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On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences
1. | Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901 |
References:
[1] |
U. G. Abdulla, On the optimal control of the free boundary problems for the second order parabolic equations. i.well-posedness and convergence of the method of lines, Inverse Problems and Imaging, 7 (2013), 307-340.
doi: 10.3934/ipi.2013.7.307. |
[2] |
O. M. Alifanov, Inverse Heat Transfer Problems, Springer-Verlag Telos, 1995. |
[3] |
J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben, ZAMM, 60 (1980), T333-T335. |
[4] |
J. B. Bell, The non-characteristic Cauchy problem for a class of equations with time dependence.I.Problem in one space dimension, SIAM J. Math. Anal., 12 (1981), 759-777.
doi: 10.1137/0512064. |
[5] |
O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral Representations of Functions and Embedding Theorems, Winston & Sons, Washington, D.C.; John Wiley & Sons, 1978. |
[6] |
B. M. Budak and V. N. Vasil'eva, On the solution of the inverse Stefan problem, Soviet Math. Dokl., 13 (1972), 811-815. |
[7] |
B. M. Budak and V. N. Vasil'eva, The solution of the inverse Stefan problem, USSR Comput. Maths. Math. Phys., 13 (1974), 130-151.
doi: 10.1016/0041-5553(74)90010-X. |
[8] |
B. M. Budak and V. N. Vasil'eva, On the solution of Stefan's converse problem. II, USSR Comput. Maths. Math. Phys., 13 (1973), 97-110.
doi: 10.1016/0041-5553(73)90069-4. |
[9] |
J. R. Cannon, A Cauchy problem for the heat equation, Ann. Math., 66 (1964), 155-165.
doi: 10.1007/BF02412441. |
[10] |
J. R. Cannon and J. Douglas, The Cauchy problem for the heat equation, SIAM. J. Numer. Anal., 4 (1967), 317-336.
doi: 10.1137/0704028. |
[11] |
A. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math., 42 (1982), 558-574.
doi: 10.1137/0142040. |
[12] |
R. E. Ewing, The Cauchy problem for a linear parabolic equation, J. Math. Anal. Appl., 71 (1979), 167-186.
doi: 10.1016/0022-247X(79)90223-3. |
[13] |
R. E. Ewing and R. S. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equation, Math. Comput., 33 (1979), 1125-1144.
doi: 10.1090/S0025-5718-1979-0537961-3. |
[14] |
A. Fasano and M. Primicerio, General free boundary problems for heat equations, J. Math. Anal. Appl., 57 (1977), 694-723.
doi: 10.1016/0022-247X(77)90256-6. |
[15] |
A. Friedman, Variational Principles and Free Boundary Problems, John Wiley, 1982. |
[16] |
N. L. Gol'dman, Inverse Stefan Problems, Mathematics and its Applications, 412, Kluwer Academic Publishers Group, Dordrecht, 1997.
doi: 10.1007/978-94-011-5488-8. |
[17] |
N. L. Gol'dman, Properties of solutions of the inverse Stefan problem, Differential Equations, 39 (2003), 66-72.
doi: 10.1023/A:1025120024905. |
[18] |
K. H. Hoffman and M. Niezgodka, Control of Parabolic Systems Involving Free Boundarie, Proc. of Int. Conf. on Free Boundary Problems, 1981. |
[19] |
K. H. Hoffman and J. Sprekels, Real time control of free boundary in a two-phase Stefan problem, Numer. Funct. Anal. Optimiz., 5 (1982), 47-76.
doi: 10.1080/01630568208816131. |
[20] |
K. H. Hoffman and J. Sprekels, On the identification of heat conductivity and latent heat conductivity ans latent heat in a one-phase Stefan problem, Control and Cybernetics, 14 (1985), 37-51. |
[21] |
P. Jochum, The Inverse Stefan problem as a problem of nonlinear approximation theory, Journal of Approximate Theorey, 30 (1980), 81-98.
