May  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).   Google Scholar

[2]

J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben,, ZAMM, 60 (1980), 335.   Google Scholar

[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.  Google Scholar

[4]

O. V. Besov, V. P. Il'in and S. M. Nikol'skii, "Integral Representations of Functions and Embedding Theorems,", Winston & Sons, (1978).   Google Scholar

[5]

B. M. Budak and V. N. Vasil'eva, On the solution of the inverse Stefan problem,, Soviet Math. Dokl, 13 (1972), 811.   Google Scholar

[6]

B. M. Budak and V. N. Vasil'eva, The solution of the inverse Stefan problem,, USSR Comput. Maths. Math. Phys, 13 (1973), 130.   Google Scholar

[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.   Google Scholar

[8]

J. R. Cannon, A Cauchy problem for the heat equation,, Ann. Math, 66 (1964), 155.  doi: 10.1007/BF02412441.  Google Scholar

[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.  Google Scholar

[10]

A. Carasso, Determining surface temperatures from interior observations,, SIAM J. Appl. Math., 42 (1982), 558.  doi: 10.1137/0142040.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Friedman, "Variational Principles and Free Boundary Problems,", John Wiley, (1982).   Google Scholar

[15]

N. L. Gol'dman, "Inverse Stefan Problems,", Mathematics and its Applications, 412 (1997).  doi: 10.1007/978-94-011-5488-8.  Google Scholar

[16]

N. L. Gol'dman, Properties of solutions of the inverse Stefan problem,, Differential Equations, 39 (2003), 66.  doi: 10.1023/A:1025120024905.  Google Scholar

[17]

K. H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundarie,, Proc. of Int. Conf. on Free Boundary Problems, (1981).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[21]

P. Jochum, The numerical solution of the inverse Stefan problem,, Numerical Mathematics, 34 (1980), 411.  doi: 10.1007/BF01403678.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[24]

K. A. Lurye, "Optimal Control in Problems of Mathematical Physics,", Nauka, (1975).   Google Scholar

[25]

A. M. Meyrmanov, "The Stefan Problem,", Walter de Gruyter, (1992).  doi: 10.1515/9783110846720.245.  Google Scholar

[26]

M. Niezgodka, Control of parabolic systems with free boundaries-application of inverse formulation,, Control and Cybernetics, 8 (1979), 213.   Google Scholar

[27]

S. M. Nikol'skii, "Approximation of Functions of Several Variables and Imbedding Theorems,", Springer-Verlag, (1975).   Google Scholar

[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.  Google Scholar

[29]

M. Primicero, The occurence of pathologies in some Stefan-like problems,, Numerical Treatment of Free Boundary-Value Problems, (1982), 233.   Google Scholar

[30]

C. Sagues, Simulation and optimal control of free boundary,, Numerical Treatment of Free Boundary-Value Problems, 58 (): 270.   Google Scholar

[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.  Google Scholar

[32]

V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type,, Trudy Mat. Inst. Steklov., 70 (1964), 133.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[35]

F. P. Vasil'ev, The existence of a solution of a certain optimal Stefan problem,, in, (1969), 110.   Google Scholar

[36]

F. P. Vasil'ev, "Methods for Solving Extremal Problems. Minimization Problems in Function Spaces, Regularization, Approximation,", (in Russian), (1981).   Google Scholar

[37]

A. D. Yurii, On an optimal Stefan problem,, Dokl. Akad. Nauk SSSR, 251 (1980), 1317.   Google Scholar

show all references

References:
[1]

O. M. Alifanov, "Inverse Heat Transfer Problems,", Springer-Verlag Telos, (1995).   Google Scholar

[2]

J. Baumeister, Zur optimal Steuerung von frien Randwertausgaben,, ZAMM, 60 (1980), 335.   Google Scholar

[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.  Google Scholar

[4]

