LMI hierarchies for stability analysis of ReLU feedback systems

We consider the stability analysis of feedback systems with rectified linear unit (ReLU) activations, and model this problem with polynomial optimization. Stability can be certified by means of copositive multipliers in the framework of integral quadratic constraints. Based on a duality argument, we show how to certify instability by considering a complete hierarchy of linear matrix inequalities. This hierarchy is obtained by leveraging the specific equality constraints arising from the ReLU encoding. We illustrate the effectiveness of the proposed approach through several numerical examples.