American Institute of Mathematical Sciences

Advanced
December  2012, 7(4): 605-615. doi: 10.3934/nhm.2012.7.605

On the optimal control for a semilinear equation with cost depending on the free boundary

Jésus Ildefonso Díaz 1, , Tommaso Mingazzini 1, and  Ángel Manuel Ramos 1,

1. 

Dept. de Matemática Aplicada, Fac. de Matemáticas, Univ. Complutense de Madrid, Madrid, 28040, Spain, Spain, Spain

Received  January 2012 Revised  September 2012 Published  December 2012

We study an optimal control problem for a semilinear elliptic boundary value problem giving rise to a free boundary. Here, the free boundary is generated due to the fact that the nonlinear term of the state equation is not differentiable. The new aspect considered in this paper, with respect to other control problems involving free boundaries, is that here the cost functional explicitly depends on the location of the free boundary. The main difficulty is to show the continuous dependence (in measure) of the free boundary with respect to the control function. The crucial tool to get such continuous dependence is to know how behaves the state solution near the free boundary, as in previous works by L.A. Caffarelli and D. Phillips among other authors. Here we improved previous results in the literature thanks to a suitable application of the Fleming-Rishel formula.
Keywords: Optimal control, semilinear equations, free boundary..
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Jésus Ildefonso Díaz, Tommaso Mingazzini, Ángel Manuel Ramos. On the optimal control for a semilinear equation with cost depending on the free boundary. Networks & Heterogeneous Media, 2012, 7 (4) : 605-615. doi: 10.3934/nhm.2012.7.605
References:
[1]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.2307/1990893. Google Scholar

[2]

L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems,, Appl. Anal., 36 (1990), 131. doi: 10.1080/00036819008839927. Google Scholar

[3]

L. Álvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems,, in, (1987), 55. Google Scholar

[4]

L. Álvarez and J. I. Díaz, On the retention of the interfaces in some elliptic and parabolic problems,, Discrete and Continuous Dynamical Systems, 25 (2009), 1. doi: 10.3934/dcds.2009.25.1. Google Scholar

[5]

R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis,", Clarendon Press, (1975). Google Scholar

[6]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations,, J. Reine Angew. Math., 368 (1986), 63. Google Scholar

[7]

V. Barbu, "Optimal Control of Variational Inequalities,", Pitman Res. Notes Math., 100 (1984). Google Scholar

[8]

A. Bermúdez, C. Rodríguez, M. E. Vázquez and A. Martínez, Mathematical modelling and optimal control methods in waste water discharges,, in, (2004), 7. Google Scholar

[9]

T. Bleninger and G. H. Jirka, Modelling and environmentally sound management of brine discharges from desalination plants,, Desalination, 221 (2008), 585. Google Scholar

[10]

F. Brezzi and L. A. Caffarelli, Convergence of the discrete free boundaries for finite element approximations,, RAIRO Anal. Numér., 17 (1983), 385. Google Scholar

[11]

L. A. Caffarelli, Compactness methods in free boundary problems,, Comm. Partial Differential Equations, 5 (1980), 427. doi: 10.1080/0360530800882144. Google Scholar

[12]

L. A. Caffarelli, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets,, Boll. Un. Mat. Ital. A (5), 18 (1981), 109. Google Scholar

[13]

X.-Y. Chen, H. Matano and M. Mimura, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption,, J. Reine Angew. Math., 459 (1995), 1. Google Scholar

[14]

S. Challal, A. Lyaghfouri and J. F. Rodrigues, On the A-obstacle problem and the Hausdorff measure of its free boundary,, Annali di Matematica, 191 (2012), 113. doi: 10.1007/s10231-010-0177-7. Google Scholar

[15]

C. Conca, J. I. Díaz, A. Liñan and C. Timofte, Homogenization in chemical Rreactive flows,, Electr. J. Diff. Eqns., 40 (2004), 1. Google Scholar

[16]

J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries,", Pitman, 106 (1985). Google Scholar

[17]

J. I. Díaz, Two problems in homogenization of porous media,, Extracta Mathematica, 14 (1999), 141. Google Scholar

[18]

