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Optimal control problem for the backward heat equation. (English. Russian original) Zbl 0843.49017

Sib. Math. J. 34, No. 1, 181-188 (1993); translation from Sib. Mat. Zh. 34, No. 1, 204-211 (1993).
The optimal control problem of minimizing a quadratic functional for the system described by the Cauchy problem for the backward heat equation in which the control appears in boundary conditions is investigated. Under suitable conditions the existence and uniqueness of the optimal solution is proved. Necessary as well as sufficient conditions for optimality in the form of variational inequalities are obtained. The results generalize investigations in this field and can be applied to a broader class of constraints on the control.

MSC:

49K20 Optimality conditions for problems involving partial differential equations
35K05 Heat equation
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[1] M. M. Lavrent’ev, V. G. Romanov, and S. P. Shishatskiî, Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980).
[2] A. V. Fursikov, ?On some control problems and on results related to unique solvability of a mixed boundary value problem for the Navier-Stockes and Euler three-dimensional systems,? Dokl. Akad. Nauk SSSR,252, No. 5, 1066-1070 (1980).
[3] A. V. Fursikov, ?Control problems and theorems related to unique solvability of a mixed boundary value problem for the three-dimensional Navier-Stockes and Euler systems,? Mat. Sb.,115, No. 2, 281-307 (1981). · Zbl 0478.49010
[4] A. V. Fursikov, ?Properties of solutions to some extremal problems connected with the Navier-Stockes systems,? Mat. Sb.,118, No. 3, 323-349 (1982). · Zbl 0512.49017
[5] A. D. Ioffe and V. M. Tikhomirov, The Theory of Extremal Problems [in Russian], Nauka, Moscow (1974).
[6] P. H. Rivera, ?Optimal control of unstable nonlinear, evolution systems,? Ann. Fac. Sci. Toulouse Math., No. 1, 33-50 (1984). · Zbl 0547.49014
[7] P. N. Rivera and C. F. Vasconcellos, ?Optimal control for a backward parabolic system,? SIAM J. Control Optim.,25, No. 5, 1163-1172 (1987). · Zbl 0631.49003
[8] A. V. Fursikov, ?Lagrange principle for problems of optimal control of ill-posed or distributed systems,? J. Math. Pures Appl.,71, No. 2, 1-57 (1992). · Zbl 0829.49001
[9] D. L. Russel, ?Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions,? SIAM Rev.,20, No. 4, 639-739 (1978). · Zbl 0397.93001
[10] G. Schmidt, ?Even more states reachable by boundary control for the heat equation,?, SIAM J. Control Optim.,24, No. 6, 1319-1322 (1986). · Zbl 0602.49026
[11] J. L. Lions, Ouelque Méthodes de Résolution des Problemes aux Limites non Linéaires [Russian translation], Mir, Moscow (1972).
[12] J. L. Lions and E. Magenes, Problems aux Limites non Homogenes et Applications. Vol. 2, Dunod, Paris (1968). · Zbl 0165.10801
[13] V. A. Solonikov, ?A priori estimates for second order equations of parabolic type,? Trudy Mat. Inst. Steklov,70, 133-212 (1966).
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