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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 and 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), 683-773. doi: 10.2307/1990893.

[2]

L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems, Appl. Anal., 36 (1990), 131-144. doi: 10.1080/00036819008839927.

[3]

L. Álvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems, in "Actas del IX CEDYA", Univ. de Valladolid, (1987), 55-59.

[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), 1-17. doi: 10.3934/dcds.2009.25.1.

[5]

R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis," Clarendon Press, Oxford, 1975.

[6]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.

[7]

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

[8]

A. Bermúdez, C. Rodríguez, M. E. Vázquez and A. Martínez, Mathematical modelling and optimal control methods in waste water discharges, in "Ocean Circulation and Pollution-A Mathematical and Numerical Investigation" (Ed. J. I. Díaz), Springer, Berlin, (2004), 7-15.

[9]

T. Bleninger and G. H. Jirka, Modelling and environmentally sound management of brine discharges from desalination plants, Desalination, 221 (2008), 585-597.

[10]

F. Brezzi and L. A. Caffarelli, Convergence of the discrete free boundaries for finite element approximations, RAIRO Anal. Numér., 17 (1983), 385-395.

[11]

L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations, 5 (1980), 427-448. doi: 10.1080/0360530800882144.

[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), 109-113.

[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), 1-36.

[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), 113-165. doi: 10.1007/s10231-010-0177-7.

[15]

C. Conca, J. I. Díaz, A. Liñan and C. Timofte, Homogenization in chemical Rreactive flows, Electr. J. Diff. Eqns., 40 (2004), 1-22.

[16]

J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries," Pitman, 106, London, 1985.

[17]

J. I. Díaz, Two problems in homogenization of porous media, Extracta Mathematica, 14 (1999), 141-155.

[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, Spain, 2011), http://www.uibcongres.org/congresos/

[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), 1575-1586. doi: 10.1016/j.camwa.2004.04.033.

[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), Perth, Western Australia, (2011).

[21]

M. G. Garroni and M. A. Vivaldi, Stability of free boundaries, Nonlinear Analysis, Theory, Methods & Applications, 12 (1998), 1339-1347. doi: 10.1016/0362-546X(88)90082-X.

[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, Físicas y Naturales. Serie A: Matemáticas (RACSAM), 102 (2008), 319-334. doi: 10.1007/BF03191826.

[23]

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

[24]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, New York, 1980.

[25]

A. Niepelt, "Development of Interfaces for the Coupling of Hydrodynamic Models for Brine Discharges from Desalination Plants," Ph.D thesis, Institute for Hydromechanics, Univ. Karlsruhe, 2007.

[26]

R. H. Nochetto, "Aproximación de Problemas Elípticos de Frontera Libre," Publicaciones del Depto. Ecuaciones Funcionales, Univ. Complutense de Madrid. 1985.

[27]

R. H. Nochetto, A note on the approximation of free boundaries by finite element methods, RAIRO Modél. Math. Anal., 20 (1986), 355-368.

[28]

D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Part. Diff. Eq., 8 (1983), 1409-1454. doi: 10.1080/03605308308820309.

[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), 451-471. doi: 10.1016/0362-546X(88)90043-0.

[30]

R. Pinsky, The interplay of nonlinear reaction and convection in dead core behavior for reaction-diffusion equations, Nonlinear Anal., 18 (1992), 1113-1123. doi: 10.1016/0362-546X(92)90156-9.

[31]

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

[32]

J. M. Rakotoson, "Rearrangement Relatif: un Instrument D'estimations dans les Problemes aux Limites," Mathematiques & Applications, no. 64, Springer, Paris, 2008. doi: 10.1007/978-3-540-69118-1.

[33]

J. F. Rodrigues, "Obstacle Problems in Mathematical Physics," North-Holland Mathematics Studies, 134. Mathematical Notes, 114. North-Holland Publishing Co., Amsterdam 1987.

[34]

J. F. Rodrigues and B. Zaltzman, Free boundary optimal control in the multidimensional Stefan problem, in "Free Boundary Problems: Theory and Applications" (Eds. J. I. Díaz, A. Liñán, M. A. Herrero and J. L. Vázquez), Pitman, 323, London (1993), 186-194.

[35]

D. Tiba, Controllability properties for elliptic systems, International Conference on Differential Equations, 1, 2 (1991), 932-936.

show all references

References:
[1]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Amer. Math. Soc., 2 (1989), 683-773. doi: 10.2307/1990893.

[2]

L. Álvarez, On the behavior of the free boundary of some nonhomogeneous elliptic problems, Appl. Anal., 36 (1990), 131-144. doi: 10.1080/00036819008839927.

