A backstepping approach to adaptive error feedback regulator design for one-dimensional linear parabolic PIDEs

This paper considers the error feedback regulator problem (EFRP) of one-dimensional parabolic partial integro-differential equations (PIDEs) in which the boundary disturbance and the tracking error are both anti-collocated with the control. Firstly, we construct the first auxiliary control system where the output is the measured tracking error. Then the observer and estimators are designed in terms of the measured error to recover the state of the auxiliary control system and estimate the unknown parameters. Secondly, we construct the second auxiliary control system in which the disturbance becomes collocated with the control. Finally, the regulator is proposed based on the backstepping method presented in [33] to make the state of the closed-loop system bounded and the tracking error asymptotically convergent to zero.