×

Tykhonov well-posedness of a heat transfer problem with unilateral constraints. (English) Zbl 07511500

Summary: We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain \(D\subset\mathbb{R}^d\) and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by \(\mathcal{P}\). We associate to Problem \(\mathcal{P}\) an optimal control problem, denoted by \(\mathcal{Q}\). Then, using appropriate Tykhonov triples, governed by a nonlinear operator \(G\) and a convex \(\widetilde{K}\), we provide results concerning the well-posedness of problems \(\mathcal{P}\) and \(\mathcal{Q}\). Our main results are Theorems 4.2 and 5.2, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem \(\mathcal{P}\), constructed with particular choices of \(G\) and \(\widetilde{K}\). We prove that Theorems 4.2 and 5.2 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results.

MSC:

49J40 Variational inequalities
49J20 Existence theories for optimal control problems involving partial differential equations
49J52 Nonsmooth analysis
49J45 Methods involving semicontinuity and convergence; relaxation
35A16 Topological and monotonicity methods applied to PDEs
35M86 Unilateral problems for nonlinear PDEs of mixed type and variational inequalities with nonlinear partial differential operators of mixed type
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baiocchi, C.; Capelo, A., Variational and Quasivariational Inequalities: Applications to Free Boundary Problems, A Wiley-Interscience Publication. John Wiley, Chichester (1984) · Zbl 0551.49007
[2] Barbu, V., Optimal Control of Variational Inequalities, Research Notes in Mathematics 100. Pitman, Boston (1984) · Zbl 0574.49005
[3] Boukrouche, M.; Tarzia, D. A., 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 · Zbl 1225.49013 · doi:10.1016/j.nonrwa.2011.01.003
[4] Boukrouche, M.; Tarzia, D. A., Convergence of distributed optimal control problems governed by elliptic variational inequalities, Comput. Optim. Appl. 53 (2012), 375-393 · Zbl 1260.49009 · doi:10.1007/s10589-011-9438-7
[5] Capatina, A., Variational Inequalities and Frictional Contact Problems, Advances in Mechanics and Mathematics 31. Springer, Cham (2014) · Zbl 1405.49001 · doi:10.1007/978-3-319-10163-7
[6] Čoban, M. M.; Kenderov, P. S.; Revalski, J. P., Generic well-posedness of optimization problems in topological spaces, Mathematika 36 (1989), 301-324 · Zbl 0679.49010 · doi:10.1112/S0025579300013152
[7] Dontchev, A. L.; Zolezzi, T., Well-Posed Optimization Problems, Lecture Notes in Mathematics 1543. Springer, Berlin (1993) · Zbl 0797.49001 · doi:10.1007/BFb0084195
[8] Friedman, A., Optimal control for variational inequalities, SIAM J. Control Optim. 24 (1986), 439-451 · Zbl 0604.49007 · doi:10.1137/0324025
[9] Glowinski, R., Numerical Methods for Nonlinear Variational Problems, Springer Series in Computational Physics. Springer, New York (1984) · Zbl 0536.65054 · doi:10.1007/978-3-662-12613-4
[10] Goeleven, D.; Mentagui, D., Well-posed hemivariational inequalities, Numer. Funct. Anal. Optim. 16 (1995), 909-921 · Zbl 0848.49013 · doi:10.1080/01630569508816652
[11] Han, W.; Reddy, B. D., Plasticity: Mathematical Theory and Numerical Analysis, Interdisciplinary Applied Mathematics 9. Springer, New York (2013) · Zbl 1258.74002 · doi:10.1007/978-1-4614-5940-8
[12] Han, W.; Sofonea, M., Numerical analysis of hemivariational inequalities in contact mechanics, Acta Numerica 28 (2019), 175-286 · Zbl 1433.65296 · doi:10.1017/S0962492919000023
[13] Haslinger, J.; Miettinen, M.; Panagiotopoulos, P. D., Finite Element Method for Hemivariational Inequalities: Theory, Methods and Applications, Nonconvex Optimization and Its Applications 35. Kluwer Academic Publishers, Dordrecht (1999) · Zbl 0949.65069 · doi:10.1007/978-1-4757-5233-5
[14] Hlaváček, I.; Haslinger, J.; Nečas, J.; Lovíšek, J., Solution of Variational Inequalities in Mechanics, Applied Mathematical Sciences 66. Springer, New York (1988) · Zbl 0654.73019 · doi:10.1007/978-1-4612-1048-1
[15] Huang, X. X., Extended and strongly extended well-posedness of set-valued optimization problems, Math. Methods Oper. Res. 53 (2001), 101-116 · Zbl 1018.49019 · doi:10.1007/s001860000100
[16] Huang, X. X.; Yang, X. Q., Generalized Levitin-Polyak well-posedness in constrained optimization, SIAM J. Optim. 17 (2006), 243-258 · Zbl 1137.49024 · doi:10.1137/040614943
[17] Kikuchi, N.; Oden, J. T., Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, SIAM Studies in Applied Mathematics 8. SIAM, Philadelphia (1988) · Zbl 0685.73002 · doi:10.1137/1.9781611970845
[18] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Classics in Applied Mathematics 31. SIAM, Philadelphia (2000) · Zbl 0988.49003 · doi:10.1137/1.9780898719451
[19] Lions, J.-L., Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles, Dunod, Paris (1968), French · Zbl 0179.41801
[20] Lucchetti, R., Convexity and Well-Posed Problems, CMS Books in Mathehmatics 22. Springer, New York (2006) · Zbl 1106.49001 · doi:10.1007/0-387-31082-7
