Controllability and Positivity Constraints in Population Dynamics with age, size Structuring and Diffusion

In this article, we consider the infinite dimensional linear control system describing the Population Models Structured by Age, Size, and Spatial Position. The control is localized in the space variable as well as with respect to the age and size. For each control support, we give an estimate of the time needed to control the system to zero. We prove the null controllability of the model, using a technique avoids the explicit use of parabolic Carleman estimates. Indeed, this method combines final-state observability estimates with the use of characteristics and with $L^{\infty}$ estimates of the associated semigroup.