O. V. Besov, V. P. Il'in and S. M. Nikol'skii, "Integral Representations of Functions and Embedding Theorems,", Winston & Sons, (1978).   Google Scholar

[5]

B. M. Budak and V. N. Vasil'eva, On the solution of the inverse Stefan problem,, Soviet Math. Dokl, 13 (1972), 811.   Google Scholar

[6]

B. M. Budak and V. N. Vasil'eva, The solution of the inverse Stefan problem,, USSR Comput. Maths. Math. Phys, 13 (1973), 130.   Google Scholar

[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.   Google Scholar

[8]

J. R. Cannon, A Cauchy problem for the heat equation,, Ann. Math, 66 (1964), 155.  doi: 10.1007/BF02412441.  Google Scholar

[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.  Google Scholar

[10]

A. Carasso, Determining surface temperatures from interior observations,, SIAM J. Appl. Math., 42 (1982), 558.  doi: 10.1137/0142040.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

A. Friedman, "Variational Principles and Free Boundary Problems,", John Wiley, (1982).   Google Scholar

[15]

N. L. Gol'dman, "Inverse Stefan Problems,", Mathematics and its Applications, 412 (1997).  doi: 10.1007/978-94-011-5488-8.  Google Scholar

[16]

N. L. Gol'dman, Properties of solutions of the inverse Stefan problem,, Differential Equations, 39 (2003), 66.  doi: 10.1023/A:1025120024905.  Google Scholar

[17]

K. H. Hoffman and M. Niezgodka, Control of parabolic systems involving free boundarie,, Proc. of Int. Conf. on Free Boundary Problems, (1981).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[21]

P. Jochum, The numerical solution of the inverse Stefan problem,, Numerical Mathematics, 34 (1980), 411.  doi: 10.1007/BF01403678.  Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[24]

K. A. Lurye, "Optimal Control in Problems of Mathematical Physics,", Nauka, (1975).   Google Scholar

[25]

A. M. Meyrmanov, "The Stefan Problem,", Walter de Gruyter, (1992).  doi: 10.1515/9783110846720.245.  Google Scholar

[26]

M. Niezgodka, Control of parabolic systems with free boundaries-application of inverse formulation,, Control and Cybernetics, 8 (1979), 213.   Google Scholar

[27]

S. M. Nikol'skii, "Approximation of Functions of Several Variables and Imbedding Theorems,", Springer-Verlag, (1975).   Google Scholar

[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.  Google Scholar

[29]

M. Primicero, The occurence of pathologies in some Stefan-like problems,, Numerical Treatment of Free Boundary-Value Problems, (1982), 233.   Google Scholar

[30]

C. Sagues, Simulation and optimal control of free boundary,, Numerical Treatment of Free Boundary-Value Problems, 58 (): 270.   Google Scholar

[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.  Google Scholar

[32]

V. A. Solonnikov, A priori estimates for solutions of second-order equations of parabolic type,, Trudy Mat. Inst. Steklov., 70 (1964), 133.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[35]

F. P. Vasil'ev, The existence of a solution of a certain optimal Stefan problem,, in, (1969), 110.   Google Scholar

[36]

F. P. Vasil'ev, "Methods for Solving Extremal Problems. Minimization Problems in Function Spaces, Regularization, Approximation,", (in Russian), (1981).   Google Scholar

[37]

A. D. Yurii, On an optimal Stefan problem,, Dokl. Akad. Nauk SSSR, 251 (1980), 1317.   Google Scholar

[1]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[2]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[3]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[4]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[5]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[6]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[7]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250

[8]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[9]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[10]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[11]

Hui Lv, Xing'an Wang. Dissipative control for uncertain singular markovian jump systems via hybrid impulsive control. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 127-142. doi: 10.3934/naco.2020020

[12]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444

[13]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[14]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[15]

M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014

[16]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[17]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[18]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[19]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020271

[20]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

2019 Impact Factor: 1.373

Metrics

  • PDF downloads (41)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]