  We investigate the stability and stabilization concepts for infinite
dimensional time fractional differential linear systems in Hilbert spaces with
Caputo derivatives. Firstly, based on a family of operators generated by
strongly continuous semigroups and on a probability density function, we
provide sufficient and necessary conditions for the exponential stability of
the considered class of systems. Then, by assuming that the system dynamics is
symmetric and uniformly elliptic and by using the properties of the
Mittag-Leffler function, we provide sufficient conditions that ensure strong
stability. Finally, we characterize an explicit feedback control that
guarantees the strong stabilization of a controlled Caputo time fractional
linear system through a decomposition approach. Some examples are presented
that illustrate the effectiveness of our results.
