  In this paper, we investigate the exact controllability properties of an
advection-diffusion equation on a bounded domain, using time- and
space-dependent velocity fields as the control parameters. This partial
differential equation (PDE) is the Kolmogorov forward equation for a reflected
diffusion process that models the spatiotemporal evolution of a swarm of
agents. We prove that if a target probability density has bounded first-order
weak derivatives and is uniformly bounded from below by a positive constant,
then it can be reached in finite time using control inputs that are bounded in
space and time. We then extend this controllability result to a class of
advection-diffusion-reaction PDEs that corresponds to a hybrid-switching
diffusion process (HSDP), in which case the reaction parameters are
additionally incorporated as the control inputs. Our proof for controllability
of the advection-diffusion equation is constructive and is based on linear
operator semigroup theoretic arguments and spectral properties of the
multiplicatively perturbed Neumann Laplacian. For the HSDP, we first
constructively prove controllability of the associated continuous-time Markov
chain (CTMC) system, in which the state space is finite. Then we show that our
controllability results for the advection-diffusion equation and the CTMC can
be combined to establish controllability of the forward equation of the HSDP.
Lastly, we provide constructive solutions to the problem of asymptotically
stabilizing an HSDP to a target non-negative stationary distribution using
time-independent state feedback laws, which correspond to spatially-dependent
coefficients of the associated system of PDEs.