J. I. Díaz, T. Mingazzini and A. M. Ramos, On an optimal control problem involving the location of a free boundary,, Proceedings of the XII Congreso de Ecuaciones Diferenciales Y Aplicaciones /Congreso de Matematica Aplicada (Palma de Mallorca, (2011). Google Scholar

[19]

J. I. Díaz and A. M. Ramos, Numerical experiments regarding the distributed control of semilinear parabolic problems,, Computers and Mathematics with Applications, 48 (2004), 1575. doi: 10.1016/j.camwa.2004.04.033. Google Scholar

[20]

J. I. Díaz, J. M. Sánchez, N. Sánchez, M. Veneros and D. Zarzo, Modeling of brine discharges using both a pilot plant and differential equations,, To appear in the proceedings of IDA World Congress - Perth Convention and Exhibition Centre (PCEC), (2011). Google Scholar

[21]

M. G. Garroni and M. A. Vivaldi, Stability of free boundaries,, Nonlinear Analysis, 12 (1998), 1339. doi: 10.1016/0362-546X(88)90082-X. Google Scholar

[22]

H. W. Gómez, I. Colominas, F. L. Navarrina and M. Casteleiro, A hyperbolic model for convection-diffusion transport problems in CFD: numerical analysis and applications,, Revista de la Real Academia de Ciencias Exactas, 102 (2008), 319. doi: 10.1007/BF03191826. Google Scholar

[23]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications,", Springer, 48 (2005). Google Scholar

[24]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980). Google Scholar

[25]

A. Niepelt, "Development of Interfaces for the Coupling of Hydrodynamic Models for Brine Discharges from Desalination Plants,", Ph.D thesis, (2007). Google Scholar

[26]

R. H. Nochetto, "Aproximación de Problemas Elípticos de Frontera Libre,", Publicaciones del Depto. Ecuaciones Funcionales, (1985). Google Scholar

[27]

R. H. Nochetto, A note on the approximation of free boundaries by finite element methods,, RAIRO Modél. Math. Anal., 20 (1986), 355. Google Scholar

[28]

D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem,, Comm. Part. Diff. Eq., 8 (1983), 1409. doi: 10.1080/03605308308820309. Google Scholar

[29]

R. Pinsky, The dead core for reaction-diffusion equations with convection and its connection with the first exit time of the related Markov diffusion process,, Nonlinear Anal., 12 (1988), 451. doi: 10.1016/0362-546X(88)90043-0. Google Scholar

[30]

R. Pinsky, The interplay of nonlinear reaction and convection in dead core behavior for reaction-diffusion equations,, Nonlinear Anal., 18 (1992), 1113. doi: 10.1016/0362-546X(92)90156-9. Google Scholar

[31]

J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992). doi: 10.1007/978-3-642-58106-9. Google Scholar

[32]

J. M. Rakotoson, "Rearrangement Relatif: un Instrument D'estimations dans les Problemes aux Limites,", Mathematiques & Applications, (2008). doi: 10.1007/978-3-540-69118-1. Google Scholar

[33]

J. F. Rodrigues, "Obstacle Problems in Mathematical Physics,", North-Holland Mathematics Studies, 134 (1987). Google Scholar

[34]

J. F. Rodrigues and B. Zaltzman, Free boundary optimal control in the multidimensional Stefan problem,, in, 323 (1993), 186. Google Scholar

[35]

D. Tiba, Controllability properties for elliptic systems,, International Conference on Differential Equations, 1 (1991), 932. Google Scholar

show all references

References:
[1]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous,, J. Amer. Math. Soc., 2 (1989), 683. doi: 10.2307/1990893. Google Scholar

[2]

L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems,, Appl. Anal., 36 (1990), 131. doi: 10.1080/00036819008839927. Google Scholar

[3]

L. Álvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems,, in, (1987), 55. Google Scholar

[4]

L. Álvarez and J. I. Díaz, On the retention of the interfaces in some elliptic and parabolic problems,, Discrete and Continuous Dynamical Systems, 25 (2009), 1. doi: 10.3934/dcds.2009.25.1. Google Scholar

[5]

R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis,", Clarendon Press, (1975). Google Scholar

[6]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations,, J. Reine Angew. Math., 368 (1986), 63. Google Scholar

[7]

V. Barbu, "Optimal Control of Variational Inequalities,", Pitman Res. Notes Math., 100 (1984). Google Scholar

[8]

