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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 641736, 9 pages
Optimal Control of Pseudoparabolic Variational Inequalities Involving State Constraint
1School of Mathematics and Physics, University of South China, Hunan 421001, China
2School of Mathematical Sciences, Fudan University, Shanghai 200433, China
Received 5 March 2013; Accepted 27 May 2013; Published 20 January 2014
Academic Editor: Victor Kovtunenko
Copyright © 2014 Youjun Xu and Shu Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- X. J. Li and J. M. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Birkhäauser, Boston, Mass, USA, 1995.
- G. S. Wang, “Optimal control of parabolic differential equations with two point boundary state constraints,” SIAM Journal on Control and Optimization, vol. 38, no. 5, pp. 1639–1654, 2000.
- G. S. Wang, “Optimal control of parabolic variational inequality involving state constraint,” Nonlinear Analysis: Theory, Methods & Applications, vol. 42, no. 5, pp. 789–801, 2000.
- V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, Boston, Mass, USA, 1993.
- M. Tsutsumi and T. Matahashi, “On some nonlinear pseudoparabolic equations,” Journal of Differential Equations, vol. 32, no. 1, pp. 65–75, 1979.
- W. Liu, Elementary Feedback Stabilization of the Linear Reaction-convection-diffusion Equation and the Wave Equation, vol. 66 of Mathmatiques et Applications, Springer, Berlin, Germany, 2010.
- S. I. Lyashko, Generalized Control in Linear Systems, Naukova Dumka, Kiev, Ukraine, 1998 (Russian).
- G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP International Science, 2003.
- V. E. Fedorov and M. V. Plekhanova, “Optimal control of Sobolev type linear equations,” Differential Equations, vol. 40, no. 11, pp. 1627–1637, 2004.
- L. W. White, “Control problems governed by a pseudo-parabolic partial differential equation,” Transactions of the American Mathematical Society, vol. 250, pp. 235–246, 1979.
- L. W. White, “Convergence properties of optimal controls of pseudoparabolic problems,” Applied Mathematics and Optimization, vol. 7, no. 2, pp. 141–147, 1981.
- I. Bock and J. Lovíšek, “On pseudoparabolic optimal control problems,” Kybernetika, vol. 29, no. 3, pp. 222–230, 1993.
- K. Kunisch and D. Wachsmuth, “Path-following for optimal control of stationary variational inequalities,” Computational Optimization and Applications, vol. 51, no. 3, pp. 1345–1373, 2012.
- J. C. de Los Reyes, “Optimal control of a class of variational inequalities of the second kind,” SIAM Journal on Control and Optimization, vol. 49, no. 4, pp. 1629–1658, 2011.
- K. Ito and K. Kunisch, “Optimal control of parabolic variational inequalities,” Journal de Mathématiques Pures et Appliquées, vol. 93, no. 4, pp. 329–360, 2010.
- K. Ito and K. Kunisch, “An augmented Lagrangian technique for variational inequalities,” Applied Mathematics and Optimization, vol. 21, no. 3, pp. 223–241, 1990.
- K. Ito and K. Kunisch, “Optimal control of elliptic variational inequalities,” Applied Mathematics and Optimization, vol. 41, no. 3, pp. 343–364, 2000.
- X. M. Guo and S. X. Zhou, “Optimal control of parabolic variational inequalities with state constraint,” Applied Mathematics and Mechanics (English Edition), vol. 24, no. 7, pp. 756–762, 2003.
- D. R. Adams and S. Lenhart, “Optimal control of the obstacle for a parabolic variational inequality,” Journal of Mathematical Analysis and Applications, vol. 268, no. 2, pp. 602–614, 2002.
- E. Casas, “Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations,” SIAM Journal on Control and Optimization, vol. 35, no. 4, pp. 1297–1327, 1997.
- Z.-X. He, “State constrained control problems governed by variational inequalities,” SIAM Journal on Control and Optimization, vol. 25, no. 5, pp. 1119–1144, 1987.
- H. Brézis, “Quelques Propriétés des Opérateurs Monotones et des Semi-groupes Non Linéaires,” in Nonlinear operators and the calculus of variations, vol. 543 of Lecture Notes in Mathematics, pp. 56–82, Springer, Berlin, Germany, 1976.
- T. B. Benjamin, J. L. Bona, and J. J. Mahony, “Model equations for long waves in nonlinear dispersive systems,” Philosophical Transactions of the Royal Society of London A, vol. 272, no. 1220, pp. 47–78, 1972.