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Optimal control of pseudoparabolic variational inequalities involving state constraint. (English) Zbl 1474.49029

Summary: We establish the necessary condition of optimality for optimal control problem governed by some pseudoparabolic differential equations involving monotone graphs. Some approximating control process and examples are given.

MSC:

49J40 Variational inequalities
49K21 Optimality conditions for problems involving relations other than differential equations
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[1] Li, X. J.; Yong, J. M., Optimal Control Theory for Infinite-Dimensional Systems (1995), Boston, Mass, USA: Birkhäauser, Boston, Mass, USA · doi:10.1007/978-1-4612-4260-4
[2] Wang, G. S., Optimal control of parabolic differential equations with two point boundary state constraints, SIAM Journal on Control and Optimization, 38, 5, 1639-1654 (2000) · Zbl 0963.49018 · doi:10.1137/S0363012998338132
[3] Wang, G. S., Optimal control of parabolic variational inequality involving state constraint, Nonlinear Analysis: Theory, Methods & Applications, 42, 5, 789-801 (2000) · Zbl 1018.49004 · doi:10.1016/S0362-546X(99)00124-8
[4] Barbu, V., Analysis and Control of Nonlinear Infinite-Dimensional Systems (1993), Boston, Mass, USA: Academic Press, Boston, Mass, USA · Zbl 0776.49005
[5] Tsutsumi, M.; Matahashi, T., On some nonlinear pseudoparabolic equations, Journal of Differential Equations, 32, 1, 65-75 (1979) · Zbl 0372.34042 · doi:10.1016/0022-0396(79)90051-2
[6] Liu, W., Elementary Feedback Stabilization of the Linear Reaction-convection-diffusion Equation and the Wave Equation. Elementary Feedback Stabilization of the Linear Reaction-convection-diffusion Equation and the Wave Equation, Mathmatiques et Applications, 66 (2010), Berlin, Germany: Springer, Berlin, Germany · Zbl 1225.93002 · doi:10.1007/978-3-642-04613-1
[7] Lyashko, S. I., Generalized Control in Linear Systems (1998 (Russian)), Kiev, Ukraine: Naukova Dumka, Kiev, Ukraine
[8] Sviridyuk, G. A.; Fedorov, V. E., Linear Sobolev Type Equations and Degenerate Semigroups of Operators (2003), VSP International Science · Zbl 1102.47061
[9] Fedorov, V. E.; Plekhanova, M. V., Optimal control of Sobolev type linear equations, Differential Equations, 40, 11, 1627-1637 (2004) · Zbl 1095.49004 · doi:10.1007/s10625-004-0013-1
[10] White, L. W., Control problems governed by a pseudo-parabolic partial differential equation, Transactions of the American Mathematical Society, 250, 235-246 (1979) · Zbl 0425.49020 · doi:10.2307/1998988
[11] White, L. W., Convergence properties of optimal controls of pseudoparabolic problems, Applied Mathematics and Optimization, 7, 2, 141-147 (1981) · Zbl 0464.49007 · doi:10.1007/BF01442112
[12] Bock, I.; Lovíšek, J., On pseudoparabolic optimal control problems, Kybernetika, 29, 3, 222-230 (1993) · Zbl 0810.49005
[13] Kunisch, K.; Wachsmuth, D., Path-following for optimal control of stationary variational inequalities, Computational Optimization and Applications, 51, 3, 1345-1373 (2012) · Zbl 1239.49010 · doi:10.1007/s10589-011-9400-8
[14] de Los Reyes, J. C., Optimal control of a class of variational inequalities of the second kind, SIAM Journal on Control and Optimization, 49, 4, 1629-1658 (2011) · Zbl 1226.49008 · doi:10.1137/090764438
[15] Ito, K.; Kunisch, K., Optimal control of parabolic variational inequalities, Journal de Mathématiques Pures et Appliquées, 93, 4, 329-360 (2010) · Zbl 1247.35062 · doi:10.1016/j.matpur.2009.10.005
[16] Ito, K.; Kunisch, K., An augmented Lagrangian technique for variational inequalities, Applied Mathematics and Optimization, 21, 3, 223-241 (1990) · Zbl 0692.49008 · doi:10.1007/BF01445164
[17] Ito, K.; Kunisch, K., Optimal control of elliptic variational inequalities, Applied Mathematics and Optimization, 41, 3, 343-364 (2000) · Zbl 0960.49003 · doi:10.1007/s002459911017
[18] Guo, X. M.; Zhou, S. X., Optimal control of parabolic variational inequalities with state constraint, Applied Mathematics and Mechanics (English Edition), 24, 7, 756-762 (2003) · Zbl 1042.49006 · doi:10.1007/BF02437807
[19] Adams, D. R.; Lenhart, S., Optimal control of the obstacle for a parabolic variational inequality, Journal of Mathematical Analysis and Applications, 268, 2, 602-614 (2002) · Zbl 1022.49009 · doi:10.1006/jmaa.2001.7833
[20] Casas, E., Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM Journal on Control and Optimization, 35, 4, 1297-1327 (1997) · Zbl 0893.49017 · doi:10.1137/S0363012995283637
[21] He, Z.-X., State constrained control problems governed by variational inequalities, SIAM Journal on Control and Optimization, 25, 5, 1119-1144 (1987) · Zbl 0631.49011 · doi:10.1137/0325061
[22] Brézis, H., Quelques Propriétés des Opérateurs Monotones et des Semi-groupes Non Linéaires, Nonlinear operators and the calculus of variations. Nonlinear operators and the calculus of variations, Lecture Notes in Mathematics, 543, 56-82 (1976), Berlin, Germany: Springer, Berlin, Germany · Zbl 0338.47026
[23] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London A, 272, 1220, 47-78 (1972) · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032
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