  In this article, we consider a simplified model of time dynamics for a
mosquito population subject to the artificial introduction of {\itshape
Wolbachia}-infected mosquitoes, in order to fight arboviruses
transmission.Indeed, it has been observed that when some mosquito populations
are infected by some {\itshape Wolbachia} bacteria, various reproductive
alterations are induced in mosquitoes, including cytoplasmic incompatibility.
Some of these {\itshape Wolbachia} bacteria greatly reduce the ability of
insects to become infected with viruses such as the dengue ones, cutting down
their vector competence and thus effectively stopping local dengue
transmission.The behavior of infected and uninfected mosquitoes is assumed to
be driven by a compartmental system enriched with the presence of an internal
control source term standing for releases of infected mosquitoes, distributed
in time. We model and design an optimal releasing control strategy with the
help of a least square problem. In a nutshell, one wants to minimize the number
of uninfected mosquitoes at a given horizon of time, under some relevant
biological constraints. We derive properties of optimal controls, highlight a
limit problem providing useful asymptotic properties of optimal controls. We
numerically illustrate the relevance of our approach.
