. J., 53, 12, 1607-1622, (2001) · Zbl 1064.35049
[17] Krutitskii, P.A., The modified jump problem for the Helmholtz equation, Ann. univ. ferrara sez. VII (N.S.), 47, 285-296, (2001) · Zbl 1010.35027
[18] Bal, G.; Ryzhik, L., Diffusion approximation of radiative transfer problems with interfaces, SIAM J. appl. math., 60, 6, 1887-1912, (2000) · Zbl 0976.45008
[19] Krutitskii, P.A., The jump problem for the Helmholtz equation and singularities at the edges, Appl. math. lett., 13, 71-76, (2000) · Zbl 0957.35038
[20] Khludnev, A.M.; Kovtunenko, V.A., Analysis of cracks in solids, (2000), WIT Press Boston
[21] Alves, C.J.S.; Ha-Duong, T., Inverse scattering for elastic plane cracks, Inverse problems, 15, 91-97, (1999) · Zbl 0929.35170
[22] Nishimura, N., Crack determination problems, Theor. appl. mech., 46, 39-57, (1997)
[23] K. Bryan, L.F. Caudill, Jr., Solvability of a parabolic boundary value problem with internal jump condition, Rose-Hulman Technical Report MSTR 00-04, http://www.rose-hulman.edu/math/researchpubs/MSTechReps.php
[24] Semenov, V.V., Solvability of a parabolic transmission problem with the condition of a generalized proper lumped source, Differential equations, 41, 6, 878-886, (2005) · Zbl 1086.35020
[25] Lyashko, S.I., Generalized optimal control of linear systems with distributed parameters, (2002), Kluwer Academic Boston · Zbl 1030.49001
[26] Lyashko, I.I.; Demchenko, L.I.; Mistetskij, G.E., Numerical solution of problems in heat and mass exchange in porous media, (1991), Naukova Dumka Kiev, in Russian · Zbl 0804.76002
[27] Nomirovskii, D.A., Approximate methods for solving the boundary value problem for a parabolic equation with inhomogeneous transmission conditions of nonideal contact type, Comput. math. math. phys., 46, 6, 995-1006, (2006)
[28] Nomirovskii, D.A., Optimization of parabolic systems with distributional coefficients, Rep. nat. acad. sci. ukraine, 12, 77-82, (2000) · Zbl 0983.49015
[29] Nomirovskii, D.A., Generalized solvability and optimization of singular parabolic systems, Rep. nat. acad. sci. ukraine, 10, 30-35, (2003) · Zbl 1097.80503
[30] Nomirovskii, D.A., Generalized solvability of parabolic systems with heterogeneous transmission conditions of imperfect contacts, Differential equations, 40, 10, 1390-1399, (2004)
[31] Sergienko, I.V.; Deineka, V.S., Optimal control of distributed systems with conjugation conditions, (2005), Kluwer Academic Boston · Zbl 1080.49001
[32] Deineka, V.S.; Sergienko, I.V.; Skopetskii, V.V., Models and methods for solving problems with conjugation conditions, (1998), Naukova Dumka Kiev, in Russian · Zbl 0947.74070
[33] F. Kikuchi, Numerical analysis of a mixed finite element method for plate buckling problems, Institute of Space and Aeronautical Science, University of Tokyo, Report No. 584, September 1980, pp. 165-190
[34] Mercier, B.; Osborn, J.; Rappaz, J.; Raviart, P.A., Eigenvalue approximation by mixed and hybrid methods, Math. comput., 36, 154, 427-453, (1981) · Zbl 0472.65080
[35] Sanchez-Palencia, E., Non-homogeneous media and vibration theory, (1980), Springer-Verlag New York · Zbl 0432.70002
[36] Lyashko, S.I.; Nomirovskii, D.A.; Sergienko, T.I., Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions, Cybernet. systems anal., 37, 5, 756-763, (2001) · Zbl 1037.93047
[37] Lyashko, S.I.; Nomirovskii, D.A., Generalized solution and optimal controls in systems describing the dynamic of a viscous fluid, Differential equations, 39, 1, 90-98, (2003) · Zbl 1175.93037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.