Regional observability of Caputo semilinear fractional systems

Abstract

This article studies the problem of regional observability of an important class of time‐fractional evolution systems involving the Caputo time‐fractional derivative. The main goal here is to check the regional observability of the considered system in the desired subregion, , of the evolution domain, . We assume that the linear part of the studied system is approximately regionally observable in the desired subregion , and we add some reasonable hypotheses on the system's dynamic, , as well as on the nonlinear operator . Then, with the help of all these assumptions, we attempt to reconstruct the initial state of our system in the desired subregion . To do that, we propose an extension of the Hilbert uniqueness method (HUM) to semilinear fractional systems. The main advantage of the HUM approach is that it transforms the reconstruction problem into a solvability one, which is more practical. We also present an algorithm for reconstructing the initial state in , which leads to some successful numerical simulations.