Skip Navigation Links
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

Riemann Liouville Fractional Spatial Derivative Stabilization of Bilinear Distributed Systems

Journal of Applied Nonlinear Dynamics 8(3) (2019) 447--461 | DOI:10.5890/JAND.2019.09.008

Hanaa Zitane, Rachid Larhrissi, Ali Boutoulout

MACS Laboratory, Department of Mathematics, Faculty of Sciences, University of Moulay Ismail, Meknes, Morocco

Download Full Text PDF



The goal of this paper is to study a fractional output stabilization problem: the stabilization of the state fractional spatial derivative of complex purely imaginary order i α with α ∈]0,1[, for bilinear distributed systems. Firstly, we develop sufficient conditions for exponential, strong and weak fractional output stabilization for the considered system, also, we offer some examples illustrating the obtained results. Moreover, we characterise the stabilizing control which minimizes an appropriate cost. Finally, an illustrating example with numerical simulations is given.


  1. [1]  Baudouin, L. and Salomon, J. (2008), Constructive solution of a bilinear optimal control problem for a Schr¨odinger equation, Systems and Control Letters, 57, 453-464.
  2. [2]  Mohler, R.R. and Kolodziej, W.J. (1980), An overview of bilinear system theory and applications, IEEE Transactions on Systems, Man and Cybernetics, 10, 683-688.
  3. [3]  Slemrod, M. (1978), Stabilization of bilinear control systems with applications to non-conservative problems in elasticity, SIAM Journal on Control and Optimization, 16, 131-141.
  4. [4]  Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993), Fractional integrals and derivatives: theory and applications, Gordon & Breach.
  5. [5]  Ball, J.M. and Slemrod, M. (1979), Feedback stabilization of distributed semilinear control systems, J. Appl. Math. Opt, 5, 169-179.
  6. [6]  Berrahmoune, L. (1999), Stabilization and decay estimate for distributed bilinear systems, Systems Control Letters, 36, 167-171.
  7. [7]  Ouzahra, M. (2009), Stabilization of infinite-dimensional bilinear systems using a quadratic feedback control, International Journal of Control, 82, 1657-1664.
  8. [8]  Quinn, J.P. (1980), Stabilization of bilinear systems by quadratic feedback controls, J. Math. Anal. Appl, 75, 66-80.
  9. [9]  Ouzahra, M. (2010), Exponential and weak stabilization of constrained bilinear systems, SIAM J. Control Optim, 48(6), 3962-3974.
  10. [10]  Ouzahra, M. (2008), Strong stabilization with decay estimate of semilinear systems, System and Control Letters, 57, 813-815.
  11. [11]  Zerrik, E., Benslimane, Y., and El Jai, A. (2013), Regional gradient stabilization for bilinear distributed systems, Int J Math Anal, 7, 195-211.
  12. [12]  Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006), Theory and applications of fractional differential equations, Elsevier Science Limited.
  13. [13]  Oldham, K.B. and Spanier, J. (1974), The fractional calculus: theory and application of differentiation and integration to arbitrary order, Academic Press, New York and London.
  14. [14]  Ross, B. (1974), Fractional calculus and its applications, Springer Verlag, in Lecture Notes in Mathematics, Proc. int. conf. New Haven, 457.
  15. [15]  Podlubny, I. (1999), Fractional differential equations, Academic press, San Diego.
  16. [16]  Li, Y., Chen, Y.Q., and Podlubny,I. (2010), Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Comput. Math. Appl., 59, 1810-1821.
  17. [17]  Podlubny, I. and Chen, Y. (2007), Adjoint fractional differential espressions and operators, ASME 2007 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, American Society Of Mechanical Engineers, 1385-1390.
  18. [18]  Pazy, A. (1983), Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York.
  19. [19]  Jichun, L. and Chen, Y. (2008), Computational partial differential equation using MATLAB, CRC press, New york.
  20. [20]  Chen, Y., Wei, Y., Zhou, X., and Wang, Y. (2017), Stability for nonlinear fractional order systems: an indirect approach, Nonlinear Dynamics, 89, 1011-1018.
  21. [21]  Gomez-Aguilar, J.F. (2017), Space-time fractional diffusion equation using a derivative with non singular and regular kernel, Phisica A: Statistical Mechanics and its Applications, 465, 562-572.
  22. [22]  Qian, D.L., Li, C.P., Agarwal, R.P., and Wong, P.J.Y. (2010), Stability analysis of fractional differential system with Riemann Liouville derivative, Math. Comput. Model., 52, 862-874.