Algorithmic Information and the Fundamental Limits of Control

This paper explores the intricate relationship between algorithmic information theory and the fundamental limits inherent in control systems. By leveraging concepts from Kolmogorov complexity and related measures of descriptive complexity, we aim to establish a theoretical framework that quantifies the informational cost associated with achieving and maintaining control over complex dynamic systems. Traditional control theory primarily focuses on performance metrics like stability, optimality, and robustness, often implicitly assuming unlimited information processing capabilities. This work explicitly introduces the notion of algorithmic information content into the analysis of control, proposing that the complexity of a system's dynamics and the desired control objective impose inherent lower bounds on the amount of information that must be acquired, processed, and encoded within a controller. We demonstrate how the minimum description length of a system's behavior or a control policy can serve as a proxy for its fundamental control difficulty, leading to a new perspective on robustness, uncertainty quantification, and the design of intelligent autonomous systems. The implications extend to understanding the trade-offs between system performance, controller complexity, and the resilience against unpredictable disturbances, offering novel insights into the irreducible informational barriers to perfect control.