doi: 10.1016/0021-9045(80)90011-8. |
[22] |
P. Jochum, The numerical solution of the inverse Stefan problem, Numerical Mathematics, 34 (1980), 411-429.
doi: 10.1007/BF01403678. |
[23] |
P. Knabner, Stability theorems for general free boundary problem of the Stefan type and applications, Meth. Ser. Numer. Meth. Verf. Math. Phys., 25 (1983), 95-116. |
[24] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of the Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American mathematical Society, Providence, R.I.,1967. |
[25] |
K. A. Lurye, Optimal Control in Problems of Mathematical Physics, Nauka, Moscow, 1975. |
[26] |
A. M. Meyrmanov, The Stefan Problem, Walter de Gruyter, 1992.
doi: 10.1515/9783110846720.245. |
[27] |
M. Niezgodka, Control of parabolic systems with free boundaries-application of inverse formulation, Control and Cybernetics, 8 (1979), 213-225. |
[28] |
S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, New York-Heidelberg, 1975. |
[29] |
R. H. Nochetto and C. Verdi, The combined use of nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems, Numer. Funct. Anal. Optimiz., 9 (1987/1988), 1177-1192.
doi: 10.1080/01630568808816279. |
[30] |
M. Primicero, The occurence of pathologies in some Stefan-like problems, Numerical Treatment of Free Boundary-Value Problems, ISNM, (1982), 233-244. |
[31] |
C. Sagues, Simulation and optimal control of free boundary, Numerical Treatment of Free Boundary-Value Problems, ISNM, 58, 270-287. |
[32] |
B. Sherman, General one-phase stefan problems and free boundary problems for the heat equation with cauchy data prescribed on the free boundary, SIAM J. Appl. Math., 20 (1971), 555-570.
doi: 10.1137/0120058. |
[33] |
V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type, Trudy Mat. Inst. Steklov., 70 (1964), 133-212. |
[34] |
V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov., 83 (1965), 3-163. |
[35] |
G. Talenti and S. Vessella, A note on an Ill-posed problem for the heat equation, J. Austral. Math. Soc., Ser. A, 32 (1982), 358-368.
doi: 10.1017/S1446788700024915. |
[36] |
F. P. Vasil'ev, The existence of a solution of a certain optimal Stefan problem, In Comput. Methods and Programming, XII(Russian), (1969), 110-114. |
[37] |
F. P. Vasil'ev, Methods for Solving Extremal Problems. Minimization Problems in Function Spaces, Regularization, Approximation, (in Russian), Moscow, Nauka, 1981. |
[38] |
A. D. Yurii, On an optimal Stefan problem, Dokl. Akad. Nauk SSSR, 251 (1980), 1317-1321. |
show all references
References:
[1] |
U. G. Abdulla, On the optimal control of the free boundary problems for the second order parabolic equations. i.well-posedness and convergence of the method of lines, Inverse Problems and Imaging, 7 (2013), 307-340.
doi: 10.3934/ipi.2013.7.307. |
[2] |
O. M. Alifanov, Inverse Heat Transfer Problems, Springer-Verlag Telos, 1995. |
[3] |
J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben, ZAMM, 60 (1980), T333-T335. |
[4] |
J. B. Bell, The non-characteristic Cauchy problem for a class of equations with time dependence.I.Problem in one space dimension, SIAM J. Math. Anal., 12 (1981), 759-777.
doi: 10.1137/0512064. |
[5] |
O. V. Besov, V. P. Il'in and S. M. Nikol'skii, Integral Representations of Functions and Embedding Theorems, Winston & Sons, Washington, D.C.; John Wiley & Sons, 1978. |
[6] |
B. M. Budak and V. N. Vasil'eva, On the solution of the inverse Stefan problem, Soviet Math. Dokl., 13 (1972), 811-815. |
[7] |
B. M. Budak and V. N. Vasil'eva, The solution of the inverse Stefan problem, USSR Comput. Maths. Math. Phys., 13 (1974), 130-151.
doi: 10.1016/0041-5553(74)90010-X. |
[8] |
B. M. Budak and V. N. Vasil'eva, On the solution of Stefan's converse problem. II, USSR Comput. Maths. Math. Phys., 13 (1973), 97-110.
doi: 10.1016/0041-5553(73)90069-4. |
[9] |
J. R. Cannon, A Cauchy problem for the heat equation, Ann. Math., 66 (1964), 155-165.
doi: 10.1007/BF02412441. |
[10] |
J. R. Cannon and J. Douglas, The Cauchy problem for the heat equation, SIAM. J. Numer. Anal., 4 (1967), 317-336.
doi: 10.1137/0704028. |
[11] |
A. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math., 42 (1982), 558-574.
doi: 10.1137/0142040. |
[12] |
R. E. Ewing, The Cauchy problem for a linear parabolic equation, J. Math. Anal. Appl., 71 (1979), 167-186.
doi: 10.1016/0022-247X(79)90223-3. |
[13] |
R. E. Ewing and R. S. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equation, Math. Comput., 33 (1979), 1125-1144.
doi: 10.1090/S0025-5718-1979-0537961-3. |
[14] |
A. Fasano and M. Primicerio, General free boundary problems for heat equations, J. Math. Anal. Appl., 57 (1977), 694-723.
