Model-based Reinforcement Learning (MBRL) has shown many desirable properties for intelligent control tasks. However, satisfying safety and stability constraints during training and rollout remains an open question. We propose a new Model-based RL framework to enable efficient policy learning with unknown dynamics based on learning model predictive control (LMPC) framework with mathematically provable guarantees of stability. We introduce and explore a novel method for adding safety constraints for model-based RL during training and policy learning. The new stability-augmented framework consists of a neural-network-based learner that learns to construct a Lyapunov function, and a model-based RL agent to consistently complete the tasks while satisfying user-specified constraints given only sub-optimal demonstrations and sparse-cost feedback. We demonstrate the capability of the proposed framework through simulated experiments.