# American Institute of Mathematical Sciences

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  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 & 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.  Google Scholar [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.  Google Scholar [3] C. Bacuta, J. 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.  Google Scholar [4] F. B. Belgacem, H. 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.   Google Scholar [5] A. Bensoussan and J.-L. 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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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [13] L. Gasiński, Z. Liu, S. Migórski, A. 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.  Google Scholar [14] L. Gasiński, S. 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.  Google Scholar [15] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman, Boston, MA, 1985.  Google Scholar [16] J.-L. Lions, Contrȏle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968.  Google Scholar [17] J.-L. Menaldi and D. A. Tarzia, A distributed parabolic control with mixed boundary conditions, Asymptot. Anal., 52 (2007), 227-241.   Google Scholar [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.  Google Scholar [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.  Google Scholar [20] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018.   Google Scholar [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.   Google Scholar [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.  Google Scholar [23] D. A. Tarzia, Sur le problème de Stefan à deux phases, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), A941–A944.  Google Scholar [24] F. Trölstzsch, Optimal Control of Partial Differetnial Equations. Theory, Methods and Applications, American Math. Soc., Providence, 2010. Google Scholar [25] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, Control Cybernet., 40 (2011), 1109-1124.   Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] C. Bacuta, J. 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.  Google Scholar [4] F. B. Belgacem, H. 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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] K. Chrysafinos, M. 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.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  Google Scholar [13] L. Gasiński, Z. Liu, S. Migórski, A. 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.  Google Scholar [14] L. Gasiński, S. 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.  Google Scholar [15] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Vol. 24, Monographs and Studies in Mathematics, Pitman, Boston, MA, 1985.  Google Scholar [16] J.-L. Lions, Contrȏle Optimal de Systèmes Gouvernés par des Équations aux Dérivées Partielles, Dunod, Paris, 1968.  Google Scholar [17] J.-L. Menaldi and D. A. Tarzia, A distributed parabolic control with mixed boundary conditions, Asymptot. Anal., 52 (2007), 227-241.   Google Scholar [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.  Google Scholar [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.  Google Scholar [20] M. Sofonea and S. Migórski, Variational-hemivariational Inequalities with Applications, CRC Press, Boca Raton, FL, 2018.   Google Scholar [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.   Google Scholar [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.  Google Scholar [23] D. A. Tarzia, Sur le problème de Stefan à deux phases, C. R. Acad. Sci. Paris Ser. A-B 288 (1979), A941–A944.  Google Scholar [24] F. Trölstzsch, Optimal Control of Partial Differetnial Equations. Theory, Methods and Applications, American Math. Soc., Providence, 2010. Google Scholar [25] S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems, Control Cybernet., 40 (2011), 1109-1124.   Google Scholar [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.  Google Scholar [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.  Google Scholar
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