  We study infinite horizon control of continuous-time non-linear branching
processes with almost sure extinction for general (positive or negative)
discount. Our main goal is to study the link between infinite horizon control
of these processes and an optimization problem involving their quasi-stationary
distributions and the corresponding extinction rates. More precisely, we obtain
an equivalent of the value function when the discount parameter is close to the
threshold where the value function becomes infinite, and we characterize the
optimal Markov control in this limit. To achieve this, we present a new proof
of the dynamic programming principle based upon a pseudo-Markov property for
controlled jump processes. We also prove the convergence to a unique
quasi-stationary distribution of non-linear branching processes controlled by a
Markov control conditioned on non-extinction.
