  In this work we construct families of anisotropic neutron stars for an
equation of state compatible with the constraints of the gravitational-wave
event GW170817 and for four anisotropy ansatze. Such stars are subjected to a
radial perturbation in order to study their stability against radial
oscillations and we develop a dynamical model to describe the non-adiabatic
gravitational collapse of the unstable anisotropic configurations whose
ultimate fate is the formation of a black hole. We find that the standard
criterion for radial stability $dM/d\rho_c >0$ is not always compatible with
the calculation of the oscillation frequencies for some anisotropy ansatze, and
each anisotropy parameter is constrained taking into account the recent
restriction of maximum mass of neutron stars. We further generalize the TOV
equations within a non-adiabatic context and we investigate the dynamical
behaviour of the equation of state, heat flux, anisotropy factor and mass
function as an unstable anisotropic star collapses. After obtaining the
evolution equations we recover, as a static limit, the background equations.
