  In this paper, we aim to explore optimal regional trajectory tracking control
problems of the anomalous subdiffusion processes governed by time-fractional
diffusion systems under the Neumann boundary conditions. Using eigenvalue
theory of the system operator and the semigroup theory, we explore the
existence and some estimates of the mild solution to the considered system. An
approach on finding solution to the optimal problem that minimizes the regional
trajectory tracking error and the corresponding control cost over a finite
space and time domain is then explored via the Hilbert uniqueness method (HUM).
The obtained results not only can be directly used to investigate the systems
that are not controllable on the whole domain, but also yield an explicit
expression of the control signal in terms of the desired trajectory. Most
importantly, it is worth noting that our results in this paper are still novel
even for the special case when the order of fractional derivative is equal to
one. Finally, we provide a numerical example to illustrate our theoretical
results.
