Averaged Control Problems Governed by a Semilinear Distributed Systems

In this work, we consider the regional averaged controllability (RAC) problem governed by a class of semilinear hyperbolic systems. We start by giving the definitions of the exact and approximate RAC systems. After that, we state the problem of RAC for semilinear systems. We propose two methods of solution: using a condition of the analytical operator to the nonlinear part of the system to characterize the optimal control via the fixed point theorem and the Hilbert Uniqueness Method (HUM) with an asymptotic condition on the nonlinear part to find the optimal control of the considered problem. Finally, we present a numerical example to show the effectiveness of the main results.