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 (TSF), leveraging their inherent sequential reasoning capabilities to model complex temporal dynamics. Existing approaches typically employ an autoregressive generation strategy to adapt LLMs for TSF. However, we identify a theoretical flaw in this paradigm: during inference, the model operates in an open-loop manner, recursively consuming its own generated outputs. This leads to error accumulation, where minor early deviations cascade into significant rollout drift over long horizons. In this paper, we reformulate autoregressive forecasting through the lens of control theory, proposing Feedback-driven LLM (F-LLM), a novel closed-loop framework. Unlike standard methods that passively propagate errors, F-LLM actively stabilizes the trajectory via a learnable residual estimator functioning as a system observer. Furthermore, we provide a mathematical proof that, under explicit contraction assumptions, this closed-loop mechanism guarantees a uniformly bounded step-wise error sequence within the local surrogate dynamics. Extensive experiments demonstrate that F-LLM significantly mitigates error propagation, achieving good performance on time series benchmarks. Our code is publicly available at https://github.com/Zh-XY22/F-LLM.