# American Institute of Mathematical Sciences

2013, 7(2): 307-340. doi: 10.3934/ipi.2013.7.307

## 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

 1 Department of Mathematics, Florida Institute of Technology, Melbourne, Florida 32901

Received  March 2012 Revised  October 2012 Published  May 2013

We develop a new variational formulation of the inverse Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundary. We employ optimal control framework, where boundary heat flux and free boundary are components of the control vector, and optimality criteria consists of the minimization of the sum of $L_2$-norm declinations from the available measurement of the temperature flux on the fixed boundary and available information on the phase transition temperature on the free boundary. This approach allows one to tackle situations when the phase transition temperature is not known explicitly, and is available through measurement with possible error. It also allows for the development of iterative numerical methods of least computational cost due to the fact that for every given control vector, the parabolic PDE is solved in a fixed region instead of full free boundary problem. We prove well-posedness in Sobolev spaces framework and convergence of discrete optimal control problems to the original problem both with respect to cost functional and control.
Citation: Ugur 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 & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
##### References:
 [1] O. M. Alifanov, "Inverse Heat Transfer Problems,", Springer-Verlag Telos, (1995). [2] J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben,, ZAMM, 60 (1980), 335. [3] 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. doi: 10.1137/0512064. [4] O. V. Besov, V. P. Il'in and S. M. Nikol'skii, "Integral Representations of Functions and Embedding Theorems,", Winston & Sons, (1978). [5] B. M. Budak and V. N. Vasil'eva, On the solution of the inverse Stefan problem,, Soviet Math. Dokl, 13 (1972), 811. [6] B. M. Budak and V. N. Vasil'eva, The solution of the inverse Stefan problem,, USSR Comput. Maths. Math. Phys, 13 (1973), 130. [7] 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. [8] J. R. Cannon, A Cauchy problem for the heat equation,, Ann. Math, 66 (1964), 155. doi: 10.1007/BF02412441. [9] J. R. Cannon and J. Douglas, The Cauchy problem for the heat equation,, SIAM. J. Numer. Anal., 4 (1967), 317. doi: 10.1137/0704028. [10] A. Carasso, Determining surface temperatures from interior observations,, SIAM J. Appl. Math., 42 (1982), 558. doi: 10.1137/0142040. [11] R. E. Ewing, The Cauchy problem for a linear parabolic equation,, J. Math. Anal. Appl., 71 (1970), 167. doi: 10.1016/0022-247X(79)90223-3. [12] R. E. Ewing and R. S. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equation,, Math. Comput., 33 (1979), 1125. doi: 10.1090/S0025-5718-1979-0537961-3. [13] A. Fasano and M. Primicerio, General free boundary problems for heat equations,, J. Math. Anal. Appl., 57 (1977), 694. doi: 10.1016/0022-247X(77)90256-6. [14] A. Friedman, "Variational Principles and Free Boundary Problems,", John Wiley, (1982). [15] N. L. Gol'dman, "Inverse Stefan Problems,", Mathematics and its Applications, 412 (1997). doi: 10.1007/978-94-011-5488-8. [16] N. L. Gol'dman, Properties of solutions of the inverse Stefan problem,, Differential Equations, 39 (2003), 66. doi: 10.1023/A:1025120024905. [17] K. H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundarie,, Proc. of Int. Conf. on Free Boundary Problems, (1981). [18] K. H. Hoffman and J. Sprekels, Real time control of free boundary in a two-phase Stefan problem,, Numer. Funct. Anal. and Optimiz, 5 (1982), 47. doi: 10.1080/01630568208816131. [19] 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, 15 (1986), 37. [20] P. Jochum, The inverse Stefan problem as a problem of nonlinear approximation theory,, Journal of Approximate Theorey, 30 (1980), 81. doi: 10.1016/0021-9045(80)90011-8. [21] P. Jochum, The numerical solution of the inverse Stefan problem,, Numerical Mathematics, 34 (1980), 411. doi: 10.1007/BF01403678. [22] 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. [23] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of the Parabolic Type,", Translations of Mathematical Monographs, 23 (1967). [24] K. A. Lurye, "Optimal Control in Problems of Mathematical Physics,", Nauka, (1975). [25] A. M. Meyrmanov, "The Stefan Problem,", Walter de Gruyter, (1992). doi: 10.1515/9783110846720.245. [26] M. Niezgodka, Control of parabolic systems with free boundaries-application of inverse formulation,, Control and Cybernetics, 8 (1979), 213. [27] S. M. Nikol'skii, "Approximation of Functions of Several Variables and Imbedding Theorems,", Springer-Verlag, (1975). [28] 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. and Optimiz., 9 (): 1987. doi: 10.1080/01630568808816279. [29] M. Primicero, The occurence of pathologies in some Stefan-like problems,, Numerical Treatment of Free Boundary-Value Problems, (1982), 233. [30] C. Sagues, Simulation and optimal control of free boundary,, Numerical Treatment of Free Boundary-Value Problems, 58 (): 270. [31] 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), 557. doi: 10.1137/0120058. [32] V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type,, Trudy Mat. Inst. Steklov., 70 (1964), 133. [33] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form,, Trudy Mat. Inst. Steklov., 83 (1965), 3. [34] G. Talenti and S. Vessella, A note on an ill-posed problem for the heat equation,, J. Austral. Math. Soc., 32 (1982), 358. doi: 10.1017/S1446788700024915. [35] F. P. Vasil'ev, The existence of a solution of a certain optimal Stefan problem,, in, (1969), 110. [36] F. P. Vasil'ev, "Methods for Solving Extremal Problems. Minimization Problems in Function Spaces, Regularization, Approximation,", (in Russian), (1981). [37] A. D. Yurii, On an optimal Stefan problem,, Dokl. Akad. Nauk SSSR, 251 (1980), 1317.

