Skip to main content
SpringerLink
Log in

Optimal Control for Fractional Diffusion Equations with Incomplete Data

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We are concerned with the optimal control of time-fractional diffusion equations with missing boundary condition. Using the notion of no-regret control and least (or low) regret control developed by Lions, we first prove that the least regret control problem associated with the boundary fractional diffusion equation has a unique solution. Then we show that this solution converges to the no-regret control which we characterize by a singular optimality system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Zamani, M., Karimi-Ghartemani, M., Sadati, N.: FOPID controller design for robust performance using particle swarm optimization. J. Fract. Calc. Appl. Anal. 10(2), 169–188 (2007)

    MathSciNet  MATH  Google Scholar 

  2. Oustaloup, A., Levron, F., Mathieu, B.: Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Trans. Circuits Syst. I. 47(1), 25–39 (2000)

    Article  Google Scholar 

  3. Anh, V.V., Leonenko, N.N.: Spectral analysis of fractional kinetic equations with random data. J. Stat. Phys. 104, 1349–1387 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion:a fractional dynamics approch. Phys. Rep. 339, 1–77 (2000)

    Article  MATH  Google Scholar 

  5. Sokolov, I.M., Klafter, J., Blumen, A.: Fractional kinetics. Phys. Today 55, 48–54 (2002)

    Article  Google Scholar 

  6. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  7. Mainardi, F., Paradisi, P.: Model of diffusion waves in viscoelasticity based on fractal calculus. Proc. IEEE Conf. Decis. Control 5, 4961–4966 (1997)

    Google Scholar 

  8. Wyss, W.: The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Metzler, R., Klafter, J.: Boundary value problems for fractional diffusion equations. Phys. A 278, 107–125 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Baeumer, B., Kurita, S., Meerschaert, M.: Inhomogeneous fractional diffusion equations. Fract. Calc. Appl. Anal. 8(4), 371–386 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38, 323–337 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. 272, 368–379 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Frederico Gastao, S.F., Torres Delfim, F.M.: Fractional optimal control in the sense of Caputo and the fractional Noether’s theorem. Int. Math. Forum 3(10), 479–493 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Ozdemir, N., Karadeniz, D., Iskender, B.B.: Fractional optimal control problem of a distributed system in cylindrical coordinates. Phys. Lett. A 373, 221–226 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Agrawal, O.P., Baleanu, D.: A central difference numerical scheme for fractional optimal control problems. J. Vib. Control 15(4), 583–597 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jelicic, Z.D., Petrovacki, N.: Optimality conditions and a solution scheme for fractional optimal control problems. Struct. Multidiscip. Optim. 38(6), 571–581 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Agrawal, O.P., Defterli, O., Baleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16(13), 1967–1976 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, J.R., Zhou, Y.: A class of fractional evolution equations and optimal controls. Nonlinear Anal. RWA 12, 262–272 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lions, J.L.: Contrôle à moindres regrets des systèmes distribués. C. R. Acad. Sci. Paris Ser. I Math. 315, 1253–1257 (1992)

    MathSciNet  MATH  Google Scholar 

  20. Lions, J.L.: No-Regret and Low-Regret Control. Environment, Economics and Their Mathematical Models. Masson, Paris (1994)

    Google Scholar 

  21. Nakoulima, O., Omrane, A., Velin, J.: Perturbations à moindres regrets dans les syst‘emes distribués à données manquantes. C. R. Acad. Sci. Paris Ser. I Math. 330, 801–806 (2000)

    Article  MATH  Google Scholar 

  22. Nakoulima, O., Omrane, A., Velin, J.: No-regret control for nonlinear distributed systems with incomplete data. J. Math. Pures Appl. 81, 1161–1189 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nakoulima, O., Omrane, A., Dorville, R.: Low-regret control of singular distributed systems: the Ill-posed backwards heat problem. Appl. Mathe. Lett. 17, 549–552 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Jacob, B., Omrane, A.: Optimal control for age-structured population dynamics of incomplete data. J. Math. Anal. Appl. 370(1), 42–48 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gabay, D., Lions, J.L.: Décisions stratégiques à moindres regrets. C. R. Acad. Sci. Paris Ser. I 319, 1249–1256 (1994)

    MATH  Google Scholar 

  26. Lions, J.L.: Least regret control, virtual control and decomposition methods. Math. Model. Numer. Anal M2AN 34(2), 409–418 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nakoulima, O., Omrane, A., Velin, J.: Low-regret perturbations in distributed systems with incomplete data. SIAM J. Control Optim. 42(4), 1167–1184 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nakoulima, O., Omrane, A., Dorville, R.: Contrôle optimal pour les problèmes de contrôlabilité des systèmes distribués à données manquantes. C. R. Acad. Sci. Paris Ser. I 338, 921–924 (2004)

    Article  MATH  Google Scholar 

  29. Lions, J.L.: Duality Arguments for Multi Agents Least-Regret Control. Collège de France, Paris (1999)

    Google Scholar 

  30. Miller, K., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  31. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integral and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Switzerland (1993)

    MATH  Google Scholar 

  32. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  33. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V, Amsterdam (2006)

    MATH  Google Scholar 

  34. Mophou, G.M.: Optimal control of fractional diffusion equation. Comput. Math. Appl. 61(1), 68–78 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Mophou, G., N’Guérékata, G.: Optimal control of a fractional diffusion equation with state constraints. Comput. Math. Appl. 62, 1413–1426 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nakagawa J., Sakamoto K., Yamamoto M.: Overview to mathematical analysis for fractional diffusion equations–new mathematical aspects motivated by industrial collaboration. J. Math. Ind. 2(A-10), 99–108 (2010)

  37. Mophou, G., Tao, S., Joseph, C.: Initial value/boundary value problem for composite fractional relaxation equation. Appl. Math. Comput. 257, 134–144 (2015)

    MathSciNet  MATH  Google Scholar 

  38. Dorville, R., Mophou, G.M., Valmorin, S.: Optimal control of a nonhomogeneous Dirichlet boundary fractional diffusion equation. Comput. Math. Appl. 62(3), 1472–1481 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications. Dunod, Paris (1968)

    MATH  Google Scholar 

Download references

Acknowledgments

The author is grateful to the referees for their valuable suggestions.

Author information

Authors and Affiliations

  1. Laboratoire CEREGMIA, Université des Antilles et de la Guyane, Campus Fouillole, 97159, Pointe-à-Pitre, Guadeloupe (FWI), France

    Gisèle Mophou

  2. Laboratoire MAINEGE, Université Ouaga 3S, 06 BP 10347, Ouagadougou 06, Burkina Faso

    Gisèle Mophou

Authors
  1. Gisèle Mophou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gisèle Mophou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mophou, G. Optimal Control for Fractional Diffusion Equations with Incomplete Data. J Optim Theory Appl 174, 176–196 (2017). https://doi.org/10.1007/s10957-015-0817-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0817-6

Keywords

Mathematics Subject Classification

Search

Navigation