Deep Variational Approach for Generating Optimal Koopman Operator of Nonlinear System

In this work, we explore the use of the Koopman operator for controlling nonlinear dynamic systems. By applying deep variational learning techniques, we probe the existence of multiple Koopman operators in an extended state space. Our investigation reveals that different operators can differently influence the success of control strategies, as shown through our experiments with pendulum systems. We find that selecting an optimal Koopman operator is crucial for effective control in settings where a model is not explicitly defined. To address this, we propose an optimization algorithm that combines offline Bayesian optimization with an online adaptive method. This dual-phase strategy not only pinpoints the most suitable operator but also guarantees the stability of the system in real-time, presenting a new direction for the analysis and model-free control of nonlinear dynamics.