Initial value methods for parabolic control problems
Author:
Ragnar Winther
Journal:
Math. Comp. 34 (1980), 115125
MSC:
Primary 65K10; Secondary 49D05, 65Mxx
MathSciNet review:
551293
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Abstract: We study iterative methods for parabolic control problems with a Neumann boundary value control and where the observation is the final state. The methods are based on transforming the original control problem (which may have constraints on the control) into an equivalent problem of minimizing a strictly convex functional (no constraints). The methods are semidiscrete in the sense that we assume that parabolic initial value problems can be solved exactly.
 [1]
J.
E. Dennis Jr. and Jorge
J. Moré, QuasiNewton methods, motivation and theory,
SIAM Rev. 19 (1977), no. 1, 46–89. MR 0445812
(56 #4146)
 [2]
Richard
S. Falk, Approximation of a class of optimal control problems with
order of convergence estimates, J. Math. Anal. Appl.
44 (1973), 28–47. MR 0686788
(58 #33347)
 [3]
Avner
Friedman, Partial differential equations, Holt, Rinehart and
Winston, Inc., New YorkMontreal, Que.London, 1969. MR 0445088
(56 #3433)
 [4]
J.L.
Lions, Optimal control of systems governed by partial differential
equations., Translated from the French by S. K. Mitter. Die
Grundlehren der mathematischen Wissenschaften, Band 170, SpringerVerlag,
New YorkBerlin, 1971. MR 0271512
(42 #6395)
 [5]
J.L.
Lions and E.
Magenes, Nonhomogeneous boundary value problems and applications.
Vol. I, SpringerVerlag, New YorkHeidelberg, 1972. Translated from
the French by P. Kenneth; Die Grundlehren der mathematischen
Wissenschaften, Band 181. MR 0350177
(50 #2670)
 [6]
J.
M. Ortega and W.
C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New YorkLondon, 1970. MR 0273810
(42 #8686)
 [7]
W. M. PATTERSON, Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space, Lecture Notes in Math. vol. 394, SpringerVerlag, New York, 1974.
 [8]
Albrecht
Pietsch and Hans
Triebel, Interpolationstheorie für Banachideale von
beschränkten linearen Operatoren, Studia Math.
31 (1968), 95–109 (German). MR 0243341
(39 #4663)
 [9]
J. R. RINGROSE, Compact NonSelfAdjoint Operators, Van Nostrand Reinhold Co., London,1971.
 [10]
M.
M. Vaĭnberg, Variational method and method of monotone
operators in the theory of nonlinear equations, Halsted Press (A
division of John Wiley & Sons), New YorkToronto, Ont.; Israel Program
for Scientific Translations, JerusalemLondon, 1973. Translated from the
Russian by A. Libin; Translation edited by D. Louvish. MR 0467428
(57 #7286b)
 [11]
R. WINTHER, A Numerical Galerkin Method for a Parabolic Control Problem, Ph.D. Thesis, Cornell University, 1977.
 [12]
Ragnar
Winther, Error estimates for a Galerkin approximation of a
parabolic control problem, Ann. Mat. Pura Appl. (4)
117 (1978), 173–206. MR 515960
(80a:49067), http://dx.doi.org/10.1007/BF02417890
 [13]
Ragnar
Winther, Some superlinear convergence results for the conjugate
gradient method, SIAM J. Numer. Anal. 17 (1980),
no. 1, 14–17. MR 559456
(81k:65060), http://dx.doi.org/10.1137/0717002
 [1]
 J. E. DENNIS & J. MORÉ, "QuasiNewton methods, motivation and theory," SIAM Rev., v. 19, 1977, pp. 4689. MR 0445812 (56:4146)
 [2]
 R. S. FALK, "Approximation of a class of optimal control problems with order of convergence estimates," J. Math. Anal. Appl., v. 44, 1973, pp. 2847. MR 0686788 (58:33347)
 [3]
 A. FRIEDMAN, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969. MR 0445088 (56:3433)
 [4]
 J. L. LIONS, Optimal Control of Systems Governed by Partial Differential Equations, SpringerVerlag, New York, 1971. MR 0271512 (42:6395)
 [5]
 J. L. LIONS & E. MAGENES, Non Homogeneous Boundary Value Problems and Applications, Vols. III, SpringerVerlag, New York, 1972. MR 0350177 (50:2670)
 [6]
 J. M. ORTEGA & W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 0273810 (42:8686)
 [7]
 W. M. PATTERSON, Iterative Methods for the Solution of a Linear Operator Equation in Hilbert Space, Lecture Notes in Math. vol. 394, SpringerVerlag, New York, 1974.
 [8]
 A. PIETSCH & H. TRIEBEL, "Interpolationstheorie für Banachideale von beschränkten linearen Operatoren," Studia Math., v. 3, 1968, pp. 95109. MR 0243341 (39:4663)
 [9]
 J. R. RINGROSE, Compact NonSelfAdjoint Operators, Van Nostrand Reinhold Co., London,1971.
 [10]
 M. M. VAINBERG, Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley, New York, 1973. MR 0467428 (57:7286b)
 [11]
 R. WINTHER, A Numerical Galerkin Method for a Parabolic Control Problem, Ph.D. Thesis, Cornell University, 1977.
 [12]
 R. WINTHER, "Error estimates for a Galerkin approximation of a parabolic control problem," Ann. Mat. Pura Appl. (4), v. 107, 1978, pp. 173206. MR 515960 (80a:49067)
 [13]
 R. WINTHER, "Some superlinear convergence results for the conjugate gradient method," SIAM J. Numer. Anal. (To appear.) MR 559456 (81k:65060)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198005512937
PII:
S 00255718(1980)05512937
Article copyright:
© Copyright 1980
American Mathematical Society
