Taiwanese Journal of Mathematics

THE EXISTENCE RESULTS FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY QUASI-VARIATIONAL INEQUALITIES IN REFLEXIVE BANACH SPACES

Zhong-bao Wang, Nan-jing Huang, and Ching-Feng Wen

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Abstract

In this paper, some existence results for optimal control problems governed by abstract quasi-variational inequalities are proved in reflexive Banach spaces. As an application, an existence of the optimal control for the bilateral obstacle optimal control problem is also given under some suitable conditions, in which the state satisfies a quasilinear elliptic variational inequality with a source term.

Article information

Source
Taiwanese J. Math., Volume 16, Number 4 (2012), 1221-1243.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406733

Digital Object Identifier
doi:10.11650/twjm/1500406733

Mathematical Reviews number (MathSciNet)
MR2951137

Zentralblatt MATH identifier
1258.49009

Subjects
Primary: 49J40: Variational methods including variational inequalities [See also 47J20] 47J20: Variational and other types of inequalities involving nonlinear operators (general) [See also 49J40] 49J27: Problems in abstract spaces [See also 90C48, 93C25]

Keywords
optimal control existence obstacle problem abstract generalized quasi-variational inequalities quasilinear elliptic variational inequality Mosco-convergence

Citation

Wang, Zhong-bao; Huang, Nan-jing; Wen, Ching-Feng. THE EXISTENCE RESULTS FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY QUASI-VARIATIONAL INEQUALITIES IN REFLEXIVE BANACH SPACES. Taiwanese J. Math. 16 (2012), no. 4, 1221--1243. doi:10.11650/twjm/1500406733. https://projecteuclid.org/euclid.twjm/1500406733


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References

  • V. Barbu, Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities, J. Math. Anal. Appl., 80 (1981), 566-597.
  • M. Bergounioux and H. Zidani, Pontragin maximum principle for optimal control of variational inequalities, SIAM J. Control Optim., 37 (1999), 1273-1290.
  • A. Friedman, Optimal control for variational inequalities, SIAM J. Control Optim., 25 (1986), 482-497.
  • Q. Chen, Indirect obstacle control problem for semilinear elliptic variational inequalities, SIAM J. Control Optim., 38 (1999), 138-158.
  • J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.
  • J. L. Lions, Some Aspects of the Optimal Control of Distributed Parameter Systems, CBMS\"CNSF Regional Conf. Ser. in Appl. Math., SIAM, Philadelphia, PA, 1972.
  • D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980.
  • K. A. Hadit, Optimal control of the obstacle problem: optimality conditions, IMA J. Math. Control Inform., 23 (2006), 325-334.
  • J. Rodrigues, Obstacle Problem in Mathematical Physics, North-Holland, 1987, p. 134.
  • Y. Y. Zhou, X. Q. Yang and K. L. Teo, The existence results for optimal control problems governed by a variational inequality, J. Math. Anal. Appl., 321 (2006), 595-608.
  • D. R. Adams, S. M. Lenhart and J. M. Yong, Optimal control of obstacle for elliptic variational inequality, Appl. Math. Optim., 38 (1998), 121-140.
  • D. R. Adams and S. M. Lenhart, An obstacle control problem with a source term, Appl. Math. Optim., 47 (2002), 59-78.
  • M. Bergounioux and S. M. Lenhart, Optimal control of bilateral obstacle problems, SIAM J. Control Optim., 43(1) (2004), 240-255.
  • Q. Chen, Optimal obstacle control for quasi-linear elliptic variational bilateral problems, SIAM J. Control Optim., 44(3) (2005), 1067-1080.
  • H. Lou, An obstacle control problem governed by quasilinear variational inequalities, SIAM J. Control Optim., 41 (2002), 1229-1253.
  • D. R. Adams and S. M. Lenhart, An obstacle control problem with a source term, Appl. Math. Optim., 47 (2003), 79-95.
  • Z. X. He, State constrained control problems governed by variational inequalities, SIAM J. Control Optim., 25 (1987), 1119-1144.
  • V. Barbu, Optimal Control of Variational Inequalities, Res. Notes Math., Vol. 100, Pitman, Boston, 1984.
  • Y. Q. Ye and Q. Chen, Optimal control of the obstacle in a quasilinear elliptic variational inequality, J. Math. Anal. Appl., 294 (2004), 258-272.
  • Y. Q. Ye, C. K. Chan and H. W. J. Lee, The existence results for obstacle optimal control problems, Appl. Math. Comput., 214 (2009), 451-456.
  • Y. Q. Ye, C. K. Chan, B. P. K. Leung and Q. Chen, Bilateral obstacle optimal control for a quasilinear elliptic variational inequality with a source term, Nonlinear Anal., 66 (2007), 1170-1184.
  • Y. Q. Ye, C. K. Chan and B. P. K. Leung, Some optimality conditions of quasilinear elliptic obstacle optimal control problems, Appl. Math. Comput., 188 (2007), 1757-1771.
  • Q. Chen and Y. Q. Ye, Bilateral obstacle optimal control for a quasilinear elliptic variational inequalities, Numer. Func. Anal. Optim., 26(3) (2005), 303-320.
  • E. Zeidler, Nonlinear functional analysis and its applications, II/B. nonlinear monotone operators, Springer-Verlag, 1990.
  • S. Adly, M. Bergounioux and M. A. Mansour, Optimal control of a quasi-variational obstacle problem, J. Glob. Optim., 47 (2010), 421-435.
  • B. T. Kien, M. M. Wong, N. C. Wong and J. C. Yao, Solution existence of variational inequalities with pseudomonotone operators in the sense of Br$\acute{e}$zis, J. Optim. Theory Appl., 140 (2009), 249-263.
  • H. Kneser, Sur un theoreme fondamantel de la theorie des jeux, C. R. Acad. Sci., 234 (1952), 2418-2420.