Gradient controllability of semilinear systems: HUM approach

The objective of this paper is to broaden the concept of regional controllability from linear systems to the semilinear hyperbolic case. Specifically, we aim to identify the control mechanism that guides our system from an initial state to a final internal state within a designated subregion $ \omega $ of the evolution domain $ \Omega $. Our methodology builds upon an extension of the Hilbert Uniqueness Method (HUM) and leverages Schauder's fixed point theorem. We demonstrate that under certain conditions, the system is weakly gradient controllable in the subregion $ \omega $, and we derive an explicit control strategy for achieving a desired gradient state. The results show that the combination of the HUM approach and fixed-point techniques provide an effective framework for controlling semilinear systems. This analysis contributes to the broader understanding of controllability in infinite-dimensional systems with nonlinear dynamics.