Differential Lyapunov matrix inequalities with matrix inertia constraints have recently emerged as a useful tool for studying properties of nonlinear systems such as k-contraction and p-dominance. However, efficient and systematic methods for designing controllers that ensure the closed-loop system satisfies these generalized Lyapunov inequalities remain underdeveloped. In this work, we propose solutions based on linear matrix inequalities (LMIs) for the design of linear dynamic output-feedback controllers addressing this challenge. Our results focus on partially linear systems whose nonlinearities satisfy a generic quadratic abstraction. Additionally, we introduce methods to handle the difficulties posed by matrix inertia constraints. Finally, we demonstrate the effectiveness and value of the newly proposed conditions by applying them within the frameworks of 2-contraction and extremum control for non-convex optimization.