  This paper considers the performance output tracking for a boundary
controlled Direct Contact Membrane Distillation (DCMD) system. First, the
mathematical properties of a recently developed mathematical model of the DCMD
system are discussed. This model consists of parabolic equations coupled at the
boundary. Then, the existence and uniqueness of the solutions are analyzed,
using the theory of operators. Some regularity results of the solution are also
established. A particular case showing the diagonal property of the principal
operator is studied. Then, based on one-side feedback law the control problem,
which consists of tracking both the feed and permeate outlet temperatures of
the membrane distillation system is formulated. A servomechanism and an output
feedback controller are proposed to solve the control problem. In addition, an
extended state observer aimed at estimating both the system state and
disturbance, based on the temperature measurements of the inlet is proposed.
Thus, by some regularity for the reference signal and when the disturbance
vanishes, we prove the exponential decay of the output tracking error.
Moreover, we show the performance of the control strategy in presence of the
flux noise.

Analysis of PDEs;Optimization and Control