A. Bermúdez, C. Rodríguez, M. E. Vázquez and A. Martínez, Mathematical modelling and optimal control methods in waste water discharges,, in, (2004), 7. Google Scholar

[9]

T. Bleninger and G. H. Jirka, Modelling and environmentally sound management of brine discharges from desalination plants,, Desalination, 221 (2008), 585. Google Scholar

[10]

F. Brezzi and L. A. Caffarelli, Convergence of the discrete free boundaries for finite element approximations,, RAIRO Anal. Numér., 17 (1983), 385. Google Scholar

[11]

L. A. Caffarelli, Compactness methods in free boundary problems,, Comm. Partial Differential Equations, 5 (1980), 427. doi: 10.1080/0360530800882144. Google Scholar

[12]

L. A. Caffarelli, A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets,, Boll. Un. Mat. Ital. A (5), 18 (1981), 109. Google Scholar

[13]

X.-Y. Chen, H. Matano and M. Mimura, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption,, J. Reine Angew. Math., 459 (1995), 1. Google Scholar

[14]

S. Challal, A. Lyaghfouri and J. F. Rodrigues, On the A-obstacle problem and the Hausdorff measure of its free boundary,, Annali di Matematica, 191 (2012), 113. doi: 10.1007/s10231-010-0177-7. Google Scholar

[15]

C. Conca, J. I. Díaz, A. Liñan and C. Timofte, Homogenization in chemical Rreactive flows,, Electr. J. Diff. Eqns., 40 (2004), 1. Google Scholar

[16]

J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries,", Pitman, 106 (1985). Google Scholar

[17]

J. I. Díaz, Two problems in homogenization of porous media,, Extracta Mathematica, 14 (1999), 141. Google Scholar

[18]

J. I. Díaz, T. Mingazzini and A. M. Ramos, On an optimal control problem involving the location of a free boundary,, Proceedings of the XII Congreso de Ecuaciones Diferenciales Y Aplicaciones /Congreso de Matematica Aplicada (Palma de Mallorca, (2011). Google Scholar

[19]

J. I. Díaz and A. M. Ramos, Numerical experiments regarding the distributed control of semilinear parabolic problems,, Computers and Mathematics with Applications, 48 (2004), 1575. doi: 10.1016/j.camwa.2004.04.033. Google Scholar

[20]

J. I. Díaz, J. M. Sánchez, N. Sánchez, M. Veneros and D. Zarzo, Modeling of brine discharges using both a pilot plant and differential equations,, To appear in the proceedings of IDA World Congress - Perth Convention and Exhibition Centre (PCEC), (2011). Google Scholar

[21]

M. G. Garroni and M. A. Vivaldi, Stability of free boundaries,, Nonlinear Analysis, 12 (1998), 1339. doi: 10.1016/0362-546X(88)90082-X. Google Scholar

[22]

H. W. Gómez, I. Colominas, F. L. Navarrina and M. Casteleiro, A hyperbolic model for convection-diffusion transport problems in CFD: numerical analysis and applications,, Revista de la Real Academia de Ciencias Exactas, 102 (2008), 319. doi: 10.1007/BF03191826. Google Scholar

[23]

A. Henrot and M. Pierre, "Variation et Optimisation de Formes. Une Analyse Géométrique, Mathématiques & Applications,", Springer, 48 (2005). Google Scholar

[24]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications,", Academic Press, (1980). Google Scholar

[25]

A. Niepelt, "Development of Interfaces for the Coupling of Hydrodynamic Models for Brine Discharges from Desalination Plants,", Ph.D thesis, (2007). Google Scholar

[26]

R. H. Nochetto, "Aproximación de Problemas Elípticos de Frontera Libre,", Publicaciones del Depto. Ecuaciones Funcionales, (1985). Google Scholar

[27]

R. H. Nochetto, A note on the approximation of free boundaries by finite element methods,, RAIRO Modél. Math. Anal., 20 (1986), 355. Google Scholar

[28]

D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem,, Comm. Part. Diff. Eq., 8 (1983), 1409. doi: 10.1080/03605308308820309. Google Scholar

[29]

R. Pinsky, The dead core for reaction-diffusion equations with convection and its connection with the first exit time of the related Markov diffusion process,, Nonlinear Anal., 12 (1988), 451. doi: 10.1016/0362-546X(88)90043-0. Google Scholar

