Sterile Insect Technique (SIT) is a promising control method against insect pests and insect vectors. It consists in releasing males previously sterilized in laboratory, in order to reduce or eliminate a specific wild population. We study in this paper the implementation by feedback control of SIT-based elimination campaign of Aedes mosquitoes. We provide state-feedback and output-feedback control laws and establish their convergence, as well as their robustness properties. In this design procedure, a pivotal role is played by the average number of secondary female insects produced by a single female insect, called basic offspring number, and by the use of properties of monotone systems. Illustrative simulations are provided.