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Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide
Numerical Analysis, Optimization and Statistical Laboratory (LANOS), Badji-Mokhtar, Annaba University, P.O. Box 12, 23000, Annaba, Algeria |
This work is deals with the numerical solution of a bilateral obstacle optimal control problem which is similar to the one given in Bergounioux et al [
References:
[1] |
Y. Achdou, G. Indragoby and O. Pironneau,
Volatility calibration with American options, Methods Appl. of Anal., 11 (2004), 533-556.
|
[2] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsivier, Academic Press, Amsterdam, 2003.
![]() ![]() |
[3] |
D. R. Adams and S. Lenhart,
An obstacle control problem with a source term, Appl. Math. Optim., 47 (2002), 79-95.
doi: 10.1007/s00245-002-0739-1. |
[4] |
G. M. Bahaa,
Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control Inform., 35 (2016), 107-122.
doi: 10.1093/imamci/dnw040. |
[5] |
V. Barbu, Optimal Control of Varitional Inequalities, Pitman, London, 1984. |
[6] |
M. Bergounioux, X. Bonnefond, T. Haberkorn and Y. Privat,
An optimal control problem in photoacoustic tomography, Math. Models Methods Appl. Sci., 24 (2014), 2525-2548.
doi: 10.1142/S0218202514500286. |
[7] |
M. Bergounioux and Y. Privat,
Shape optimization with Stokes constraints over the set of axisymmetric domains, SIAM J. Control Optim., 51 (2013), 599-628.
doi: 10.1137/100818133. |
[8] |
M. Bergounioux and S. Lenhart, Optimal control of the obstacle in semilinear variational inequalities, Positivity, 8 (2004), 229–242.
doi: 10.1007/s11117-004-5009-9. |
[9] |
M. Bergounioux and S. Lenhart,
Optimal control of the bilateral obstacle problems, SIAM J. Control Optim., 43 (2004), 249-255.
doi: 10.1137/S0363012902416912. |
[10] |
T. Betz, Optimal Control of Two Variational Inequalities Arising in Solid Mechanics, Ph.D Thesis, Universitätsbibliothek Dortmund, 2015. |
[11] |
Bock, Igor and Kečkemétyová, Mária,
Regularized optimal control problem for a beam vibrating against an elastic foundation, Tatra Mt. Math. Publ., 63 (2015), 53-71.
doi: 10.1515/tmmp-2015-0020. |
[12] |
H. Brzis and D. Kinderlehrer,
The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1974), 831-844.
doi: 10.1512/iumj.1974.23.23069. |
[13] |
P. Colli, G. Gilardi, E. Rocca and J. Sprekels,
Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.
doi: 10.1088/1361-6544/aa6e5f. |
[14] |
M. Chipot, Variational Inequalities and Flow in Porous Media, Springer-Verlag, New York, 52 (1984).
doi: 10.1007/978-1-4612-1120-4. |
[15] |
J. C. De Los Reyes,
On the optimal control of some nonsmooth distributed parameter systems arising in mechanics, GAMM-Mitt., 40 (2018), 268-286.
doi: 10.1002/gamm.201740002. |
[16] |
S. Desong, Z. Zhongding and Y. Fuxin,
A variational inequality principle in solid mechanics and application in physically non-linear problems, Communications in Applied Numerical Methods, 6 (1990), 35-45.
doi: 10.1002/cnm.1630060106. |
[17] |
R. Ghanem,
Optimal control of unilateral obstacle problem with a source term, Positivity, 13 (2009), 321-338.
doi: 10.1007/s11117-008-2241-8. |
[18] |
R. Ghanem and B. Zireg,
On the numerical study of an obstacle optimal control problem with source term, J. Appl. Math. Comput., 45 (2014), 375-409.
doi: 10.1007/s12190-013-0728-3. |
[19] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[20] |
I. Hlaváček, I. Bock and J. Lovíšek,
Optimal control of a variational inequality with applications to structural analysis. I. Optimal design of a beam with unilateral supports, Appl. Math. Optim., 11 (1984), 111-143.
