In this paper, we deal with the null controllability result of a degenerate Schrödinger 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.

在本文中，我们处理了退化的Schrödinger方程的零可控制性结果。零可控性现象是这样的事实，即在适当的力（称为控制力）和精确的时间（称为控制时间）的帮助下，将给定系统从其初始状态变为零平衡。经典地，为了解决这个问题，我们必须证明相关联的伴随模型的可观察性，并且这种趋势采用了各种技术来建立这样的不等式。但是首先，我们将通过相关框架证明我们的模型是正确的，其主要开拓者是根据扩散系数加权的Sobolev空间。然后，基于精心选择的加权函数，建立相关联的伴随系统的相关卡尔曼估计。众所周知，这种不等式是解及其导数的加权估计。作为Carleman不等式的结果，我们证明了相关的可观性不等式，这使我们可以推断出控制的存在。为此，我们使用众所周知的HUM方法，该方法基于与研究模型匹配的成本函数，并且可以显示为连续，强制和凸面。我们着重指出，这种控制是给定子序列的弱限制。