  This paper considers a class of space fractional partial differential
equations (FPDEs) that describe gas pressures in fractured media. First, the
well-posedness, uniqueness, and the stability in $L_(\infty{R})$of the
considered FPDEs are investigated. Then, the reference tracking problem is
studied to track the pressure gradient at a downstream location of a channel.
This requires manipulation of gas pressure at the downstream location and the
use of pressure measurements at an upstream location. To achiever this, the
backstepping approach is adapted to the space FPDEs. The key challenge in this
adaptation is the non-applicability of the Lyapunov theory which is typically
used to prove the stability of the target system as, the obtained target system
is fractional in space. In addition, a backstepping adaptive observer is
designed to jointly estimate both the system's state and the disturbance. The
stability of the closed loop (reference tracking controller/observer) is also
investigated. Finally, numerical simulations are given to evaluate the
efficiency of the proposed method.