[3]

L. Álvarez and J. I. Díaz, On the behaviour near the free boundary of solutions of some non homogeneous elliptic problems, in "Actas del IX CEDYA", Univ. de Valladolid, (1987), 55-59.

[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), 1-17. doi: 10.3934/dcds.2009.25.1.

[5]

R. Aris, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysis," Clarendon Press, Oxford, 1975.

[6]

H. W. Alt and D. Phillips, A free boundary problem for semilinear elliptic equations, J. Reine Angew. Math., 368 (1986), 63-107.

[7]

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

[8]

A. Bermúdez, C. Rodríguez, M. E. Vázquez and A. Martínez, Mathematical modelling and optimal control methods in waste water discharges, in "Ocean Circulation and Pollution-A Mathematical and Numerical Investigation" (Ed. J. I. Díaz), Springer, Berlin, (2004), 7-15.

[9]

T. Bleninger and G. H. Jirka, Modelling and environmentally sound management of brine discharges from desalination plants, Desalination, 221 (2008), 585-597.

[10]

F. Brezzi and L. A. Caffarelli, Convergence of the discrete free boundaries for finite element approximations, RAIRO Anal. Numér., 17 (1983), 385-395.

[11]

L. A. Caffarelli, Compactness methods in free boundary problems, Comm. Partial Differential Equations, 5 (1980), 427-448. doi: 10.1080/0360530800882144.

[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), 109-113.

[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), 1-36.

[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), 113-165. doi: 10.1007/s10231-010-0177-7.

[15]

C. Conca, J. I. Díaz, A. Liñan and C. Timofte, Homogenization in chemical Rreactive flows, Electr. J. Diff. Eqns., 40 (2004), 1-22.

[16]

J. I. Díaz, "Nonlinear Partial Differential Equations and Free Boundaries," Pitman, 106, London, 1985.

[17]

J. I. Díaz, Two problems in homogenization of porous media, Extracta Mathematica, 14 (1999), 141-155.

[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, Spain, 2011), http://www.uibcongres.org/congresos/

[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), 1575-1586. doi: 10.1016/j.camwa.2004.04.033.

[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), Perth, Western Australia, (2011).

[21]

M. G. Garroni and M. A. Vivaldi, Stability of free boundaries, Nonlinear Analysis, Theory, Methods & Applications, 12 (1998), 1339-1347. doi: 10.1016/0362-546X(88)90082-X.

[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, Físicas y Naturales. Serie A: Matemáticas (RACSAM), 102 (2008), 319-334. doi: 10.1007/BF03191826.

[23]

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

[24]

D. Kinderlehrer and G. Stampacchia, "An Introduction to Variational Inequalities and Their Applications," Academic Press, New York, 1980.

[25]

A. Niepelt, "Development of Interfaces for the Coupling of Hydrodynamic Models for Brine Discharges from Desalination Plants," Ph.D thesis, Institute for Hydromechanics, Univ. Karlsruhe, 2007.

[26]

R. H. Nochetto, "Aproximación de Problemas Elípticos de Frontera Libre," Publicaciones del Depto. Ecuaciones Funcionales, Univ. Complutense de Madrid. 1985.

[27]

R. H. Nochetto, A note on the approximation of free boundaries by finite element methods, RAIRO Modél. Math. Anal., 20 (1986), 355-368.

[28]

D. Phillips, Hausdorff measure estimates of a free boundary for a minimum problem, Comm. Part. Diff. Eq., 8 (1983), 1409-1454. doi: 10.1080/03605308308820309.

[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), 451-471. doi: 10.1016/0362-546X(88)90043-0.

[30]

R. Pinsky, The interplay of nonlinear reaction and convection in dead core behavior for reaction-diffusion equations, Nonlinear Anal., 18 (1992), 1113-1123. doi: 10.1016/0362-546X(92)90156-9.

[31]

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

[32]

J. M. Rakotoson, "Rearrangement Relatif: un Instrument D'estimations dans les Problemes aux Limites," Mathematiques & Applications, no. 64, Springer, Paris, 2008. doi: 10.1007/978-3-540-69118-1.

[33]

J. F. Rodrigues, "Obstacle Problems in Mathematical Physics," North-Holland Mathematics Studies, 134. Mathematical Notes, 114. North-Holland Publishing Co., Amsterdam 1987.

[34]

J. F. Rodrigues and B. Zaltzman, Free boundary optimal control in the multidimensional Stefan problem, in "Free Boundary Problems: Theory and Applications" (Eds. J. I. Díaz, A. Liñán, M. A. Herrero and J. L. Vázquez), Pitman, 323, London (1993), 186-194.

[35]

D. Tiba, Controllability properties for elliptic systems, International Conference on Differential Equations, 1, 2 (1991), 932-936.

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