# American Institute of Mathematical Sciences

September  2017, 6(3): 319-344. doi: 10.3934/eect.2017017

## On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems

 Florida Institute of Technology, 150 W. University Blvd, Melbourne, FL 32901, USA

Received  November 2016 Revised  April 2017 Published  June 2017

Fund Project: This work was supported by NSF Grant #1359074.

We consider the inverse Stefan type free boundary problem, where the coefficients, boundary heat flux, and density of the sources are missing and must be found along with the temperature and the free boundary. We pursue an optimal control framework where boundary heat flux, density of sources, and free boundary are components of the control vector. The optimality criteria consists of the minimization of the $L_2$-norm declinations of the temperature measurements at the final moment, phase transition temperature, and final position of the free boundary. We prove the Frechet differentiability in Besov-Hölder spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov-Hölder spaces for the numerical solution of the inverse Stefan problem.

Citation: Ugur G. Abdulla, Evan Cosgrove, Jonathan Goldfarb. On the Frechet differentiability in optimal control of coefficients in parabolic free boundary problems. Evolution Equations & Control Theory, 2017, 6 (3) : 319-344. doi: 10.3934/eect.2017017
##### 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.  Google Scholar [2] U. G. Abdulla, On the optimal control of the free boundary problems for the second order parabolic equations. Ⅱ. Convergence of the method of finite differences, Inverse Problems and Imaging, 10 (2016), 869-898.  doi: 10.3934/ipi.2016025.  Google Scholar [3] U. G. Abdulla and J. Goldfarb, Frechet differentability in Besov spaces in the optimal control of parabolic free boundary problems, arXiv: 1604.00057, to appear in Inverse and Ill-posed Problems, https://arxiv.org/abs/1604.00057. Google Scholar [4] J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben, ZAMM, 60 (1980), T333-T335.   Google Scholar [5] J. Bell, The non-characteristic Cauchy problem for a class of equations with time dependence. I. Problem in one space dimension, SIAM Journal on Mathematical Analysis, 12 (1981), 759-777.  doi: 10.1137/0512064.  Google Scholar [6] O. Besov, V. Ilin and S. Nikolskii, Integral Representations of Functions and Imbedding Theorems vol. 1, John Wiley & Sons, 1978.  Google Scholar [7] O. Besov, V. Ilin and S. Nikolskii, Integral Representations of Functions and Imbedding Theorems vol. 2, John Wiley & Sons, 1979. Google Scholar [8] B. M. Budak and V. N. Vasileva, On the solution of the inverse Stefan problem, Soviet Mathematics Doklady, 13 (1972), 811-815.   Google Scholar [9] B. M. Budak and V. N. Vasileva, On the solution of Stefan's converse problem Ⅱ, USSR Computational Mathematics and Mathematical Physics, 13 (1973), 97-110.   Google Scholar [10] B. M. Budak and V. N. Vasileva, The solution of the inverse Stefan problem, USSR Computational Mathematics and Mathematical Physics, 13 (1974), 130-151.   Google Scholar [11] J. R. Cannon, A Cauchy problem for the heat equation, Annali di Matematica Pura Ed Applicata, 66 (1964), 155-165.  doi: 10.1007/BF02412441.  Google Scholar [12] J. R. Cannon and J. Douglas, The Cauchy problem for the heat equation, SIAM Journal on Numerical Analysis, 4 (1967), 317-336.  doi: 10.1137/0704028.  Google Scholar [13] A. Carasso, Determining surface temperatures from interior observations, SIAM Journal on Applied Mathematics, 42 (1982), 558-574.  doi: 10.1137/0142040.  Google Scholar [14] R. Ewing, The Cauchy problem for a linear parabolic equation, Journal of Mathematical Analysis and Applications, 71 (1979), 167-186.  doi: 10.1016/0022-247X(79)90223-3.  Google Scholar [15] R. Ewing and R. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equations, Mathematics of Computation, 33 (1979), 1125-1144.  