Regional gradient compensation with minimum energy

In this paper we interest to the regional gradient remediability or compensation problem with minimum energy. That is, when a system is subjected to disturbances, then one of the objectives becomes to find the optimal control which compensates regionally the effect of the disturbances of the system, with respect to the regional gradient observation. Therefore, we show how to find the optimal control ensuring the effect compensation of any known or unknown disturbance distributed only on a subregion of the geometrical evolution domain, with respect to the observation of the gradient on any given subregion of the evolution domain and this in finite time. Under convenient hypothesis, the minimum energy problem is studied using an extension of the Hilbert Uniqueness Method (HUM). Approximations, numerical simulations, appropriate algorithm, and illustrative examples are also presented.