Controllability of a Family of Nonlinear Population Dynamics Models

Abstract

Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operator $\mathbf{A}$ and an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a time $T$ dependent on the constraints on the age and the observability minimal time $T_0$ of the pair $\left(\mathbf{A},\mathbf{B}\right)$ ( $\mathbf{B}$ is the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..