doi: 10.1007/BF01442173. |
[21] |
C. U. Huy, P. J. Mckenna and W. Walter,
Finite difference approximations to the Dirichlet problem for elliptic systems, Numer. Math., 49 (1986), 227-237.
doi: 10.1007/BF01389626. |
[22] |
K. Ito and K. Kunisch,
Optimal control of elliptic variational inequalities, Appl. Math. Optim., 41 (2000), 343-364.
doi: 10.1007/s002459911017. |
[23] |
K. Ito and K. Kunisch,
Optimal control of obstacle problems by $H^{1}-$obstacles, Appl. Math. Optim., 56 (2007), 1-17.
doi: 10.1007/s00245-007-0877-6. |
[24] |
K. Kunisch and D. Wachsmuth,
Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control Optim. Calc. Var., 18 (2012), 520-547.
doi: 10.1051/cocv/2011105. |
[25] |
J. L. Lions and G. Stampacchia,
Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.
doi: 10.1002/cpa.3160200302. |
[26] |
J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, (French), Dunod, Paris, 1 (1968). |
[27] |
F. Mignot and J. P. Puel,
Optimal control in some variational inequalities, SIAM J. Control Optim., 22 (1984), 466-476.
doi: 10.1137/0322028. |
[28] |
F. Mignot,
Contrôle dans les inéquatons variationelles elliptiques, (French), J. Funct. Anal., 22 (1976), 466-476.
doi: 10.1016/0022-1236(76)90017-3. |
[29] |
S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165–172.
doi: 10.1155/2013/692879. |
[30] |
Z. Peng and K. Kunisch,
Optimal control of elliptic variational–hemivariational inequalities, J. Optim. Theory Appl., 178 (2018), 1-25.
doi: 10.1007/s10957-018-1303-8. |
[31] |
J. F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier, New york, 1987. |
[32] |
V. Shcherbakov,
Shape optimization of rigid inclusions for elastic plates with cracks, Z. Angew. Math. Phys., 67 (2016), 71-76.
doi: 10.1007/s00033-016-0666-7. |
[33] |
M. Sofonea, A. Benraouda and H. Hechaichi,
Optimal control of a two-dimensional contact problem, Appl. Anal., 97 (2018), 1281-1298.
doi: 10.1080/00036811.2017.1337895. |
show all references
References:
[1] |
Y. Achdou, G. Indragoby and O. Pironneau,
Volatility calibration with American options, Methods Appl. of Anal., 11 (2004), 533-556.
|
[2] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsivier, Academic Press, Amsterdam, 2003.
![]() ![]() |
[3] |
D. R. Adams and S. Lenhart,
An obstacle control problem with a source term, Appl. Math. Optim., 47 (2002), 79-95.
doi: 10.1007/s00245-002-0739-1. |
[4] |
G. M. Bahaa,
Fractional optimal control problem for variational inequalities with control constraints, IMA J. Math. Control Inform., 35 (2016), 107-122.
doi: 10.1093/imamci/dnw040. |
[5] |
V. Barbu, Optimal Control of Varitional Inequalities, Pitman, London, 1984. |
[6] |
M. Bergounioux, X. Bonnefond, T. Haberkorn and Y. Privat,
An optimal control problem in photoacoustic tomography, Math. Models Methods Appl. Sci., 24 (2014), 2525-2548.
doi: 10.1142/S0218202514500286. |
[7] |
M. Bergounioux and Y. Privat,
Shape optimization with Stokes constraints over the set of axisymmetric domains, SIAM J. Control Optim., 51 (2013), 599-628.
doi: 10.1137/100818133. |
[8] |
M. Bergounioux and S. Lenhart, Optimal control of the obstacle in semilinear variational inequalities, Positivity, 8 (2004), 229–242.
doi: 10.1007/s11117-004-5009-9. |
[9] |
M. Bergounioux and S. Lenhart,
Optimal control of the bilateral obstacle problems, SIAM J. Control Optim., 43 (2004), 249-255.
doi: 10.1137/S0363012902416912. |
[10] |
T. Betz, Optimal Control of Two Variational Inequalities Arising in Solid Mechanics, Ph.D Thesis, Universitätsbibliothek Dortmund, 2015. |
[11] |
Bock, Igor and Kečkemétyová, Mária,
Regularized optimal control problem for a beam vibrating against an elastic foundation, Tatra Mt. Math. Publ., 63 (2015), 53-71.
