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This paper considers a class of space fractional partial differentialequations (FPDEs) that describe gas pressures in fractured media. First, thewell-posedness, uniqueness, and the stability in $L_(\infty{R})$of theconsidered FPDEs are investigated. Then, the reference tracking problem isstudied to track the pressure gradient at a downstream location of a channel.This requires manipulation of gas pressure at the downstream location and theuse of pressure measurements at an upstream location. To achiever this, thebackstepping approach is adapted to the space FPDEs. The key challenge in thisadaptation is the non-applicability of the Lyapunov theory which is typicallyused to prove the stability of the target system as, the obtained target systemis fractional in space. In addition, a backstepping adaptive observer isdesigned to jointly estimate both the system's state and the disturbance. Thestability of the closed loop (reference tracking controller/observer) is alsoinvestigated. Finally, numerical simulations are given to evaluate theefficiency of the proposed method.

Reference Tracking AND Observer Design for Space-Fractional Partial Differential Equation Modeling Gas Pressures in Fractured Media