On the control cost via the null controllability concept: The case of an interior set degenerate Schrödinger equation with finite delay

Abstract

In this paper, we are interested in the null controllability property of linear degenerate Schrödinger equation with a finite delay and a degeneracy that occurs in an interior subset of $(0,1),\text{i.e}\exists W_1\subset \subset (0,1),\text{such}\text{that}\forall x\in W_1,k\left(x\right)=0$, where $k$ stands for quantum diffusion. More precisely, we are looking for a suitable control, called the control time, which steers the studied wave to zero equilibrium in a finite time. To this end, we wield the classical procedure based on a adequate Carleman estimate involving a new weight functions and afterwards we get our observability inequality matched with the full adjoint system.