Nonlinear Mittag-Leffler stability of nonlinear fractional partial differential equations

Volume 10, Issue 10, pp 5182--5200 http://dx.doi.org/10.22436/jnsa.010.10.06

Publication Date: October 12, 2017 Submission Date: February 14, 2017

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Authors

Ibtisam Kamil Hanan - Institute of Engineering Mathematics, Universiti Malaysia Perlis, Pauh Putra Main Campus, 02600 Arau, Perlis, Malaysia. - Department of Mathematics and Computer Applications, College of Science, Al-Nahrain University, P. O. Box 47077, Baghdad, Iraq. 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.

Abstract

This paper focuses on the application of fractional backstepping control scheme for nonlinear fractional partial differential equation (FPDE). Two types of fractional derivatives are considered in this paper, Caputo and the Grünwald-Letnikov fractional derivatives. Therefore, obtaining highly accurate approximations for this derivative is of a great importance. Here, the discretized approach for the space variable is used to transform the FPDE into a system of fractional differential equations. The convergence of the closed loop system is guaranteed in the sense of Mittag-Leffler stability. An illustrative example is given to demonstrate the effectiveness of the proposed control scheme.

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Keywords

• Backstepping method
• fractional Lyapunov function
• fractional derivative
• boundary control
• fractional partial differential equation

MSC

•  37B25
•  26A33
•  35R11

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