[21] Lucchetti, R.; Patrone, F., A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities, Numer. Funct. Anal. Optim. 3 (1981), 461-476 · Zbl 0479.49025 · doi:10.1080/01630568108816100
[22] Lucchetti, R.; Patrone, F., Some properties of “well-posed” variational inequalities governed by linear operators, Numer. Funct. Anal. Optim. 5 (1983), 349-361 · Zbl 0517.49007 · doi:10.1080/01630568308816145
[23] Matei, A.; Micu, S., Boundary optimal control for nonlinear antiplane problems, Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 1641-1652 · Zbl 1428.74162 · doi:10.1016/j.na.2010.10.034
[24] Matei, A.; Micu, S.; Niţă, C., Optimal control for antiplane frictional contact problems involving nonlinearly elastic materials of Hencky type, Math. Mech. Solids 23 (2018), 308-328 · Zbl 1404.74114 · doi:10.1177/1081286517718605
[25] Mignot, F., Contrôle dans les inéquations variationnelles elliptiques, J. Funct. Anal. 22 (1976), 130-185 French · Zbl 0364.49003 · doi:10.1016/0022-1236(76)90017-3
[26] Mignot, F.; Puel, J.-P., Optimal control in some variational inequalities, SIAM J. Control Optim. 22 (1984), 466-476 · Zbl 0561.49007 · doi:10.1137/0322028
[27] Migórski, S.; Ochal, A.; Sofonea, M., Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26. Springer, New York (2013) · Zbl 1262.49001 · doi:10.1007/978-1-4614-4232-5
[28] Naniewicz, Z.; Panagiotopoulos, P. D., Mathematical Theory of Hemivariational Inequalities and Applications, Pure and Applied Mathematics, Marcel Dekker 188. Marcel Dekker, New York (1994) · Zbl 0968.49008
[29] Neitaanmäki, P.; Sprekels, J.; Tiba, D., Optimization of Elliptic Systems: Theory and Applications, Springer Monographs in Mathematics. Springer, New York (2006) · Zbl 1106.49002 · doi:10.1007/b138797
[30] Panagiotopoulos, P. D., Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions, Birkhäuser, Boston (1985) · Zbl 0579.73014 · doi:10.1007/978-1-4612-5152-1
[31] Panagiotopoulos, P. D., Hemivariational Inequalities: Applications in Mechanics and Engineering, Springer, Berlin (1993) · Zbl 0826.73002 · doi:10.1007/978-3-642-51677-1
[32] Peng, Z., Optimal obstacle control problems involving nonsmooth functionals and quasilinear variational inequalities, SIAM J. Control Optim. 58 (2020), 2236-2255 · Zbl 1454.49017 · doi:10.1137/19M1249102
[33] Peng, Z.; Kunisch, K., Optimal control of elliptic variational-hemivariational inequalities, J. Optim. Theory Appl. 178 (2018), 1-25 · Zbl 06931878 · doi:10.1007/s10957-018-1303-8
[34] Sofonea, M., Optimal control of a class of variational-hemivariational inequalities in reflexive Banach spaces, Appl. Math. Optim. 79 (2019), 621-646 · Zbl 07068063 · doi:10.1007/s00245-017-9450-0
[35] Sofonea, M., Optimal control of quasivariational inequalities with applications to contact mechanics, Current Trends in Mathematical Analysis and Its Interdisciplinary Applications Birkhäuser, Cham (2019), 445-489 · Zbl 1447.35319 · doi:10.1007/978-3-030-15242-0_13
[36] Sofonea, M.; Bollati, J.; Tarzia, D. A., Optimal control of differential quasivariational inequalities with applications in contact mechanics, J. Math. Anal. Appl. 493 (2021), Article ID 124567, 23 pages · Zbl 1471.49008 · doi:10.1016/j.jmaa.2020.124567
[37] Sofonea, M.; Migórski, S., Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2018) · Zbl 1384.49002 · doi:10.1201/9781315153261
[38] Sofonea, M.; Tarzia, D. A., Convergence results for optimal control problems governed by elliptic quasivariational inequalities, Numer. Func. Anal. Optim. 41 (2020), 1326-1351 · Zbl 1512.49009 · doi:10.1080/01630563.2020.1772288
[39] Sofonea, M.; Tarzia, D. A., On the Tykhonov well-posedness of an antiplane shear problem, Mediterr. J. Math. 17 (2020), Article ID 150, 21 pages · Zbl 1447.74037 · doi:10.1007/s00009-020-01577-5
[40] Sofonea, M.; Xiao, Y.-B., On the well-posedness concept in the sense of Tykhonov, J. Optim. Theory Appl. 183 (2019), 139-157 · Zbl 1425.49005 · doi:10.1007/s10957-019-01549-0
[41] Sofonea, M.; Xiao, Y.-B., Tykhonov well-posedness of elliptic variational-hemivariational inequalities, Electron. J. Differ. Equ. 2019 (2019), Article ID 64, 19 pages · Zbl 1415.49009
[42] Tikhonov, A. N., On the stability of functional optimization problems, U.S.S.R. Comput. Math. Math. Phys. 6 (1966), 28-33 translation from Zh. Vychisl. Mat. Mat. Fiz. 6 1966 631-634 · Zbl 0212.23803 · doi:10.1016/0041-5553(66)90003-6
[43] Xiao, Y.-B.; Huang, N.-J.; Wong, M.-M., Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Math. 15 (2011), 1261-1276 · Zbl 1239.49013 · doi:10.11650/twjm/1500406298
[44] Xiao, Y.-B.; Sofonea, M., Generalized penalty method for elliptic variational-hemivariational inequalities, Appl. Math. Optim. 83 (2021), 789-812 · Zbl 1461.49013 · doi:10.1007/s00245-019-09563-4
[45] Zolezzi, T., Extended well-posedness of optimization problems, J. Optim. Theory Appl. 91 (1996), 257-266 · Zbl 0873.90094 · doi:10.1007/BF02192292
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.