Advertisement

Applied Mathematics & Optimization

, Volume 79, Issue 1, pp 41–67 | Cite as

Optimal Control Problems for a Semilinear Evolution System with Infinite Delay

  • Fatima Zahra Mokkedem
  • Xianlong FuEmail author
Article
  • 141 Downloads

Abstract

In this paper we study the standard optimal control and time optimal control problems for a class of semilinear evolution systems with infinite delay. We first establish the results of existence and uniqueness of mild solution and the compactness of the solution operator for the control system. Then, based on these results, we investigate the optimal control problem with integral cost function and the time optimal control problem respectively. Under some conditions we show the existence of optimal controls for the both cases of bounded and unbounded admissible control sets. We also obtain the existence of time optimal control to a target set. In addition, a convergence theorem of time optimal controls to a point target set is proved. Finally, an example is given to show the application of the main results.

Keywords

Evolution system Infinite delay Fundamental solution Optimal control Time optimal control 

Mathematics Subject Classification

34K30 34K35 35R10 93B05 93C10 

Notes

Acknowledgements

This work is supported by NNSF of China (Nos. 11671142 and 11371087), STCSM ( No. 13dz2260400), Shanghai Leading Academic Discipline Project (No. B407).

References

  1. 1.
    Ahmed, N.U., Teo, K.L.: Optimal Control of Distributed Parameter Systems. North-Holland, New York (1981)zbMATHGoogle Scholar
  2. 2.
    Caicedo, A., Cuevas, C., Mophow, G.M., N’Gu\(\acute{e}\)r\(\acute{e}\)kata, G.M.: Asymptotic behavior of solutions of some semilinear functional differential and integro-differential equations with infinite delay in Banach spaces. J. Frank. Inst. 349, 1–24 (2012)Google Scholar
  3. 3.
    Curtain, R.F., Pritchard, A.J.: Infinite Dimensional Linear Systems Theory. Lecture Notes in Control and Information Science, vol. 8. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dong, Q., Li, G.: Existence of solutions for nonlinear evolution equations with infinite delay. Bull. Korean Math. Soc. 51, 43–54 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Guliyev, H.F., Tagiyev, H.T.: An optimal control problem with nonlocal conditions for the weakly nonlinear hyperbolic equation. Optim. Control Appl. Methods 34, 216–235 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hale, J., Kato, J.: Phase space for retarded equations with infinite delay. Funk. Ekvac. 21, 11–41 (1978)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Henriquez, H.R.: Periodic solutions of quasi-linear partial functional differential equations with unbounded delay. Funkc. Ekvac. 37, 329–343 (1994)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Henriquez, H.R.: Regularity of solutions of abstract retarded functional differential equations with unbounded delay. Nonlinear Anal. 28, 513–531 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Jeong, J.M., Ju, E.Y., Cheon, S.J.: Optimal control problems for evolution equations of parabolic type with nonlinear perturbations. J. Optim. Theory Appl. 151, 573–588 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jeong, J.M., Son, S.J.: Time optimal control of semilinear control systems involving time delays. J. Optim. Theory Appl. 165, 793–811 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Krakowiak, A.: Time optimal control of retarded parabolic systems. IMA J. Math. Control Inf. 24, 357–369 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kunisch, K., Wang, L.: Bang-bang property of time optimal controls of semilinear parabolic equation. Discret. Contin. Dyn. Syst. 36, 279–302 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  16. 16.
    Liu, K.: The fundamental solution and its role in the optimal control of infinite dimensional neutral systems. Appl. Math. Optim. 609, 1–38 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Liu, J., Xiao, M.: A leapfrog semi-smooth Newton-multigrid method for semilinear parabolic optimal control problems. Comput. Optim. Appl. 63, 69–95 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lunardi, A.: On the linear heat equation with fading memory. SIAM J. Math. Anal. 21, 1213–1224 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Mokkedem, F.Z., Fu, X.: Approximate controllability for a semilinear evolution system with infinite delay. J. Dyn. Control Sys. 22, 71–89 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mordukhovich, B.S., Wang, D., Wang, L.: Optimal control of delay-differential inclusions with functional endpoint constraints in infinite dimensions. Nonlinear Anal. 71, 2740–2749 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nakagiri, S.: Optimal control of linear retarded systems in Banach spaces. J. Math. Anal. Appl. 120, 169–210 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nunziato, J.W.: On heat conduction in materials with memory. Q. Appl. Math. 29, 187–304 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-, New York (1983)CrossRefzbMATHGoogle Scholar
  24. 24.
    Shin, J.S.: On the uniqueness of solutions for functional differential equations with infinite delay. Funkc. Ekvac. 30, 225–236 (1987)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Travis, C.C., Webb, G.F.: Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395–418 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vinter R.B.: Optimal Control. Birkh\(\ddot{a}\)user, Boston (2000)Google Scholar
  27. 27.
    Wang, L.: Approximate controllability results of semilinear integrodifferential equations with infinite delays. Sci. China Ser. F-Inf. Sci. 52, 1095–1102 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wang, P.K.C.: Optimal control of parabolic systems with boundary conditions involving time delays. SIAM J. Control 13, 274–293 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Wang, J., Zhou, Y., Medved’, M.: On the solvability and optimal controls of fractional integrodifferential evolution systems with infinite delay. J. Optim. Theory Appl. 152, 31–50 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Witayakiattilerd, W.: Nonlinear fuzzy differential equation with time delay and optimal control problem. Abstr. Appl. Anal. (2015), Art. ID 659072Google Scholar
  31. 31.
    Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  32. 32.
    Xiaoling, X., Huawu, K.: Delay systems and optimal control. Acta Math. Appl. Sin. 16, 27–35 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Laboratoire Système Dynamique et ApplicationsUniversité de Tlemcen (Aboubekr Belkaid University)TlemcenAlgeria
  2. 2.Department of Mathematics, Shanghai Key Laboratory of PMMPEast China Normal UniversityShanghaiPeople’s Republic of China

Personalised recommendations