American Institute of Mathematical Sciences

Advanced
September  2020, 10(3): 275-300. doi: 10.3934/naco.2020002

Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide

Radouen Ghanem and  Billel Zireg

Numerical Analysis, Optimization and Statistical Laboratory (LANOS), Badji-Mokhtar, Annaba University, P.O. Box 12, 23000, Annaba, Algeria

Received  December 2018 Revised  August 2019 Published  February 2020

Fund Project: The authors would like to thank the anonymous referee for careful reading and the suggestions of some improvements in presentation that have been implemented in the final version of the manuscript

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 [9] with some modifications. It can be regarded as an extension of our previous work [18], where the main feature of the present work is that the controls and the two obstacles are the same. For the numerical resolution we follow the idea of our previous work [18]. We begin by discretizing the optimality system of the underlying problem by using finite differences schemes, then we propose an iterative algorithm. Finally, numerical examples are provides to show the efficiency of the proposed algorithm and the used scheme.

Keywords: Optimal control, Bilateral obstacle problem, Finite differences method.
Mathematics Subject Classification: Primary: 65K15 49K10, 35J87; Secondary: 49J20, 65L12, 49M15.
Citation: Radouen Ghanem, Billel Zireg. Numerical solution of bilateral obstacle optimal control problem, where the controls and the obstacles coincide. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 275-300. doi: 10.3934/naco.2020002
References:
[1]

Y. Achdou, G. Indragoby and O. Pironneau, Volatility calibration with American options, Methods Appl. of Anal., 11 (2004), 533-556.   Google Scholar

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsivier, Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[5]

V. Barbu, Optimal Control of Varitional Inequalities, Pitman, London, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Betz, Optimal Control of Two Variational Inequalities Arising in Solid Mechanics, Ph.D Thesis, Universitätsbibliothek Dortmund, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Chipot, Variational Inequalities and Flow in Porous Media, Springer-Verlag, New York, 52 (1984). doi: 10.1007/978-1-4612-1120-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, Appl. Math. Optim., 41 (2000), 343-364.  doi: 10.1007/s002459911017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[26]

J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, (French), Dunod, Paris, 1 (1968).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165–172. doi: 10.1155/2013/692879.  Google Scholar

[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.  Google Scholar

[31]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier, New york, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Elsivier, Academic Press, Amsterdam, 2003.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[5]

V. Barbu, Optimal Control of Varitional Inequalities, Pitman, London, 1984.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

T. Betz, Optimal Control of Two Variational Inequalities Arising in Solid Mechanics, Ph.D Thesis, Universitätsbibliothek Dortmund, 2015. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

M. Chipot, Variational Inequalities and Flow in Porous Media, Springer-Verlag, New York, 52 (1984). doi: 10.1007/978-1-4612-1120-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, Appl. Math. Optim., 41 (2000), 343-364.  doi: 10.1007/s002459911017.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math., 20 (1967), 493-519.  doi: 10.1002/cpa.3160200302.  Google Scholar

[26]

J. L. Lions and E. Magenes, Problèmes Aux Limites Non Homogènes et Applications, (French), Dunod, Paris, 1 (1968).  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

S. A. Morris, The Schauder-Tychonoff fixed point theorem and applications, Matematický Časopis, 25 (1975), 165–172. doi: 10.1155/2013/692879.  Google Scholar

[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.  Google Scholar

[31]

J. F. Rodrigues, Obstacle Problems in Mathematical Physics, Elsevier, New york, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Left (error $ \epsilon _{n} $, continuous line; $ \omega = 0.25 $, dash line; $ \omega = 0.5 $, dash-dot line; $ \omega = 0.75 $), right (state function $ y $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Left (obstacle function $ \psi $), right (obstacle function $ \varphi $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Left (error $ \epsilon _{n} $, continuous line; $ N = 30 $, dash line; $ N = 35 $, dash-dot line; $ N = 40 $), right (state function $ y $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Left (obstacle function $ \psi $), right (obstacle function $ \varphi $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Left(error $ \epsilon _{n} $, continuous line $ \nu = 0.1 $; dash line $ \nu = 0.5 $; dash-dot line $ \nu = 1 $), right (state function $ y $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Left(obstacle function $ \varphi $), right (obstacle function $ \psi $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Left(error $ \epsilon_{n} $, continuous line; $ \delta = h^{2} $, dash line; $ \delta = h^{3} $, dash-dot line; $ \delta = h^{4} $), right (state function $ y $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Left (obstacle function $ \psi $), right (right(obstacle function $ \varphi $))
Figure Options

