The field of Koopman operator theory is experiencing significant growth, with recent developments focusing on improving the accuracy and efficiency of data-driven modeling and control techniques. Researchers are exploring innovative approaches to address challenges in complex dynamical systems, such as nonlinear systems with input delays and control-affine systems. The integration of deep learning methods, such as LSTM networks, with Koopman operator theory is showing promising results in capturing historical dependencies and encoding time-delayed system dynamics. Furthermore, the application of Koopman-based Economic Model Predictive Control (EMPC) is demonstrating improvements in resource-efficient control of thermal-intensive plants. Noteworthy papers include:

• EchoLSTM, which proposes a self-reflective recurrent network for stabilizing long-range memory, achieving significant performance gains on challenging benchmarks.
• Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit, which presents a deep Koopman-based EMPC for efficient operation of a laboratory-scale pasteurization unit, achieving a 32% reduction in total economic cost.
• Deep Dictionary-Free Method for Identifying Linear Model of Nonlinear System with Input Delay, which introduces a novel approach to approximate the Koopman operator using an LSTM-enhanced Deep Koopman model, enabling linear representations of nonlinear systems with time delays.