Abstract
We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov 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 spaces for the numerical solution of the inverse Stefan problem.
Funding source: National Science Foundation
Award Identifier / Grant number: 1359074
Funding statement: This research was funded by National Science Foundation (grant no. 1359074), REU Site: Partial Differential Equations and Dynamical Systems at Florida Institute of Technology.
Acknowledgements
Two REU students, Jessica Pillow and Dylanger Pittman, worked on part of the project restricted to heuristic derivation of the Frechet gradient with the intention to implement the gradient type method for numerical analysis.
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 Probl. Imaging 7 (2013), no. 2, 307–340. 10.3934/ipi.2013.7.307Search in 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 Probl. Imaging 10 (2016), no. 4, 869–898. 10.3934/ipi.2016025Search in Google Scholar
[3] J. Baumeister, Zur optimalen Steuerung von freien Randwertaufgaben, Z. Angew. Math. Mech. 60 (1980), no. 7, T333–T335. Search in Google Scholar
[4] J. B. Bell, The noncharacteristic Cauchy problem for a class of equations with time dependence. I. Problems in one space dimension, SIAM J. Math. Anal. 12 (1981), no. 5, 759–777. 10.1137/0512064Search in Google Scholar
[5] O. V. Besov, V. P. Il’in and S. M. Nikol’skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. 1, John Wiley & Sons, Hoboken, 1979. Search in Google Scholar
[6] O. V. Besov, V. P. Il’in and S. M. Nikol’skiĭ, Integral Representations of Functions and Imbedding Theorems. Vol. 2, John Wiley & Sons, Hoboken, 1979. Search in Google Scholar
[7] B. M. Budak and V. N. Vasil’eva, On the solution of the inverse Stefan problem, Sov. Math. Dokl. 13 (1972), 811–815. 10.1016/0041-5553(74)90010-XSearch in Google Scholar
[8] B. M. Budak and V. N. Vasil’eva, Solution of the inverse Stefan problem (in Russian), Ž. Vyčisl. Mat. i Mat. Fiz. 13 (1973), 103–118, 268. 10.1016/0041-5553(74)90010-XSearch in Google Scholar
[9] B. M. Budak and V. N. Vasil’eva, Solution of the inverse Stefan problem. II (in Russian), Ž. Vyčisl. Mat. i Mat. Fiz. 13 (1973), 897–906, 1090. Search in Google Scholar
[10] J. R. Cannon, A Cauchy problem for the heat equation, Ann. Mat. Pura Appl. (4) 66 (1964), 155–165. 10.1007/BF02412441Search in Google Scholar
[11] J. R. Cannon and J. Douglas, Jr., The Cauchy problem for the heat equation, SIAM J. Numer. Anal. 4 (1967), 317–336. 10.1137/0704028Search in Google Scholar
[12] A. Carasso, Determining surface temperatures from interior observations, SIAM J. Appl. Math. 42 (1982), no. 3, 558–574. 10.1137/0142040Search in Google Scholar
[13] R. E. Ewing, The Cauchy problem for a linear parabolic partial differential equation, J. Math. Anal. Appl. 71 (1979), no. 1, 167–186. 10.1016/0022-247X(79)90223-3Search in Google Scholar
[14] R. E. Ewing and R. S. Falk, Numerical approximation of a Cauchy problem for a parabolic partial differential equation, Math. Comp. 33 (1979), no. 148, 1125–1144. 10.1090/S0025-5718-1979-0537961-3Search in Google Scholar
[15] A. Fasano and M. Primicerio, General free-boundary problems for the heat equation. I, J. Math. Anal. Appl. 57 (1977), no. 3, 694–723. 10.1016/0022-247X(77)90256-6Search in Google Scholar
[16] N. L. Gol’dman, Inverse Stefan Problems, Kluwer Academic, Dodrecht, 1997. 10.1007/978-94-011-5488-8Search in Google Scholar
