Distributed Parameter Control Strategies for the Wave Equation with Fractional Order Derivative

In this paper, the control strategies are studied for the fractional order uncertain wave equation subject to persistent external disturbances in Hilbert spaces. The twisting and super-twisting fractional order sliding mode controllers (SMCs) are designed for the infinite dimensional setting and they are applied for addressing the asymptotic state tracking of the fractional order perturbed wave equation. Furthermore, by introducing the adaptive control law to the twisting controller, the bound of the external disturbances which is unknown is dealt with, and for the design of the super-twisting SMC, a fractional order sliding mode manifold is utilized which results in a continuous input control and a chattering free signal. Both of the controllers are associated with the fractional order parameter, which influences the convergence rate of the proposed control algorithms. In addition, the relative theorem involved in the paper for the proof of the stability is proved. Then, the control algorithms are extended to globally asymptotically stabilize the fractional order uncertain wave equation by choosing the appropriate Lyapunov functional. Finally, numerical simulations are presented to verify the viability and efficiency of the proposed fractional order controllers.