doi: 10.1515/tmmp-2015-0020. |
[12] |
H. Brzis and D. Kinderlehrer,
The smoothness of solutions to nonlinear variational inequalities, Indiana Univ. Math. J., 23 (1974), 831-844.
doi: 10.1512/iumj.1974.23.23069. |
[13] |
P. Colli, G. Gilardi, E. Rocca and J. Sprekels,
Optimal distributed control of a diffuse interface model of tumor growth, Nonlinearity, 30 (2017), 2518-2546.
doi: 10.1088/1361-6544/aa6e5f. |
[14] |
M. Chipot, Variational Inequalities and Flow in Porous Media, Springer-Verlag, New York, 52 (1984).
doi: 10.1007/978-1-4612-1120-4. |
[15] |
J. C. De Los Reyes,
On the optimal control of some nonsmooth distributed parameter systems arising in mechanics, GAMM-Mitt., 40 (2018), 268-286.
doi: 10.1002/gamm.201740002. |
[16] |
S. Desong, Z. Zhongding and Y. Fuxin,
A variational inequality principle in solid mechanics and application in physically non-linear problems, Communications in Applied Numerical Methods, 6 (1990), 35-45.
doi: 10.1002/cnm.1630060106. |
[17] |
R. Ghanem,
Optimal control of unilateral obstacle problem with a source term, Positivity, 13 (2009), 321-338.
doi: 10.1007/s11117-008-2241-8. |
[18] |
R. Ghanem and B. Zireg,
On the numerical study of an obstacle optimal control problem with source term, J. Appl. Math. Comput., 45 (2014), 375-409.
doi: 10.1007/s12190-013-0728-3. |
[19] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[20] |
I. Hlaváček, I. Bock and J. Lovíšek,
Optimal control of a variational inequality with applications to structural analysis. I. Optimal design of a beam with unilateral supports, Appl. Math. Optim., 11 (1984), 111-143.
doi: 10.1007/BF01442173. |
[21] |
C. U. Huy, P. J. Mckenna and W. Walter,
Finite difference approximations to the Dirichlet problem for elliptic systems, Numer. Math., 49 (1986), 227-237.
doi: 10.1007/BF01389626. |
[22] |
K. Ito and K. Kunisch,
Optimal control of elliptic variational inequalities, Appl. Math. Optim., 41 (2000), 343-364.
doi: 10.1007/s002459911017. |
[23] |
K. Ito and K. Kunisch,
Optimal control of obstacle problems by $H^{1}-$obstacles, Appl. Math. Optim., 56 (2007), 1-17.
doi: 10.1007/s00245-007-0877-6. |
[24] |
K. Kunisch and D. Wachsmuth,
Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control Optim. Calc. Var., 18 (2012), 520-547.
doi: 10.1051/cocv/2011105. |
[25] |
J. L. Lions and G. Stampacchia,
Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.
doi: 10.1002/cpa.3160200302. |
[26] |
J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, (French), Dunod, Paris, 1 (1968). |
[27] |
F. Mignot and J. P. Puel,
Optimal control in some variational inequalities, SIAM J. Control Optim., 22 (1984), 466-476.
doi: 10.1137/0322028. |
[28] |
F. Mignot,
Contrôle dans les inéquatons variationelles elliptiques, (French), J. Funct. Anal., 22 (1976), 466-476.
doi: 10.1016/0022-1236(76)90017-3. |
[29] |
S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165–172.
doi: 10.1155/2013/692879. |
[30] |
Z. Peng and K. Kunisch,
Optimal control of elliptic variational–hemivariational inequalities, J. Optim. Theory Appl., 178 (2018), 1-25.
doi: 10.1007/s10957-018-1303-8. |
[31] |
J. F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier, New york, 1987. |
[32] |
V. Shcherbakov,
Shape optimization of rigid inclusions for elastic plates with cracks, Z. Angew. Math. Phys., 67 (2016), 71-76.
