In this work, we address the challenge of perpetual generation, which involves producing coherent, infinite-length sequences by learning dynamics from a single data stream online, without external inputs or storing prior information. We introduce a framework grounded in control theory and system dynamics, proposing effective strategies that utilize eigenvalue projection to achieve marginal stability in linear state-space models. Through extensive experiments on different streams, we demonstrate the effectiveness of our approach, with models leveraging spectral properties significantly outperforming traditional baselines. Our findings highlight the critical role of eigenvalue control in enabling perpetual generation, providing a robust foundation for future research in this domain.