On the Hardness of Learning to Stabilize Linear Systems
This addresses a fundamental challenge in control theory for researchers and practitioners, revealing inherent statistical hardness in learning-based stabilization.
The paper tackles the problem of learning to stabilize linear time-invariant systems, showing that even when systems are easy to identify, the sample complexity for stabilization increases exponentially with system dimension.
Inspired by the work of Tsiamis et al. \cite{tsiamis2022learning}, in this paper we study the statistical hardness of learning to stabilize linear time-invariant systems. Hardness is measured by the number of samples required to achieve a learning task with a given probability. The work in \cite{tsiamis2022learning} shows that there exist system classes that are hard to learn to stabilize with the core reason being the hardness of identification. Here we present a class of systems that can be easy to identify, thanks to a non-degenerate noise process that excites all modes, but the sample complexity of stabilization still increases exponentially with the system dimension. We tie this result to the hardness of co-stabilizability for this class of systems using ideas from robust control.