Skip to main content
SpringerLink
Log in

Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we establish a bang-bang principle of time optimal controls for a controlled parabolic equation of fractional order evolved in a bounded domain Ω of ℝn, with a controller ω to be any given nonempty open subset of ω. The problem is reduced to a new controllability property for this equation, i.e. the null controllability of the system at any given time T > 0 when the control is restricted to be active in ω × E, where E is any given subset of [0, T] with positive (Legesgue) measure. The desired controllability result is established by means of a sharp observability estimate on the eigenfunctions of the Dirichlet Laplacian due to Lebeau and Robbiano, and a delicate result in the measure theory due to Lions.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Li, X., Yong, J.: Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995

    Google Scholar 

  2. Phung, K., Wang, G., Zhang, X.: On the existence of time optimal control of some linear evolution equations. Discrete and Continuous Dynamical Systems, Series B, 8, 925–941 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Wang, G.: L ∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim., 47, 1701–1720 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Li, X., Yao, Y.: Time optimal control for distributed parameter systems. Scientia Sinica, 24, 455–465 (1981)

    MATH  MathSciNet  Google Scholar 

  5. Yong, J.: Time optimal control for semilinear distributed parameter systems-existence theory and necessary conditions. Kodai Math. J., 14, 239–253 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  6. Fattorini, H. O.: Time optimal control of solutions of operational differential equations. SIAM J. Control Optim., 2, 54–49 (1964)

    MATH  MathSciNet  Google Scholar 

  7. Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1971

    MATH  Google Scholar 

  8. Schmidt, E. J. P. G.: The ‘bang-bang’ principle for time-optimal problem in boundary control of the heat equation. SIAM J. Control Optim., 18, 101–107 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  9. Mizel, V. J., Seidman, T. I.: An abstract ‘bang-bang principle’ and time-optimal boundary control of the heat equation. SIAM J. Control Optim., 35, 1204–1216 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  10. Wang, G.: The existence of optimal control of semilinear parabolic equations. System and Control Letters, 53, 171–175 (2004)

    Article  MATH  Google Scholar 

  11. Duyckaerts, T., Zhang, X., Zuazua, E.: On the optimality of the observability inequalities for parabolic and hyperbolic systems with potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, 1–41 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fu, X.: A weighted identity for partial differential operators of second order and its applications. C. R. Acad. Sci. Paris Série I, 342, 579–584 (2006)

    MATH  Google Scholar 

  13. Lebeau, G., Robbiano, L.: Contróle exact de l’équation de la chaleur. Comm. PDE, 20, 335–356 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Li, W., Zhang, X.: Controllability of parabolic and hyperbolic equations: towards a unified theory. In: Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 242, 2005, 157–174

    Chapter  Google Scholar 

  15. Tang, Y., Zhang, X.: A note on the singular limit of the exact internal controllability of dissipative wave equations. Acta Mathematica Sinica, English Series, 16, 601–612 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhang, X.: A remark on null exact controllability of the heat equation. SIAM J. Control Optim., 40, 39–53 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  17. Zuazua, E.: Controllability and observability of partial differential equations: some results and open problems. In: Handbook of Differential Equations: Evolutionary Differential Equations, Elsevier Science, 3, 2006, 527–621

    Article  Google Scholar 

  18. Micu, S., Zuazua, E.: On the controllability of a fractional order parabolic equation. SIAM J. Control Optim., 44, 1950–1972 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Miller, L.: On the controllability of anomalous diffusions generated by the fractional laplacian. Mathematics of Control, Signals, and Systems, 18, 260–271 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  20. Lebeau, G., Zuazua, E.: Null-controllability of a system of linear thermoelasticity. Arch. Rational Mech. Anal., 141, 297–329 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lopéz, A., Zhang, X., Zuazua, E.: Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equation. J. Math. Pures Appl., 79, 741–808 (2000)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Sichuan University, Chengdu, 610064, P. R. China

    Qi Lü

Authors
  1. Qi Lü
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Lü.

Additional information

Partially supported by National Natural Science Foundation of China (Grant No. 10525105) and the NCET of China (Grant No. 04-0882)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lü, Q. Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations. Acta. Math. Sin.-English Ser. 26, 2377–2386 (2010). https://doi.org/10.1007/s10114-010-9051-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-010-9051-1

Keywords

MR(2000) Subject Classification

Search

Navigation