doi: 10.1016/0022-247X(77)90256-6. |
[15] |
A. Friedman, Variational Principles and Free Boundary Problems, John Wiley, 1982. |
[16] |
N. L. Gol'dman, Inverse Stefan Problems, Mathematics and its Applications, 412, Kluwer Academic Publishers Group, Dordrecht, 1997.
doi: 10.1007/978-94-011-5488-8. |
[17] |
N. L. Gol'dman, Properties of solutions of the inverse Stefan problem, Differential Equations, 39 (2003), 66-72.
doi: 10.1023/A:1025120024905. |
[18] |
K. H. Hoffman and M. Niezgodka, Control of Parabolic Systems Involving Free Boundarie, Proc. of Int. Conf. on Free Boundary Problems, 1981. |
[19] |
K. H. Hoffman and J. Sprekels, Real time control of free boundary in a two-phase Stefan problem, Numer. Funct. Anal. Optimiz., 5 (1982), 47-76.
doi: 10.1080/01630568208816131. |
[20] |
K. H. Hoffman and J. Sprekels, On the identification of heat conductivity and latent heat conductivity ans latent heat in a one-phase Stefan problem, Control and Cybernetics, 14 (1985), 37-51. |
[21] |
P. Jochum, The Inverse Stefan problem as a problem of nonlinear approximation theory, Journal of Approximate Theorey, 30 (1980), 81-98.
doi: 10.1016/0021-9045(80)90011-8. |
[22] |
P. Jochum, The numerical solution of the inverse Stefan problem, Numerical Mathematics, 34 (1980), 411-429.
doi: 10.1007/BF01403678. |
[23] |
P. Knabner, Stability theorems for general free boundary problem of the Stefan type and applications, Meth. Ser. Numer. Meth. Verf. Math. Phys., 25 (1983), 95-116. |
[24] |
O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of the Parabolic Type, Translations of Mathematical Monographs, Vol. 23, American mathematical Society, Providence, R.I.,1967. |
[25] |
K. A. Lurye, Optimal Control in Problems of Mathematical Physics, Nauka, Moscow, 1975. |
[26] |
A. M. Meyrmanov, The Stefan Problem, Walter de Gruyter, 1992.
doi: 10.1515/9783110846720.245. |
[27] |
M. Niezgodka, Control of parabolic systems with free boundaries-application of inverse formulation, Control and Cybernetics, 8 (1979), 213-225. |
[28] |
S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, New York-Heidelberg, 1975. |
[29] |
R. H. Nochetto and C. Verdi, The combined use of nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems, Numer. Funct. Anal. Optimiz., 9 (1987/1988), 1177-1192.
doi: 10.1080/01630568808816279. |
[30] |
M. Primicero, The occurence of pathologies in some Stefan-like problems, Numerical Treatment of Free Boundary-Value Problems, ISNM, (1982), 233-244. |
[31] |
C. Sagues, Simulation and optimal control of free boundary, Numerical Treatment of Free Boundary-Value Problems, ISNM, 58, 270-287. |
[32] |
B. Sherman, General one-phase stefan problems and free boundary problems for the heat equation with cauchy data prescribed on the free boundary, SIAM J. Appl. Math., 20 (1971), 555-570.
doi: 10.1137/0120058. |
[33] |
V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type, Trudy Mat. Inst. Steklov., 70 (1964), 133-212. |
[34] |
V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Trudy Mat. Inst. Steklov., 83 (1965), 3-163. |
[35] |
G. Talenti and S. Vessella, A note on an Ill-posed problem for the heat equation, J. Austral. Math. Soc., Ser. A, 32 (1982), 358-368.
doi: 10.1017/S1446788700024915. |
[36] |
F. P. Vasil'ev, The existence of a solution of a certain optimal Stefan problem, In Comput. Methods and Programming, XII(Russian), (1969), 110-114. |
[37] |
F. P. Vasil'ev, Methods for Solving Extremal Problems. Minimization Problems in Function Spaces, Regularization, Approximation, (in Russian), Moscow, Nauka, 1981. |
[38] |
A. D. Yurii, On an optimal Stefan problem, Dokl. Akad. Nauk SSSR, 251 (1980), 1317-1321. |
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