doi: 10.2307/2006451.  Google Scholar [16] A. Fasano and M. Primicerio, General free boundary problems for heat equations, Journal of Mathematical Analysis and Applications, 57 (1977), 694-723.  doi: 10.1016/0022-247X(77)90256-6.  Google Scholar [17] N. L. Gol'dman, Inverse Stefan Problems, Kluwer Academic Publishers Group, Dodrecht, 1997. doi: 10.1007/978-94-011-5488-8.  Google Scholar [18] K. H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundaries, in Proceedings of the International Conference on Free Boundary Problems, 1981. Google Scholar [19] K. H. Hoffman and J. Sprekels, Real time control of free boundary in a two-phase Stefan problem, Numerical Functional Analysis and Optimization, 5 (1982), 47-76.  doi: 10.1080/01630568208816131.  Google Scholar [20] K. H. Hoffman and J. Sprekels, On the identification of heat conductivity and latent heat conductivity as latent heat in a one-phase Stefan problem, Control and Cybernetics, 14 (1985), 37-51.   Google Scholar [21] P. Jochum, The inverse Stefan problem as a problem of nonlinear approximation theory, Journal of Approximation Theory, 30 (1980), 81-98.  doi: 10.1016/0021-9045(80)90011-8.  Google Scholar [22] P. Jochum, The numerical solution of the inverse Stefan problem, Numerical Mathematics, 34 (1980), 411-429.  doi: 10.1007/BF01403678.  Google Scholar [23] P. Knabner, Stability theorems for general free boundary problems of the Stefan type and applications, Applied Nonlinear Functional Analysis, Methoden und Verfahren der Mathematischen Physik, 25 (1983), 95-116.   Google Scholar [24] A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1977.  Google Scholar [25] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Translations of Mathematical Monographs, American Mathematical Society, Providence, R. I., 1968.  Google Scholar [26] K. A. Lurye, Optimal Control in Problems of Mathematical Physics, Moscow, Nauka, 1975.  Google Scholar [27] M. Niezgodka, Control of parabolic systems with free boundaries -application of inverse formulation, Control and Cybernetics, 8 (1979), 213-225.   Google Scholar [28] S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar [29] R. Nochetto and C. Verdi, The combined use of nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems, Numerical Functional Analysis and Optimization, 9 (1987/88), 1177-1192.  doi: 10.1080/01630568808816279.  Google Scholar [30] M. Primicerio, The occurence of pathologies in some Stefan-like problems, in Numerical Treatment of Free Boundary-Value problems (eds. J. Albrecht, L. Collatz and K. H. Hoffman), vol. 58, ISNM, Birkhauser Verlag, Basel, (1982), 233-244.   Google Scholar [31] C. Sagues, Simulation and optimal control of free boundary, in Numerical Treatment of Free Boundary-Value problems (eds. J. Albrecht, L. Collatz and K. H. Hoffman), ISNM, Birkhauser Verlag, Basel, 58 (1982), 270-287.   Google Scholar [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), 557-570.  doi: 10.1137/0120058.  Google Scholar [33] V. A. Solonnikov, A-priori estimates for solutions of second-order equations of parabolic type, Trudy Mat. Inst. Steklov., 70 (1964), 133-212.   Google Scholar [34] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations in general form, Proceedings of the Steklov Institute of Mathematics, 83 (1965), 3-163.   Google Scholar [35] G. Talenti and S. Vessella, A note on an ill-posed problem for the heat equation, Journal of the Austrailian Mathematical Society, 32 (1982), 358-368.   Google Scholar [36] F. P. Vasil'ev, The existence of a solution to a certain optimal Stefan problem, Computational Methods and Programming, (1969), 110-114.   Google Scholar [37] K. Yosida, Functional Analysis, Classics in Mathematics, Springer, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar [38] A. D. Yurii, On an optimal Stefan problem, Doklady Akademii nauk SSSR, 251 (1980), 1317-1321.   Google Scholar