doi: 10.1007/s00033-016-0666-7. |
[33] |
M. Sofonea, A. Benraouda and H. Hechaichi,
Optimal control of a two-dimensional contact problem, Appl. Anal., 97 (2018), 1281-1298.
doi: 10.1080/00036811.2017.1337895. |




Algorithm 2 Implemented continuous algorithm |
1: Begin
2: Input : 3: Calculate 4: Solve 5: Calculate 6: Solve 7: Calculate 8: Solve 9: Calculate 10: Solve 11: Calculate 12: Calculate 13: If 14: Ensure : 15: Else; 16: End if 17: End |
Algorithm 2 Implemented continuous algorithm |
1: Begin
2: Input : 3: Calculate 4: Solve 5: Calculate 6: Solve 7: Calculate 8: Solve 9: Calculate 10: Solve 11: Calculate 12: Calculate 13: If 14: Ensure : 15: Else; 16: End if 17: End |
0.25 | 62 | 29.460551e-4 | 9.601694e-16 | 3.982865e-11 |
0.5 | 28 | 29.460551e-4 | 6.392456e-16 | 1.180507e-11 |
0.75 | 15 | 29.460551e-4 | 5.212844e-16 | 5.253112e-12 |
1 | 5 | 29.460551e-4 | 6.878178e-16 | 9.642406e-10 |
0.25 | 62 | 29.460551e-4 | 9.601694e-16 | 3.982865e-11 |
0.5 | 28 | 29.460551e-4 | 6.392456e-16 | 1.180507e-11 |
0.75 | 15 | 29.460551e-4 | 5.212844e-16 | 5.253112e-12 |
1 | 5 | 29.460551e-4 | 6.878178e-16 | 9.642406e-10 |
30 | 26 | 1.632e-3 | 5.193328e-16 | 7.355250e-12 |
35 | 27 | 2.241e-3 | 5.165139e-16 | 6.513361e-12 |
40 | 28 | 2.946e-3 | 6.392456e-16 | 1.180507e-11 |
45 | 30 | 3.746e-3 | 7.350890e-16 | 2.389886e-11 |
30 | 26 | 1.632e-3 | 5.193328e-16 | 7.355250e-12 |
35 | 27 | 2.241e-3 | 5.165139e-16 | 6.513361e-12 |
40 | 28 | 2.946e-3 | 6.392456e-16 | 1.180507e-11 |
45 | 30 | 3.746e-3 | 7.350890e-16 | 2.389886e-11 |
0.001 | 300 | 29.460541e-4 | 1.227802e-13 | 3.695397e-05 |
0.01 | 29 | 29.460541e-4 | 5.676882e-16 | 5.966039e-11 |
0.1 | 29 | 29.460544e-4 | 5.676882e-16 | 5.966039e-11 |
0.5 | 28 | 29.460549e-4 | 7.650130e-16 | 2.529592e-11 |
1 | 28 | 29.460551e-4 | 6.392456e-16 | 1.180507e-11 |
0.001 | 300 | 29.460541e-4 | 1.227802e-13 | 3.695397e-05 |
0.01 | 29 | 29.460541e-4 | 5.676882e-16 | 5.966039e-11 |
0.1 | 29 | 29.460544e-4 | 5.676882e-16 | 5.966039e-11 |
0.5 | 28 | 29.460549e-4 | 7.650130e-16 | 2.529592e-11 |
1 | 28 | 29.460551e-4 | 6.392456e-16 | 1.180507e-11 |
27 | 29.460541e-4 | 8.135853e-16 | 8.627484e-12 | |
27 | 29.460543e-4 | 8.270294e-16 | 8.809409e-12 | |
28 | 29.460551e-4 | 6.392456e-16 | 1.180507e-11 | |
33 | 29.460566e-4 | 9.358833e-16 | 1.016003e-10 |
27 | 29.460541e-4 | 8.135853e-16 | 8.627484e-12 | |
27 | 29.460543e-4 | 8.270294e-16 | 8.809409e-12 | |
28 | 29.460551e-4 | 6.392456e-16 | 1.180507e-11 | |
33 | 29.460566e-4 | 9.358833e-16 | 1.016003e-10 |
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