Acta Mathematica Sinica

, Volume 22, Issue 2, pp 607–624 | Cite as

Optimal Control of Variational Inequalities with Delays in the Highest Order Spatial Derivatives



In this paper, an optimal control problem for parabolic variational inequalities with delays in the highest order spatial derivatives is investigated. The well–posedness of such kinds of variational inequalities is established. The existence of optimal controls under a Cesari–type condition is proved, and the necessary conditions of Pontryagin type for optimal controls is derived.


Variational inequalities Time–delay operator Optimal control Maximum principle 

MR (2000) Subject Classification

49J40 49K25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adams, D. R., Lenhart, S. M., Yong, J.: Optimal control of the obstaclfor an elliptic variational inequality. Appl. Math. Optim., 38, 121–140 (1998)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Barbu, V.: Necessary conditions for distributed control problems governed by parabolic variational inequalities. SIAM J. Control Optim., 19, 64–86 (1981)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Chen, Q.: Indirect obstacle control problem for semilinear elliptic variational inequalities. SIAM J. Control Optim., 38, 138–158 (1999)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Friedman, A.: Optimal control for parabolic variational inequalities. SIAM J. Control Optim., 25, 482–497 (1987)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Li, X., Yong, J.: Necessary conditions of optimal control for distributed parameter systems. SIAM J. Control Optim., 29, 895–908 (1991)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Li, X., Yong, J.: Optimal Control Theory For Infinite Dimensional Systems, Birkhäuser, Boston, 1995Google Scholar
  7. 7.
    Gurtin, M. E., Pipkin, A. C.: A general theory of heat conduction with finite wave speeds. Arch. Rat. Mech. Anal., 31, 113–126 (1968)MATHMathSciNetGoogle Scholar
  8. 8.
    Wilmott, P.: Derivatives: The Theory and Practice of Financial Engineering, John Wiley & Sons, University edition, Chichester, New York, 1998Google Scholar
  9. 9.
    Barbu, V.: Optimal Control of Variational Inequalities, Pitman, London, New York, 1984Google Scholar
  10. 10.
    Kinderlehrer, D., Stampacchia, dna G.: Introduction to Variational Inequalities and Their Applications, Academic Press, New York, London, 198Google Scholar
  11. 11.
    Ardito, A., Ricciardi, P.: Existence and regularity for linear delay partial differential equations. Nonlinear Anal., 4, 411–414 (1980)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Di Blasio, G., Kunish, K., Sinestrari, E.: L2-regularity for parabolic partial integro differential equations with delay in the highest-order derivatives. J. Math. Anal. Appl., 102, 38–57 (1984)MATHMathSciNetGoogle Scholar
  13. 13.
    Li, X., Yao, Y.: Maximum principle of distributed parameter systems with time lags, in Distributed parameter systems, Lecture Notes in Control and Inform. Sci., 75, Springer, 410–427, 1985Google Scholar
  14. 14.
    Nunziato, J. W.: On heat conduction in materials with memory. Quart. Appl. Math., 29, 187–204 (1971)MATHMathSciNetGoogle Scholar
  15. 15.
    Yong, J., Pan, L.: Quasi-linear parabolic partial differential equations with delays in the highest-order spatial derivatives. J. Austral. Math. Soc., Ser. A, 54, 174–203 (1993)MATHMathSciNetGoogle Scholar
  16. 16.
    Pan, L., Yong, J.: Optimal control for quasilinear retarded parabolic systems. ANZIAM J., 42, 532–551 (2001)MATHMathSciNetGoogle Scholar
  17. 17.
    Comincioli, V.: A result concerning a variational inequality of evolution for operators of first order in t with retarded terms. Ann. Mat. Pura Appl., IV. Ser., 88, 357–378 (1971)MATHMathSciNetGoogle Scholar
  18. 18.
    Ekeland, I.: Nonconvex minimization problems. Bull. Amer. Math. Soc. (NS), 1, 443–474 (1979)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Institute of MathematicsFudan UniversityShanghai 200433P. R. China
  2. 2.Department of Applied MatematicsShanxi Finance & Economics UniversityTaiyuan 030006P. R. China

Personalised recommendations