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##### 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.  Google Scholar [2] U. G. Abdulla, On the optimal control of the free boundary problems for the second order parabolic equations. Ⅱ. Convergence of the method of finite differences, Inverse Problems and Imaging, 10 (2016), 869-898.  doi: 10.3934/ipi.2016025.  Google Scholar [3] U. G. Abdulla and J. Goldfarb, Frechet differentability in Besov spaces in the optimal control of parabolic free boundary problems, arXiv: 1604.00057, to appear in Inverse and Ill-posed Problems, https://arxiv.org/abs/1604.00057. Google Scholar [4] J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben, ZAMM, 60 (1980), T333-T335.   Google Scholar [5] J. Bell, The non-characteristic Cauchy problem for a class of equations with time dependence. I. Problem in one space dimension, SIAM Journal on Mathematical Analysis, 12 (1981), 759-777.  doi: 10.1137/0512064.  Google Scholar [6] O. Besov, V. Ilin and S. Nikolskii, Integral Representations of Functions and Imbedding Theorems vol. 1, John Wiley & Sons, 1978.  Google Scholar [7] O. Besov, V. Ilin and S. Nikolskii, Integral Representations of Functions and Imbedding Theorems vol. 2, John Wiley & Sons, 1979. Google Scholar [8] B. M. Budak and V. N. Vasileva, On the solution of the inverse Stefan problem, Soviet Mathematics Doklady, 13 (1972), 811-815.   Google Scholar [9] B. M. Budak and V. N. Vasileva, On the solution of Stefan's converse problem Ⅱ, USSR Computational Mathematics and Mathematical Physics, 13 (1973), 97-110.   Google Scholar [10] B. M. Budak and V. N. Vasileva, The solution of the inverse Stefan problem, USSR Computational Mathematics and Mathematical Physics, 13 (1974), 130-151.   Google Scholar [11] J. R. Cannon, A Cauchy problem for the heat equation, Annali di Matematica Pura Ed Applicata, 66 (1964), 155-165.  doi: 10.1007/BF02412441.  Google Scholar [12] J. R. Cannon and J. Douglas, The Cauchy problem for the heat equation, SIAM Journal on Numerical Analysis, 4 (1967), 317-336.  doi: 10.1137/0704028.  Google Scholar [13] A. Carasso, Determining surface temperatures from interior observations, SIAM Journal on Applied Mathematics, 42 (1982), 558-574.  doi: 10.1137/0142040.  Google Scholar [14] R. Ewing, The Cauchy problem for a linear parabolic equation, Journal of Mathematical Analysis and Applications, 71 (1979), 167-186.  doi: 10.1016/0022-247X(79)90223-3.  Google Scholar [15] R. Ewing and R. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equations, Mathematics of Computation, 33 (1979), 1125-1144.  doi: 10.2307/2006451.  Google Scholar [16] A. Fasano and M. Primicerio, General free boundary problems for heat equations, Journal of Mathematical Analysis and Applications, 57 (1977), 694-723.  doi: 10.1016/0022-247X(77)90256-6.  Google Scholar [17] N. L. Gol'dman, Inverse Stefan Problems, Kluwer Academic Publishers Group, Dodrecht, 1997. doi: 10.1007/978-94-011-5488-8.  Google Scholar [18] K. H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundaries, in Proceedings of the International Conference on Free Boundary Problems, 1981. Google Scholar [19] K. H. Hoffman and J. Sprekels, Real time control of free boundary in a two-phase Stefan problem, Numerical Functional Analysis and Optimization, 5 (1982), 47-76.  doi: 10.1080/01630568208816131.  Google Scholar [20] K. H. Hoffman and J. Sprekels, On the identification of heat conductivity and latent heat conductivity as latent heat in a one-phase Stefan problem, Control and Cybernetics, 14 (1985), 37-51.   Google Scholar [21] P. Jochum, The inverse Stefan problem as a problem of nonlinear approximation theory, Journal of Approximation Theory, 30 (1980), 81-98.  doi: 10.1016/0021-9045(80)90011-8.  Google Scholar [22] P. Jochum, The numerical solution of the inverse Stefan problem, Numerical Mathematics, 34 (1980), 411-429.  doi: 10.1007/BF01403678.  Google Scholar [23] P. Knabner, Stability theorems for general free boundary problems of the Stefan type and applications, Applied Nonlinear Functional Analysis, Methoden und Verfahren der Mathematischen Physik, 25 (1983), 95-116.   Google Scholar [24] A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, The Netherlands, 1977.  Google Scholar [25] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, vol. 23 of Translations of Mathematical Monographs, American Mathematical Society, Providence, R. I., 1968.  Google Scholar [26] K. A. Lurye, Optimal Control in Problems of Mathematical Physics, Moscow, Nauka, 1975.  Google Scholar [27] M. Niezgodka, Control of parabolic systems with free boundaries -application of inverse formulation, Control and Cybernetics, 8 (1979), 213-225.   Google Scholar [28] S. M. Nikol'skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer-Verlag, New York-Heidelberg, 1975.  Google Scholar [29] R. Nochetto and C. Verdi, The combined use of nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems, Numerical Functional Analysis and Optimization, 9 (1987/88), 1177-1192.  doi: 10.1080/01630568808816279.  Google Scholar [30] M. Primicerio, The occurence of pathologies in some Stefan-like problems, in Numerical Treatment of Free Boundary-Value problems (eds. J. Albrecht, L. Collatz and K. H. Hoffman), vol. 58, ISNM, Birkhauser Verlag, Basel, (1982), 233-244.   Google Scholar [31] C. Sagues, Simulation and optimal control of free boundary, in Numerical Treatment of Free Boundary-Value problems (eds. J. Albrecht, L. Collatz and K. H. Hoffman), ISNM, Birkhauser Verlag, Basel, 58 (1982), 270-287.   Google Scholar [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), 557-570.  doi: 10.1137/0120058.  Google Scholar [33] V. A. Solonnikov, A-priori estimates for solutions of second-order equations of parabolic type, Trudy Mat. Inst. Steklov., 70 (1964), 133-212.   Google Scholar [34] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations in general form, Proceedings of the Steklov Institute of Mathematics, 83 (1965), 3-163.   Google Scholar [35] G. Talenti and S. Vessella, A note on an ill-posed problem for the heat equation, Journal of the Austrailian Mathematical Society, 32 (1982), 358-368.   Google Scholar [36] F. P. Vasil'ev, The existence of a solution to a certain optimal Stefan problem, Computational Methods and Programming, (1969), 110-114.   Google Scholar [37] K. Yosida, Functional Analysis, Classics in Mathematics, Springer, 1995. doi: 10.1007/978-3-642-61859-8.  Google Scholar [38] A. D. Yurii, On an optimal Stefan problem, Doklady Akademii nauk SSSR, 251 (1980), 1317-1321.   Google Scholar
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