Backstepping in infinite dimensional for the time fractional order partial differential equations


  • Ibtisam Kamil Hanan Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia
  • Muhammad Zaini Ahmad Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia
  • Fadhel Subhi Fadhel Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, P. O. Box 47077, Baghdad, Iraq
  • Ghazwan Raheem Mohammed Mayoralty of Baghdad, Rasheed Municipality, Khilany squarel Al-Kholfaa street, Baghdad, Iraq



Boundary control, Backstepping, Stabilisation, Coordinate transformation, Fractional order PDE.


This paper focuses on the application of backstepping control scheme for the time fractional order partial differential equation (FPDE). The fractional derivative is presented by using Caputo fractional derivative. The design technique here can exhaust systems with an arbitrary finite number of open loop unstable eigenvalues and is not limited to a certain kind of boundary actuation. We show how the FPDE is converted into a Mittag-Leffler stability by designing invertible coordinate transformation. Numerical simulation is given to demonstrate the effectiveness of the proposed control scheme.


R. Triggiani (1980). Boundary feedback stabilizability of parabolic equations. Applied Mathematics & Optimization, 6(1), 201-220.

Lasiecka & R. Triggiani. (1983). Stabilization and structural assignment of Dirichlet boundary feedback parabolic equations. SIAM journal on control and optimization, 21(5), 766-803.

T. Nambu. (1984). On the stabilization of diffusion equations: boundary observation and feedback. Journal of differential equations, 52(2), 204-233.

A. Bensoussan, G. Da Prato, M. C. Delfour, & S. K. Mitter. (1993). Representation and control of infinite dimensional systems. Boston: Birkhäuser.

H. Chao, D. Lu, L. Tian. (2006). Boundary control of the Kuramoto-Sivashinsky equation with an exteral excitation. International Journal of Nonlinear Science, 1(2), 67-81.

Z. Zhao. (2006). Optimal control of Kuramto-Sivashing equation. International Journal of Nonlinear Science, 1(1), 54-57.

M. Zhu, & Z. Zhao. (2006). Optimal control of nonlinear strength Burgers equation under the Neumann boundary condition. International Journal of Nonlinear Science, 1(2), 111-118.

D. Ding, & L. Tian. (2006). The study of solution of dissipative Camassa-Holm equation on total space. International Journal of Nonlinear Science, 1(1), 37-42.

L. Tian, & J. Yin. (2004). New compacton solutions and solitary wave solutions of fully nonlinear generalized Camassa–Holm equations. Chaos, Solitons & Fractals, 20(2), 289-299.

J. A. Burns, & D. Rubio. (1997). A distributed parameter control approach to sensor location for optimal feedback control of thermal processes. Proceedings of the 36th IEEE Conference on Decision and Control. IEEE.

A. Shidfar, & G. R. Karamali. (2005). Numerical solution of inverse heat conduction problem with nonstationary measurements. Applied Mathematics and Computation, 168(1), 540-548.

Doubova, E. Fernández-Cara, & M. González-Burgos, (2004). On the controllability of the heat equation with nonlinear boundary Fourier conditions. Journal of Differential Equations, 196(2), 385-417.

D. M. Boskovic, M. Krstic, & W. Liu. (2001). Boundary control of an unstable heat equation via measurement of domain-averaged temperature. IEEE Transactions on Automatic Control, 46(12), 2022-2028.

X. Wang, L. Tian, & L. Yu. (2006). Linear feedback controlling and synchronization of the Chen’s chaotic system. International Journal of Nonlinear Science, 2(1), 43-49.

W. Liu. (2003). Boundary feedback stabilization of an unstable heat equation. SIAM journal on control and optimization, 42(3), 1033-1043.

M. Krstic, & A. Smyshlyaev. (2008). Boundary control of PDEs: A course on backstepping designs. Siam. USA.

A. Smyshlyaev, & M. Krstic. (2010). Adaptive control of parabolic PDEs. Princeton University Press.

Balogh, & M. Krstic. (2004). Stability of partial difference equations governing control gains in infinite-dimensional backstepping. Systems & control letters, 51(2), 151-164.

Z. Zhou, & C. Guo. (2013). Stabilization of linear heat equation with a heat source at intermediate point by boundary control. Automatica, 49(2), 448-456.

J. Liang, Y. Chen, & R. Fullmer. (2004). Simulation studies on the boundary stabilization and disturbance rejection for fractional diffusion-wave equation. In American Control Conference, 2004. IEEE.

J. Liang, Y. Chen, Y., & B. Z. Guo. (2004). A hybrid symbolic-numerical simulation method for some typical boundary control problems. Simulation, 80(11), 635-643.

P. J. Torvik, & R. L. Bagley. (1984). On the appearance of the fractional derivative in the behavior of real materials. Journal of Applied Mechanics, 51(2), 294-298.

B.Mandelbrot, & R. Pignoni. (1983). The fractal geometry of nature. New York: WH freeman.

A. A. Kilbas, H.M. Srivastava, J.J. Trujillo. (2006). Theory and

Applications of Fractional Differential Equations. Elsevier Science Limited.

I. Henry, & S. L. Wearne. (2000). Fractional reaction–diffusion. Physica A: Statistical Mechanics and its Applications, 276(3), 448-455.

S. B. Yuste, & K. Lindenberg. (2002). Subdiffusion-limited reactions. Chemical physics, 284(1), 169-180.

V. Uchaikin, R. Sibatov. (2013). Fractional kinetics in solids: anomalous charge transport in semiconductors. World Science.

J. Sung, E. Barkai, R.J. Silbey, S. Lee. (2002). Fractional dynamics approach to diffusion-assisted reactions in disordered media. The Journal of chemical physics, 116(6), 2338-2341.

A. Balogh, & M. Krstic. (2002). Infinite dimensional backstepping-style feedback transformations for a heat equation with an arbitrary level of instability. European journal of control, 8(2), 165-175.

I. K. Hanan, M. Z. Ahmad, & F. S. Fadhel. (2017). The backstepping method for stabilizing time fractional order partial differential equation. Journal of Theoretical & Applied Information Technology, 95 (6), 1318-1328.

I. K. Hanan, M. Z. Ahmad, & F. S. Fadhel. (2017). Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations. Journal of Nonlinear Sciences and Applications, 10(2017), 5182-5200.

I. K. Hanan, M. Z. Ahmad, & F. S. Fadhel. (2017). Stability of fractional order parabolic partial differential equations using discretised backstepping method. Malaysian Journal of Fundamental and Applied

Sciences, 13(4). (Article in Press).

F. Ge, Y. Chen, & C. Kou. (2016). Boundary feedback stabilisation for the time fractional-order anomalous diffusion system. IET Control Theory & Applications, 10(11), 1250-1257.

M. Bošković, A. Balogh, & M. Krstić. (2003). Backstepping in infinite dimension for a class of parabolic distributed parameter systems. Mathematics of Control, Signals and Systems, 16(1), 44-75.

N. Aguila-Camacho, M. A. Duarte-Mermoud, & J. A. Gallegos. (2014). Lyapunov functions for fractional order systems. Communications in Nonlinear Science and Numerical Simulation, 19(9), 2951-2957.

Podlubny. (1999). Fractional Differential Equations. Academic Press, San Diego, California, USA.

Y. N. Zhang, Z. Z. Sun, & H. L. Liao. (2014). Finite difference methods for the time fractional diffusion equation on non-uniform meshes. Journal of Computational Physics, 265, 195-210.