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Descripción

This work presents some controllability results for two kinds of systems: systems with classical derivatives and others with non-integer derivatives. We are interested in the controllability problems with output constraints, also called enlarged controllability. We are focused on the computation of the control with minimal cost steering the studied system from an initial state to a state between two well-defined functions in a subregion $\omega$ of the evolution domain $\Omega$. To do so, we adopt two techniques: the first based on convex analysis, more accurately, on the sub-differential notion and the second is based on the technique of Lagrange multipliers. Subsequently, we also study the case where the target region is a part of the boundary, and we characterize the minimal energy control.

Generally, a mathematical model of a real system does not represent exactly the dynamic of a physical process, there can be many reasons for this: the parameters may not be known precisely, the model can be of reduced order, also, in order to simplify the calculations, it can be a linear approximation of a nonlinear process, and since the calculated control is based only on an approximate model, the final state of the original system will not be accurate. That's why we were interested to study the enlarged controllability of the gradient for both cases the internal and the boundary one.