Null Controllability of a Degenerate Schrödinger Equation

Abstract

In this paper, we deal with the null controllability result of a degenerate Schrodinger equation. The null controllability phenomenon is the fact to bring a given system from its initial state to the zero equilibrium with the help of a suitable force called control and in a precise time called control time. Classically, to grapple such a question we have to prove the observability property of the associated adjoint model and this trend various techniques are employed to establish such an inequality. But firstly, we will prove that our model is well posed via a pertinent framework whose main pioneers are a weighted Sobolev spaces depending on the diffusion coefficient. Afterwards, a relevant Carleman estimate of the associated adjoint system is established based on a well chosen weighted functions. It is well-known that this kind of inequality is a weighted estimate of the solution and their derivatives. As an outcome of Carleman inequality, we prove our relevant observability inequality which allows us to deduce the existence of our control. To this end, we use the well-known HUM method laying on a cost function matched with the studied model and which can be shown continuous, coercive and convex. We highlight that such a control is a weak limit of a given subsequence.