Closing the Loop: A Control-Theoretic Framework for Provably Stable Time Series Forecasting with LLMs

Large Language Models (LLMs) have recently shown exceptional potential in time series forecasting, leveraging their inherent sequential reasoning capabilities to model complex temporal dynamics. However, existing approaches typically employ a naive autoregressive generation strategy. We identify a critical theoretical flaw in this paradigm: during inference, the model operates in an open-loop manner, consuming its own generated outputs recursively. This leads to inevitable error accumulation (exposure bias), where minor early deviations cascade into significant trajectory drift over long horizons. In this paper, we reformulate autoregressive forecasting through the lens of control theory, proposing \textbf{F-LLM} (Feedback-driven LLM), a novel closed-loop framework. Unlike standard methods that passively propagate errors, F-LLM actively stabilizes the trajectory via a learnable residual estimator (Observer) and a feedback controller. Furthermore, we provide a theoretical guarantee that our closed-loop mechanism ensures uniformly bounded error, provided the base model satisfies a local Lipschitz constraint. Extensive experiments demonstrate that F-LLM significantly mitigates error propagation, achieving good performance on time series benchmarks.