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##### References:
 [1] O. M. Alifanov, "Inverse Heat Transfer Problems,", Springer-Verlag Telos, (1995). [2] J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben,, ZAMM, 60 (1980), 335. [3] 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. doi: 10.1137/0512064. [4] O. V. Besov, V. P. Il'in and S. M. Nikol'skii, "Integral Representations of Functions and Embedding Theorems,", Winston & Sons, (1978). [5] B. M. Budak and V. N. Vasil'eva, On the solution of the inverse Stefan problem,, Soviet Math. Dokl, 13 (1972), 811. [6] B. M. Budak and V. N. Vasil'eva, The solution of the inverse Stefan problem,, USSR Comput. Maths. Math. Phys, 13 (1973), 130. [7] 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. [8] J. R. Cannon, A Cauchy problem for the heat equation,, Ann. Math, 66 (1964), 155. doi: 10.1007/BF02412441. [9] J. R. Cannon and J. Douglas, The Cauchy problem for the heat equation,, SIAM. J. Numer. Anal., 4 (1967), 317. doi: 10.1137/0704028. [10] A. Carasso, Determining surface temperatures from interior observations,, SIAM J. Appl. Math., 42 (1982), 558. doi: 10.1137/0142040. [11] R. E. Ewing, The Cauchy problem for a linear parabolic equation,, J. Math. Anal. Appl., 71 (1970), 167. doi: 10.1016/0022-247X(79)90223-3. [12] R. E. Ewing and R. S. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equation,, Math. Comput., 33 (1979), 1125. doi: 10.1090/S0025-5718-1979-0537961-3. [13] A. Fasano and M. Primicerio, General free boundary problems for heat equations,, J. Math. Anal. Appl., 57 (1977), 694. doi: 10.1016/0022-247X(77)90256-6. [14] A. Friedman, "Variational Principles and Free Boundary Problems,", John Wiley, (1982). [15] N. L. Gol'dman, "Inverse Stefan Problems,", Mathematics and its Applications, 412 (1997). doi: 10.1007/978-94-011-5488-8. [16] N. L. Gol'dman, Properties of solutions of the inverse Stefan problem,, Differential Equations, 39 (2003), 66. doi: 10.1023/A:1025120024905. [17] K. H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundarie,, Proc. of Int. Conf. on Free Boundary Problems, (1981). [18] K. H. Hoffman and J. Sprekels, Real time control of free boundary in a two-phase Stefan problem,, Numer. Funct. Anal. and Optimiz, 5 (1982), 47. doi: 10.1080/01630568208816131. [19] 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, 15 (1986), 37. [20] P. Jochum, The inverse Stefan problem as a problem of nonlinear approximation theory,, Journal of Approximate Theorey, 30 (1980), 81. doi: 10.1016/0021-9045(80)90011-8. [21] P. Jochum, The numerical solution of the inverse Stefan problem,, Numerical Mathematics, 34 (1980), 411. doi: 10.1007/BF01403678. [22] 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. [23] O. A. Ladyzenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of the Parabolic Type,", Translations of Mathematical Monographs, 23 (1967). [24] K. A. Lurye, "Optimal Control in Problems of Mathematical Physics,", Nauka, (1975). [25] A. M. Meyrmanov, "The Stefan Problem,", Walter de Gruyter, (1992). doi: 10.1515/9783110846720.245. [26] M. Niezgodka, Control of parabolic systems with free boundaries-application of inverse formulation,, Control and Cybernetics, 8 (1979), 213. [27] S. M. Nikol'skii, "Approximation of Functions of Several Variables and Imbedding Theorems,", Springer-Verlag, (1975). [28] 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. and Optimiz., 9 (): 1987. doi: 10.1080/01630568808816279. [29] M. Primicero, The occurence of pathologies in some Stefan-like problems,, Numerical Treatment of Free Boundary-Value Problems, (1982), 233. [30] C. Sagues, Simulation and optimal control of free boundary,, Numerical Treatment of Free Boundary-Value Problems, 58 (): 270. [31] 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), 557. doi: 10.1137/0120058. [32] V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type,, Trudy Mat. Inst. Steklov., 70 (1964), 133. [33] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form,, Trudy Mat. Inst. Steklov., 83 (1965), 3. [34] G. Talenti and S. Vessella, A note on an ill-posed problem for the heat equation,, J. Austral. Math. Soc., 32 (1982), 358. doi: 10.1017/S1446788700024915. [35] F. P. Vasil'ev, The existence of a solution of a certain optimal Stefan problem,, in, (1969), 110. [36] F. P. Vasil'ev, "Methods for Solving Extremal Problems. Minimization Problems in Function Spaces, Regularization, Approximation,", (in Russian), (1981). [37] A. D. Yurii, On an optimal Stefan problem,, Dokl. Akad. Nauk SSSR, 251 (1980), 1317.
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