[17] K.-H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundaries, Free Boundary Problems. Vol. 2, Pitman Research Notes in Math. 79, Pitman, Boston (1983), 431–462. Search in Google Scholar
[18] K.-H. Hoffman and J. Sprekels, Real-time control of the free boundary in a two-phase Stefan problem, Numer. Funct. Anal. Optim. 5 (1982), no. 1, 47–76. 10.1080/01630568208816131Search in Google Scholar
[19] K.-H. Hoffmann and J. Sprekels, On the identification of heat conductivity and latent heat in a one-phase Stefan problem, Control Cybernet. 14 (1985), no. 1–3, 37–51. Search in Google Scholar
[20] P. Jochum, The inverse Stefan problem as a problem of nonlinear approximation theory, J. Approx. Theory 30 (1980), no. 2, 81–98. 10.1016/0021-9045(80)90011-8Search in Google Scholar
[21] P. Jochum, The numerical solution of the inverse Stefan problem, Numer. Math. 34 (1980), no. 4, 411–429. 10.1007/BF01403678Search in Google Scholar
[22] A. D. Juriĭ, An optimal problem of Stefan type, Dokl. Akad. Nauk SSSR 251 (1980), no. 6, 1317–1321. Search in Google Scholar
[23] P. Knabner, Stability theorems for general free boundary problems of the Stefan type and applications, Applied Nonlinear Functional Analysis (Berlin 1981), Methoden Verfahren Math. Phys. 25, Lang, Frankfurt (1983), 95–116. Search in Google Scholar
[24] A. Kufner, O. John and S. Fučík, Function Spaces, Noordhoff International, Leyden, 1977. Search in Google Scholar
[25] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968. 10.1090/mmono/023Search in Google Scholar
[26] K. A. Lurye, Optimal Control in Problems of Mathematical Physics, “Nauka”, Moscow, 1975. Search in Google Scholar
[27] M. Niezgódka, Control of parabolic systems with free boundaries—Application of inverse formulations, Control Cybernet. 8 (1979), no. 3, 213–225. Search in Google Scholar
[28] S. M. Nikol’skii, Approximation of Functions of Several Variables and Imbedding Theorems, Springer, Heidelberg, 1975. 10.1007/978-3-642-65711-5Search in Google Scholar
[29] R. H. Nochetto and C. Verdi, The combined use of a nonlinear Chernoff formula with a regularization procedure for two-phase Stefan problems, Numer. Funct. Anal. Optim. 9 (1987/88), no. 11–12, 1177–1192. 10.1080/01630568808816279Search in Google Scholar
[30] M. Primicerio, The Occurence of pathologies in some Stefan-like problems, Numerical Treatment of Free Boundary-Value Problems, Internat. Ser. Numer. Math. 58, Birkhäuser, Basel (1982), 233–244. 10.1007/978-3-0348-6563-0_18Search in Google Scholar
[31] C. Sagues, Simulation and optimal control of free boundary, Numerical Treatment of Free Boundary-Value Problems, Internat. Ser. Numer. Math. 58, Birkhäuser, Basel (1982), 270–287. 10.1007/978-3-0348-6563-0_21Search in 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), 555–570. 10.1137/0120058Search in Google Scholar
[33] V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type, Tr. Mat. Inst. Steklova 70 (1964), 133–212. Search in Google Scholar
[34] V. A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form, Tr. Mat. Inst. Steklova 83 (1965), 3–163. Search in Google Scholar
[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), no. 3, 358–368. 10.1017/S1446788700024915Search in Google Scholar
[36] F. P. Vasil’ev, The existence of a solution to a certain optimal Stefan problem (in Russian), Comput. Meth. Programm. 12 (1969), 110–114. Search in Google Scholar
[37] K. Yosida, Functional Analysis, Reprint of the 6th ed., Classics Math., Springer, Berlin, 1995. 10.1007/978-3-642-61859-8Search in Google Scholar
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