Startseite Null-controllability of a fractional order diffusion equation
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Null-controllability of a fractional order diffusion equation

  • Xiangdong Yang EMAIL logo
Veröffentlicht/Copyright: 19. Februar 2017
Veröffentlichen auch Sie bei De Gruyter Brill
Informationen für Autor*innen Erkunden Sie dieses Fachgebiet
Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 20 Heft 1

Abstract

The article considers the controllability of a fractional order diffusion equation. We show that the resulting fractional order diffusion equation is null-controllable. Our method reduces essentially to the study of a moment problem related to the Mittag-Leffler functions. Paley-Wiener type theorems are applied to construct biorthogonal sequence to a family of complex Mittag-Leffler functions.

MSC 2010: Primary 34A08, 30E05; Secondary 35E20, 41A75
Key Words and Phrases: null-controllability; Mittag-Leffler functions; Paley-Wiener type theorems; fractional order diffusion equation

Acknowledgements

The author thanks the editor and referees for suggestions that improved the original manuscript. The author also thanks National Natural Science Foundation of China for the support, under Grant No. 11261024.

References

[1] S. Avdonin, S. Lenhart and V. Protopopescu, Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method. Inverse Probl. 18 (2002), 349–361.10.1088/0266-5611/18/2/304Suche in Google Scholar

[2] K. Balachandran, Y. Zhou and J. Kokila, Relative controllability of fractional dynamical systems with delays in control. Commun. Nonlinear. Sci. Numer. Simulat. 17 (2012), 3201–3209.10.1016/j.cnsns.2011.12.018Suche in Google Scholar

[3] M.M. Djrbashian, Harmonic Analysis and Boundary Value Problems in the Complex Domain, Springer, Basel AG (1993).10.1007/978-3-0348-8549-2Suche in Google Scholar

[4] J. Dong and M. Xu, Space-time fractional Schrödinger equation with time-indepedent potentials. J. Math. Anal. Appl. 344 (2008), 1005–1017.110.1016/j.jmaa.2008.03.061Suche in Google Scholar

[5] O. Glass, A complex-analytic approach to the problem ofuniform controllability of a transport equation in the vanishing viscosity limit. J. Funct. Annal. 258 (2010), 852–868.10.1016/j.jfa.2009.06.035Suche in Google Scholar

[6] R. Gorenflo, A.A. Kilbas, F. Mainardi and S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin Heidelberg (2014).10.1007/978-3-662-43930-2Suche in Google Scholar

[7] T. Kaczorek, Selected Problems of Fractional Systems Theory. Springer, Berlin-Heidelberg (2011).10.1007/978-3-642-20502-6Suche in Google Scholar

[8] J. Liang and H. Yang, Controllability of fractional integro-differential evolution equations with nonlocal conditions. Appl. Math. Comput. 254 (2015), 20–29.10.1016/j.amc.2014.12.145Suche in Google Scholar

[9] P. Martin, L. Rosier and P. Rouchon, Null controllability of thestructually damped wave equation with moving point control. SIAM J. Control Optim. 51 (2011), 660–684.10.1137/110856150Suche in Google Scholar

[10] S. Micu and E. Zuazua, On the controllability of a fractional order parabolic equation. SIAM J. Control Optim. 17 (2006), 1950–1972.10.1137/S036301290444263XSuche in Google Scholar

[11] S. Micu, J.H. Ortega and A.F. Pazoto, Null-controllability of a hyperbolic equation as singular limit of parabolic ones. J. Fourier Anal. Appl. 17 (2011), 991–1007.10.1007/s00041-010-9168-8Suche in Google Scholar

[12] M. Mirrahimi, Lyapunov control of a quantum particle in a decaying potential. Ann. I. H. Poincaré26 (2009), 1743–1765.10.1016/j.anihpc.2008.09.006Suche in Google Scholar

[13] J. Pöschel and E. Trubowitz, Inverse Spectral Theory, Academic Press, Orlando (1987).Suche in Google Scholar

[14] D.L. Russell, Nonharmonic Fourier sereis in the control theory of distributed parameter systems. J. Math. Anal. Appl. 18 (1967), 542–560.10.1016/0022-247X(67)90045-5Suche in Google Scholar

[15] Vu Kim Tuan, Inverse problem for fractional diffusion equations. Fract. Calc. Appl. Anal. 14, No 1 (2011), 31–55; DOI: 10.2478/s13540-011-0004-x; https://www.degruyter.com/view/j/fca.2011.14.issue-1/issue-files/fca.2011.14.issue-1.xml.Suche in Google Scholar

[16] Vu Kim Tuan and N. Si Hoang, An inverse problem for a multidimensional fractional diffusion equations. Analysis (Berlin) 36 (2016), 107–122.10.1515/anly-2015-5010Suche in Google Scholar

Received: 2016-3-14
Revised: 2016-11-29
Published Online: 2017-2-19
Published in Print: 2017-2-1

© 2017 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 20–1–2017)
  4. Survey paper
  5. Ten equivalent definitions of the fractional laplace operator
  6. Research paper
  7. Consensus of fractional-order multi-agent systems with input time delay
  8. Research paper
  9. Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives
  10. Research paper
  11. A preconditioned fast finite difference method for space-time fractional partial differential equations
  12. Research paper
  13. On existence and uniqueness of solutions for semilinear fractional wave equations
  14. Research paper
  15. Computational solutions of the tempered fractional wave-diffusion equation
  16. Research paper
  17. Completeness on the stability criterion of fractional order LTI systems
  18. Research paper
  19. Wavelet convolution product involving fractional fourier transform
  20. Research paper
  21. Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method
  22. Research paper
  23. A foundational approach to the Lie theory for fractional order partial differential equations
  24. Research paper
  25. Null-controllability of a fractional order diffusion equation
  26. Research paper
  27. New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions
  28. Research paper
  29. The stretched exponential behavior and its underlying dynamics. The phenomenological approach
  30. Short Paper
  31. Lyapunov-type inequality for an anti-periodic fractional boundary value problem
https://doi.org/10.1515/fca-2017-0012
Schlagwörter für diesen Artikel
null-controllability; Mittag-Leffler functions; Paley-Wiener type theorems; fractional order diffusion equation

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 20–1–2017)
  4. Survey paper
  5. Ten equivalent definitions of the fractional laplace operator
  6. Research paper
  7. Consensus of fractional-order multi-agent systems with input time delay
  8. Research paper
  9. Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-Type Hadamard derivatives
  10. Research paper
  11. A preconditioned fast finite difference method for space-time fractional partial differential equations
  12. Research paper
  13. On existence and uniqueness of solutions for semilinear fractional wave equations
  14. Research paper
  15. Computational solutions of the tempered fractional wave-diffusion equation
  16. Research paper
  17. Completeness on the stability criterion of fractional order LTI systems
  18. Research paper
  19. Wavelet convolution product involving fractional fourier transform
  20. Research paper
  21. Solutions of the main boundary value problems for the time-fractional telegraph equation by the green function method
  22. Research paper
  23. A foundational approach to the Lie theory for fractional order partial differential equations
  24. Research paper
  25. Null-controllability of a fractional order diffusion equation
  26. Research paper
  27. New results in stability analysis for LTI SISO systems modeled by GL-discretized fractional-order transfer functions
  28. Research paper
  29. The stretched exponential behavior and its underlying dynamics. The phenomenological approach
  30. Short Paper
  31. Lyapunov-type inequality for an anti-periodic fractional boundary value problem
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 16.11.2025 von https://www.degruyterbrill.com/document/doi/10.1515/fca-2017-0012/html?lang=de
Button zum nach oben scrollen