We address the stabilization of linear time-varying parabolic PDEs using model predictive control (MPC) based on reduced-order models (ROMs). We first prove exponential stability and suboptimality of the full-order MPC scheme in Hilbert spaces. Since MPC requires the repeated solution of finite-horizon optimal control subproblems, and since the dynamics typically become simpler close to the equilibrium, model order reduction is a natural approach to accelerate the online computations.

We then introduce a Galerkin reduced-order approach together with a rigorous a posteriori error analysis for the associated finite-horizon optimal control problems. This leads to a ROM-based MPC algorithm that adaptively constructs reduced-order controls, ensures exponential stability of the full-order closed-loop state, and provides computable performance bounds with respect to the infinite-horizon full-order control problem. The resulting ROM-MPC scheme combines error estimation, adaptive optimization, and closed-loop control in a certified pipeline, and can efficiently update the ROM when the dynamics change unpredictably online.

Numerical experiments with a nonsmooth cost functional involving the squared $\ell^1$-norm confirm the effectiveness of the method, showing substantial computational savings together with low approximation errors, even for exponentially unstable systems.