[30]

R. Pinsky, The interplay of nonlinear reaction and convection in dead core behavior for reaction-diffusion equations,, Nonlinear Anal., 18 (1992), 1113. doi: 10.1016/0362-546X(92)90156-9. Google Scholar

[31]

J. Sokolowski and J. P. Zolésio, "Introduction to Shape Optimization. Shape Sensitivity Analysis,", Springer Series in Computational Mathematics, 16 (1992). doi: 10.1007/978-3-642-58106-9. Google Scholar

[32]

J. M. Rakotoson, "Rearrangement Relatif: un Instrument D'estimations dans les Problemes aux Limites,", Mathematiques & Applications, (2008). doi: 10.1007/978-3-540-69118-1. Google Scholar

[33]

J. F. Rodrigues, "Obstacle Problems in Mathematical Physics,", North-Holland Mathematics Studies, 134 (1987). Google Scholar

[34]

J. F. Rodrigues and B. Zaltzman, Free boundary optimal control in the multidimensional Stefan problem,, in, 323 (1993), 186. Google Scholar

[35]

D. Tiba, Controllability properties for elliptic systems,, International Conference on Differential Equations, 1 (1991), 932. Google Scholar

[1]

Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control & Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018
[2]

Ugur 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 & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[3]

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
[4]

Lorena Bociu, Lucas Castle, Kristina Martin, Daniel Toundykov. Optimal control in a free boundary fluid-elasticity interaction. Conference Publications, 2015, 2015 (special) : 122-131. doi: 10.3934/proc.2015.0122
[5]

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
[6]

Sébastien Court, Karl Kunisch, Laurent Pfeiffer. Hybrid optimal control problems for a class of semilinear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1031-1060. doi: 10.3934/dcdss.2018060
[7]

Xavier Fernández-Real, Xavier Ros-Oton. On global solutions to semilinear elliptic equations related to the one-phase free boundary problem. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-15. doi: 10.3934/dcds.2019238
[8]

Chonghu Guan, Fahuai Yi, Xiaoshan Chen. A fully nonlinear free boundary problem arising from optimal dividend and risk control model. Mathematical Control & Related Fields, 2019, 9 (3) : 425-452. doi: 10.3934/mcrf.2019020
[9]

Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003
[10]

Xiaoshan Chen, Fahuai Yi. Free boundary problem of Barenblatt equation in stochastic control. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1421-1434. doi: 10.3934/dcdsb.2016003
[11]

Andrzej Nowakowski. Variational analysis of semilinear plate equation with free boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3133-3154. doi: 10.3934/dcds.2015.35.3133
[12]

Avner Friedman. Free boundary problems for systems of Stokes equations. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1455-1468. doi: 10.3934/dcdsb.2016006
[13]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801
[14]

Shakoor Pooseh, Ricardo Almeida, Delfim F. M. Torres. Fractional order optimal control problems with free terminal time. Journal of Industrial & Management Optimization, 2014, 10 (2) : 363-381. doi: 10.3934/jimo.2014.10.363
[15]

Ahmad Ahmad Ali, Klaus Deckelnick, Michael Hinze. Error analysis for global minima of semilinear optimal control problems. Mathematical Control & Related Fields, 2018, 8 (1) : 195-215. doi: 10.3934/mcrf.2018009
[16]

Constantin Christof, Christian Meyer, Stephan Walther, Christian Clason. Optimal control of a non-smooth semilinear elliptic equation. Mathematical Control & Related Fields, 2018, 8 (1) : 247-276. doi: 10.3934/mcrf.2018011
[17]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
[18]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011
[19]

Volodymyr O. Kapustyan, Ivan O. Pyshnograiev, Olena A. Kapustian. Quasi-optimal control with a general quadratic criterion in a special norm for systems described by parabolic-hyperbolic equations with non-local boundary conditions. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1243-1258. doi: 10.3934/dcdsb.2019014
[20]

Daniela De Silva, Fausto Ferrari, Sandro Salsa. On two phase free boundary problems governed by elliptic equations with distributed sources. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 673-693. doi: 10.3934/dcdss.2014.7.673

2018 Impact Factor: 0.871

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences