December  2020, 9(4): 1187-1201. doi: 10.3934/eect.2020045

Convergence of simultaneous distributed-boundary parabolic optimal control problems

1. 

Depto. Matemática-CONICET, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina

2. 

Depto. Matemática, FCEFQyN, Univ. Nac. de Río Cuarto, Ruta 36 Km 601, 5800 Río Cuarto, Argentina

* Corresponding author: DTarzia@austral.edu.ar

Received  October 2019 Published  December 2020 Early access  March 2020

Fund Project: The first and third authors is supported by the European Union's Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement 823731 CONMECH

We consider a heat conduction problem $ S $ with mixed boundary conditions in a n-dimensional domain $ \Omega $ with regular boundary $ \Gamma $ and a family of problems $ S_{\alpha} $, where the parameter $ \alpha>0 $ is the heat transfer coefficient on the portion of the boundary $ \Gamma_{1} $. In relation to these state systems, we formulate simultaneous distributed-boundary optimal control problems on the internal energy $ g $ and the heat flux $ q $ on the complementary portion of the boundary $ \Gamma_{2} $. We obtain existence and uniqueness of the optimal controls, the first order optimality conditions in terms of the adjoint state and the convergence of the optimal controls, the system and the adjoint states when the heat transfer coefficient $ \alpha $ goes to infinity. Finally, we prove estimations between the simultaneous distributed-boundary optimal control and the distributed optimal control problem studied in a previous paper of the first author.

Citation: Domingo Tarzia, Carolina Bollo, Claudia Gariboldi. Convergence of simultaneous distributed-boundary parabolic optimal control problems. Evolution Equations and Control Theory, 2020, 9 (4) : 1187-1201. doi: 10.3934/eect.2020045
References:
[1]

A. Azzam and E. Kreyszig, On solutions of elliptic equations satisfying mixed boundary conditions, SIAM J. Math. Anal., 13 (1982), 254-262.  doi: 10.1137/0513018.

[2]

I. Babŭska and S. Ohnimus, A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations, Comput. Methods Appl. Mech. Engrg., 190 (2001), 4691-4712.  doi: 10.1016/S0045-7825(00)00340-6.

[3]

C. BacutaJ. H. Bramble and J. E. Pasciak, Using finite element tools in proving shift theorems for elliptic boundary value problems, Numer. Linear Algebra Appl., 10 (2003), 33-64.  doi: 10.1002/nla.311.

[4]

F. B. BelgacemH. E. Fekih and J. P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptot. Anal., 34 (2003), 121-136. 

[5]

A. Bensoussan and J.-L. Lions, Applications of Variational Inequalities in Stochastic Control, Vol. 12, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1982.

[6]

M. Bergounioux and F. Tröltzsch, Optimal control of semilinear parabolic equations with state-constraints of bottleneck type, ESAIM: Control Optim. Calc. Var., 4 (1999), 595-608.  doi: 10.1051/cocv:1999124.

[7]

M. Boukrouche and D. A. Tarzia, Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind, Nonlinear Anal. Real World Appl., 12 (2011), 2211-2224.  doi: 10.1016/j.nonrwa.2011.01.003.

[8]

M. Boukrouche and D. A. Tarzia, Convergence of optimal control problems governed by second kind parabolic variational inequalities, J. Control Theory Appl., 11 (2013), 422-427.  doi: 10.1007/s11768-013-2155-2.

[9]

K. ChrysafinosM. D. Gunzburger and L. S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE, J. Math. Anal. Appl., 323 (2006), 891-912.  doi: 10.1016/j.jmaa.2005.10.053.

[10]

C. M. Gariboldi and D. A. Tarzia, Convergence of boundary optimal control problems with restrictions in mixed elliptic Stefan-like problems, Adv. Differ. Equ. Control Process., 1 (2008), 113-132. 

[11]

C. M. Gariboldi and D. A. Tarzia, Existence, uniqueness and convergence of simultaneous distributed-boundary optimal control problems, Control Cybernet., 44 (2015), 5-17. 

[12]

C. M. Gariboldi and E. L. Schwindt, Simultaneous optimal controls for non-stationary Stokes systems, Anal. Theory Appl., 33 (2017), 229-239.  doi: 10.4208/ata.2017.v33.n3.4.

[13]

L. GasińskiZ. LiuS. MigórskiA. Ochal and Z. Peng, Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets, J. Optim. Theory Appl., 164 (2015), 514-533.  doi: 10.1007/s10957-014-0587-6.

[14]

L. GasińskiS. Migórski and A. Ochal, Existence results for evolutionary inclusions and variational-hemivariational inequalities, Appl. Anal., 94 (2015), 1670-1694.  doi: 10.1080/00036811.2014.940920.

[15]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman, Boston, MA, 1985.

[16]

J.-L. Lions, Contrȏle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968.

[17]

J.-L. Menaldi and D. A. Tarzia, A distributed parabolic control with mixed boundary conditions, Asymptot. Anal., 52 (2007), 227-241. 

[18]

Ş. S. Şener and M. Subaşi, On a Neumann boundary control in a parabolic system, Bound. Value Probl., 2015 (2015), 16 pp. doi: 10.1186/s13661-015-0430-5.

[19]

M. Sofonea and A. Benraouda, Convergence results for elliptic quasivariational inequalities, Z. Angew. Math. Phys., 68 (2017), Art. 10, 19 pp. doi: 10.1007/s00033-016-0750-z.

[20] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018. 
[21]

N. H. Sweilam and L. F. Abd-Elal, A computational approach for optimal control systems governed by parabolic variational inequalities, J. Comput. Math., 21 (2003), 815-824. 

[22]

E. D. Tabacman and D. A. Tarzia, Sufficient and/ or necessary condition for the heat transfer coefficient on $\Gamma_1$ and the heat flux on $\Gamma_2$ to obtain a steady-state two-phase Stefan problem, J. Differential Equations, 77 (1989), 16-37.  doi: 10.1016/0022-0396(89)90155-1.

[23]

D. A. Tarzia, Sur le problème de Stefan à deux phases, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), A941–A944.

[24]

F. Trölstzsch, Optimal Control of Partial Differetnial Equations. Theory, Methods and Applications, American Math. Soc., Providence, 2010.

[25]

S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, Control Cybernet., 40 (2011), 1109-1124. 

[26]

L. Wang and Q. Yan, Optimal control problem for exact synchronization of parabolic system, Math. Control Relat. Fields, 9 (2019), 411-424.  doi: 10.3934/mcrf.2019019.

[27]

Y. Zhu, R. Du and L. Bao, Approximate controllability of a class of coupled degenerate systems, Bound. Value Probl., 2016 (2016), 7 pp. doi: 10.1186/s13661-016-0637-0.

show all references

References:
[1]

A. Azzam and E. Kreyszig, On solutions of elliptic equations satisfying mixed boundary conditions, SIAM J. Math. Anal., 13 (1982), 254-262.  doi: 10.1137/0513018.

[2]

I. Babŭska and S. Ohnimus, A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations, Comput. Methods Appl. Mech. Engrg., 190 (2001), 4691-4712.  doi: 10.1016/S0045-7825(00)00340-6.

[3]

C. BacutaJ. H. Bramble and J. E. Pasciak, Using finite element tools in proving shift theorems for elliptic boundary value problems, Numer. Linear Algebra Appl., 10 (2003), 33-64.  doi: 10.1002/nla.311.

[4]

F. B. BelgacemH. E. Fekih and J. P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptot. Anal., 34 (2003), 121-136. 

[5]

A. Bensoussan and J.-L. Lions, Applications of Variational Inequalities in Stochastic Control, Vol. 12, Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1982.

[6]

M. Bergounioux and F. Tröltzsch, Optimal control of semilinear parabolic equations with state-constraints of bottleneck type, ESAIM: Control Optim. Calc. Var., 4 (1999), 595-608.  doi: 10.1051/cocv:1999124.

[7]

M. Boukrouche and D. A. Tarzia, Existence, uniqueness, and convergence of optimal control problems associated with parabolic variational inequalities of the second kind, Nonlinear Anal. Real World Appl., 12 (2011), 2211-2224.  doi: 10.1016/j.nonrwa.2011.01.003.

[8]

M. Boukrouche and D. A. Tarzia, Convergence of optimal control problems governed by second kind parabolic variational inequalities, J. Control Theory Appl., 11 (2013), 422-427.  doi: 10.1007/s11768-013-2155-2.

[9]

K. ChrysafinosM. D. Gunzburger and L. S. Hou, Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE, J. Math. Anal. Appl., 323 (2006), 891-912.  doi: 10.1016/j.jmaa.2005.10.053.

[10]

C. M. Gariboldi and D. A. Tarzia, Convergence of boundary optimal control problems with restrictions in mixed elliptic Stefan-like problems, Adv. Differ. Equ. Control Process., 1 (2008), 113-132. 

[11]

C. M. Gariboldi and D. A. Tarzia, Existence, uniqueness and convergence of simultaneous distributed-boundary optimal control problems, Control Cybernet., 44 (2015), 5-17. 

[12]

C. M. Gariboldi and E. L. Schwindt, Simultaneous optimal controls for non-stationary Stokes systems, Anal. Theory Appl., 33 (2017), 229-239.  doi: 10.4208/ata.2017.v33.n3.4.

[13]

L. GasińskiZ. LiuS. MigórskiA. Ochal and Z. Peng, Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets, J. Optim. Theory Appl., 164 (2015), 514-533.  doi: 10.1007/s10957-014-0587-6.

[14]

L. GasińskiS. Migórski and A. Ochal, Existence results for evolutionary inclusions and variational-hemivariational inequalities, Appl. Anal., 94 (2015), 1670-1694.  doi: 10.1080/00036811.2014.940920.

[15]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman, Boston, MA, 1985.

[16]

J.-L. Lions, Contrȏle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968.

[17]

J.-L. Menaldi and D. A. Tarzia, A distributed parabolic control with mixed boundary conditions, Asymptot. Anal., 52 (2007), 227-241. 

[18]

Ş. S. Şener and M. Subaşi, On a Neumann boundary control in a parabolic system, Bound. Value Probl., 2015 (2015), 16 pp. doi: 10.1186/s13661-015-0430-5.

[19]

M. Sofonea and A. Benraouda, Convergence results for elliptic quasivariational inequalities, Z. Angew. Math. Phys., 68 (2017), Art. 10, 19 pp. doi: 10.1007/s00033-016-0750-z.

[20] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018. 
[21]

N. H. Sweilam and L. F. Abd-Elal, A computational approach for optimal control systems governed by parabolic variational inequalities, J. Comput. Math., 21 (2003), 815-824. 

[22]

E. D. Tabacman and D. A. Tarzia, Sufficient and/ or necessary condition for the heat transfer coefficient on $\Gamma_1$ and the heat flux on $\Gamma_2$ to obtain a steady-state two-phase Stefan problem, J. Differential Equations, 77 (1989), 16-37.  doi: 10.1016/0022-0396(89)90155-1.

[23]

D. A. Tarzia, Sur le problème de Stefan à deux phases, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), A941–A944.

[24]

F. Trölstzsch, Optimal Control of Partial Differetnial Equations. Theory, Methods and Applications, American Math. Soc., Providence, 2010.

[25]

S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, Control Cybernet., 40 (2011), 1109-1124. 

[26]

L. Wang and Q. Yan, Optimal control problem for exact synchronization of parabolic system, Math. Control Relat. Fields, 9 (2019), 411-424.  doi: 10.3934/mcrf.2019019.

[27]

Y. Zhu, R. Du and L. Bao, Approximate controllability of a class of coupled degenerate systems, Bound. Value Probl., 2016 (2016), 7 pp. doi: 10.1186/s13661-016-0637-0.

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