Download full-size image

Download as PowerPoint slide

Table202 
Algorithm 2 Implemented continuous algorithm
1: Begin
2: Input :$ \left\{ y_{0}^{\delta }, p_{0}^{\delta }, \varphi _{0}^{\delta }, \lambda _{0}^{\delta },\psi _{0}^{\delta },\delta ,\nu ,\omega_{y},\omega_{\varphi},\omega_{\psi},\varepsilon\right\} $ choose $ \varphi _{0}^{\delta } $ and $ \psi _{0}^{\delta }\in \mathcal{W},\varepsilon $ and $ \delta $ in $ \mathbb{R}_{+}^{\ast }; $
3: Calculate $ J_{n-1} \leftarrow J_{n-1}\left(y^{\delta}_{n-1}, \varphi^{\delta}_{n-1}, \psi^{\delta}_{n-1}\right) $
4:   Solve $ \left( A+\beta _{\delta }^{\prime }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left(\psi _{n-1}^{\delta }- y_{n-1}^{\delta }\right) \right)r_{n}^{\delta }= $ $ -\omega_{y}\left( A y_{n-1}^{\delta }+\beta _{\delta }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta } \right)-\beta _{\delta }\left( \psi _{n-1}^{\delta } -y_{n-1}^{\delta }\right) -f \right) $ on $ r_{n}^{\delta } $.
5: Calculate $ y_{n}^{\delta }=y_{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
6: Solve $ \left( A +\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta } -y_{n}^{\delta }\right) \right) p_{n}^{\delta }=y_{n}^{\delta }-z $ on $ p_{n}^{\delta } $.
7: Calculate $ \lambda_{n}^{\delta } = \nu \Delta \varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta } $.
8: Solve$ \left( \nu \Delta+\beta _{\delta }^{\prime \prime }\left(\psi _{n-1}^{\delta } -y_{n}^{\delta }\right) p_{n}^{\delta }\right)r_{n}^{\delta }=-\omega_{\psi}\left(\nu A_{h}^{d}\psi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta }-y_{n}^{\delta }\right) p_{n}^{\delta }+\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
9: Calculate $ \psi _{n}^{\delta }=\psi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
10: Solve$ \left( \nu \Delta-\beta _{\delta }^{\prime \prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta } \right) p_{n}^{\delta }\right) r_{n}^{\delta }=-\omega_{\varphi}\left(\nu A_{h}^{d}\varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta }-\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
11: Calculate $ \varphi _{n}^{\delta }=\varphi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
12: Calculate $ J_{n} \leftarrow J_{n-1}\left(y^{\delta}_{n}, \varphi^{\delta}_{n},\psi^{\delta}_{n}\right) $.
13: If $ |J_{n}-J_{n-1}| \leq \varepsilon $ Stop.
14: Ensure : $ s_{n}^{\delta }=\left( y_{n}^{\delta },\varphi_{n}^{\delta },\psi_{n}^{\delta },p_{n}^{\delta }\right) $ is a solution.
15:      Else; $ n\leftarrow n+1 $, go to Begin.
16: End if
17: End
Table Options

Download as excel

Algorithm 2 Implemented continuous algorithm
1: Begin
2: Input :$ \left\{ y_{0}^{\delta }, p_{0}^{\delta }, \varphi _{0}^{\delta }, \lambda _{0}^{\delta },\psi _{0}^{\delta },\delta ,\nu ,\omega_{y},\omega_{\varphi},\omega_{\psi},\varepsilon\right\} $ choose $ \varphi _{0}^{\delta } $ and $ \psi _{0}^{\delta }\in \mathcal{W},\varepsilon $ and $ \delta $ in $ \mathbb{R}_{+}^{\ast }; $
3: Calculate $ J_{n-1} \leftarrow J_{n-1}\left(y^{\delta}_{n-1}, \varphi^{\delta}_{n-1}, \psi^{\delta}_{n-1}\right) $
4:   Solve $ \left( A+\beta _{\delta }^{\prime }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left(\psi _{n-1}^{\delta }- y_{n-1}^{\delta }\right) \right)r_{n}^{\delta }= $ $ -\omega_{y}\left( A y_{n-1}^{\delta }+\beta _{\delta }\left( y_{n-1}^{\delta }-\varphi _{n-1}^{\delta } \right)-\beta _{\delta }\left( \psi _{n-1}^{\delta } -y_{n-1}^{\delta }\right) -f \right) $ on $ r_{n}^{\delta } $.
5: Calculate $ y_{n}^{\delta }=y_{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
6: Solve $ \left( A +\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right)+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta } -y_{n}^{\delta }\right) \right) p_{n}^{\delta }=y_{n}^{\delta }-z $ on $ p_{n}^{\delta } $.
7: Calculate $ \lambda_{n}^{\delta } = \nu \Delta \varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta } $.
8: Solve$ \left( \nu \Delta+\beta _{\delta }^{\prime \prime }\left(\psi _{n-1}^{\delta } -y_{n}^{\delta }\right) p_{n}^{\delta }\right)r_{n}^{\delta }=-\omega_{\psi}\left(\nu A_{h}^{d}\psi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( \psi _{n-1}^{\delta }-y_{n}^{\delta }\right) p_{n}^{\delta }+\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
9: Calculate $ \psi _{n}^{\delta }=\psi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
10: Solve$ \left( \nu \Delta-\beta _{\delta }^{\prime \prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta } \right) p_{n}^{\delta }\right) r_{n}^{\delta }=-\omega_{\varphi}\left(\nu A_{h}^{d}\varphi _{n-1}^{\delta }+\beta _{\delta }^{\prime }\left( y_{n}^{\delta }-\varphi _{n-1}^{\delta }\right) p_{n}^{\delta }-\lambda_{n}^{\delta }\right) $ on $ r_{n}^{\delta } $.
11: Calculate $ \varphi _{n}^{\delta }=\varphi _{n-1}^{\delta }+ $ $ r_{n}^{\delta } $.
12: Calculate $ J_{n} \leftarrow J_{n-1}\left(y^{\delta}_{n}, \varphi^{\delta}_{n},\psi^{\delta}_{n}\right) $.
13: If $ |J_{n}-J_{n-1}| \leq \varepsilon $ Stop.
14: Ensure : $ s_{n}^{\delta }=\left( y_{n}^{\delta },\varphi_{n}^{\delta },\psi_{n}^{\delta },p_{n}^{\delta }\right) $ is a solution.
15:      Else; $ n\leftarrow n+1 $, go to Begin.
16: End if
17: End
Table 1.  Numerical results for two dimensional space while varying $ \omega $
$ \omega $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
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
Table Options

