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.
This paper focuses on the application of backstepping control scheme for fractional order partial differential equations (FPDEs) of order with . Therefore to obtain highly accurate approximations for this derivative is of great importance. Here the discretised approach for the space variable is used to transform the FPDEs into a system of differential equations. These approximations arise mainly from the Caputo definition and the Grünwald-Letnikov definition. A Lyapunov function is defined at each stage and the negativity of an overall Lyapunov function is ensured by proper selection of the control law. Illustrative example is given to demonstrate the effectiveness of the proposed control scheme.
In this paper we use the homotopy analysis method to solve special types of the initial value problems that consist of multi-fractional order integro differential equation in which the fractional derivative and fractional integral in them are described in the Caputo sense and Riemann-Liouville sense respectively. Numerical examples are solved by using this method. These examples shows that high accuracy, simplicity and efficiency of this method.
In this article, the backstepping control scheme is proposed to stabilize the fractional order Riccati matrix differential equation with retarded arguments in which the fractional derivative is presented using Caputo's definition of fractional derivative. The results are established using Mittag-Leffler stability. The fractional Lyapunov function is defined at each stage and the negativity of an overall fractional Lyapunov function is ensured by the proper selection of the control law. Numerical simulation has been used to demonstrate the effectiveness of the proposed control scheme for stabilizing such type of Riccati matrix differential equations.
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