Revived transformation-based transfer learning safety control: A geometric interpretation

Ensuring safety during the learning process of reinforcement learning (RL) control poses a significant challenge. To address this issue, our previous work proposed a novel transfer learning approach, allowing controllers to be trained in a non-dangerous source system and then transferred to the dangerous target system. This approach is built upon a system transformation–termed the revived transformation–that maps the source system to the target system. In this work, we develop a rigorous geometric framework to characterize the intrinsic manifold structures induced by the revived transformation, which link the source and target systems. These geometric insights provide a precise theoretical foundation for analyzing how control policies can be reliably transferred between systems. Specifically, we show that the revived transformation induces a diffeomorphic mapping between the state manifolds of the source and target systems, thereby yielding a structured relationship between their respective vector fields. Furthermore, this relationship establishes a connection between the one-parameter transformation groups acting on the two state manifolds. These geometric structures ultimately ensure the safety and stability of the target system under the transferred controller. Finally, simulation results validate the correctness of the geometric framework as well as the transfer learning approach.