The problem of reducing a partial differential equation (PDE) to a system of finite dimensional ordinary differential equations (ODE), is of paramount importance in engineering and physics where solving PDE models is often too time consuming. The idea of being able to reduce the PDE model to a simple ODE model without loosing the main characteristics of the original model, such as stability and prediction precision, is appealing for any real-time model-based estimation and control applications. However, this problem remains challenging since model reduction can introduce stability loss and prediction degradation. To remedy these problems many methods have been developed aiming at what is known as stable model reduction. In this talk, we focus on the so-called closure models and their application in reduced order model (ROM) stabilization. We present some results on robust stabilization for reduced order models (ROM) of partial differential equations using Lyapunov theory. Stabilization is achieved via closure models for ROMs where we use Lyapunov theory to design a new closure model, which is robust with respect to model structured uncertainties. Furthermore, we use an extremum-seeking algorithm to optimally tune the closure models' parameters for optimal ROM stabilization. The Burgers' equation and a 3D Boussinesq equation examples are employed as a test-bed for the proposed stabilization method.