Download as excel

$ \omega $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
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
Table 2.  Numerical results for two dimensional space while varying $ N $
$ N $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
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
Table Options

Download as excel

$ N $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
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
Table 3.  Numerical results for two dimensional space while varying $ \nu $
$ \nu $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
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
Table Options

Download as excel

$ \nu $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
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
Table 4.  Numerical results for two dimensional space while varying $ \delta $
$ \delta $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
$ h^{3} $ 27 29.460541e-4 8.135853e-16 8.627484e-12
$ h^{3.5} $ 27 29.460543e-4 8.270294e-16 8.809409e-12
$ h^{4} $ 28 29.460551e-4 6.392456e-16 1.180507e-11
$ h^{4.5} $ 33 29.460566e-4 9.358833e-16 1.016003e-10
Table Options

Download as excel

$ \delta $ $ \sharp $ Iteration $ {J} $ $ \mid J_{n}-J_{n-1}\mid $ $ {\epsilon}_{n} $
$ h^{3} $ 27 29.460541e-4 8.135853e-16 8.627484e-12
$ h^{3.5} $ 27 29.460543e-4 8.270294e-16 8.809409e-12
$ h^{4} $ 28 29.460551e-4 6.392456e-16 1.180507e-11
$ h^{4.5} $ 33 29.460566e-4 9.358833e-16 1.016003e-10
[1]

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
[2]

Sie Long Kek, Kok Lay Teo, Mohd Ismail Abd Aziz. Filtering solution of nonlinear stochastic optimal control problem in discrete-time with model-reality differences. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 207-222. doi: 10.3934/naco.2012.2.207
[3]

Ali Fuat Alkaya, Dindar Oz. An optimal algorithm for the obstacle neutralization problem. Journal of Industrial & Management Optimization, 2017, 13 (2) : 835-856. doi: 10.3934/jimo.2016049
[4]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo. A gradient algorithm for optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 251-266. doi: 10.3934/naco.2015.5.251
[5]

Marcus Wagner. A direct method for the solution of an optimal control problem arising from image registration. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 487-510. doi: 10.3934/naco.2012.2.487
[6]

Laurent Bourgeois, Dmitry Ponomarev, Jérémi Dardé. An inverse obstacle problem for the wave equation in a finite time domain. Inverse Problems & Imaging, 2019, 13 (2) : 377-400. doi: 10.3934/ipi.2019019
[7]

Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183
[8]

Sie Long Kek, Mohd Ismail Abd Aziz, Kok Lay Teo, Rohanin Ahmad. An iterative algorithm based on model-reality differences for discrete-time nonlinear stochastic optimal control problems. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 109-125. doi: 10.3934/naco.2013.3.109
[9]

Sie Long Kek, Mohd Ismail Abd Aziz. Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 275-288. doi: 10.3934/naco.2015.5.275
[10]

Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967
[11]

Georg Vossen, Torsten Hermanns. On an optimal control problem in laser cutting with mixed finite-/infinite-dimensional constraints. Journal of Industrial & Management Optimization, 2014, 10 (2) : 503-519. doi: 10.3934/jimo.2014.10.503
[12]

Qun Lin, Ryan Loxton, Kok Lay Teo. The control parameterization method for nonlinear optimal control: A survey. Journal of Industrial & Management Optimization, 2014, 10 (1) : 275-309. doi: 10.3934/jimo.2014.10.275
[13]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020019
[14]

Simone Göttlich, Patrick Schindler. Optimal inflow control of production systems with finite buffers. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 107-127. doi: 10.3934/dcdsb.2015.20.107
[15]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311
[16]

Zhongyi Huang. Tailored finite point method for the interface problem. Networks & Heterogeneous Media, 2009, 4 (1) : 91-106. doi: 10.3934/nhm.2009.4.91
[17]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential. Evolution Equations & Control Theory, 2017, 6 (1) : 35-58. doi: 10.3934/eect.2017003
[18]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020213
[19]

Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477
[20]

Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027

 Impact Factor: 

Tools

Metrics

  • PDF downloads (66)
  • HTML views (253)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences