Providing stability guarantees for controllers that use neural networks can be challenging. Robust control theoretic tools are used to derive a framework for providing nominal stability guarantees – stability guarantees for a known nominal system – controlled by a learning-based neural network controller. The neural network controller is trained using data from an existing baseline controller that achieves desirable closed-loop performance which might, however, not provide provable properties such as stability. Examples of possible applications are human-driver-data-based controllers for autonomous driving, or the learning of control strategies for chemical plants based on the control actions of human operators. To provide stability guarantees for the learning-based controller, the controller is reformulated in form of diagonal nonlinear differential form. This representation exploits the fact that the neural network activation functions are sector-bounded and that their slopes are globally bounded. Based on this representation, sufficient closed-loop stability conditions are established in form of Linear Matrix Inequalities for the nominal system, as well as for the disturbed system controlled by the learning-based controller. For nonlinear activation functions that do not satisfy the necessary conditions, a loop transformation is outlined that allows the application of the presented stability certificate.