Stabilizing an unknown dynamical system is one of the central problems in control theory. In this paper, we study the sample complexity of the learn-to-stabilize problem in Linear Time-Invariant (L TI) systems on a single trajectory. Current state-of-the-art approaches require a sample complexity linear in n, the state dimension, which incurs a state norm that blows up exponentially in n. We propose a novel algorithm based on spectral decomposition that only needs to learn "a small part" of the dynamical matrix acting on its unstable subspace. We show that, under proper assumptions, our algorithm stabilizes an L TI system on a single trajectory with O ( k log n) samples, where k is the instability index of the system. This represents the first sub-linear sample complexity result for the stabilization of L TI systems under the regime when k = o (n).

We study the problem of learning to stabilize (LTS) a linear time-invariant (LTI) system. Policy gradient (PG) methods for control assume access to an initial stabilizing policy. However, designing such a policy for an unknown system is one of the most fundamental problems in control, and it may be as hard as learning the optimal policy itself. Existing work on the LTS problem requires large data as it scales quadratically with the ambient dimension. We propose a two-phase approach that first learns the left unstable subspace of the system and then solves a series of discounted linear quadratic regulator (LQR) problems on the learned unstable subspace, targeting to stabilize only the system's unstable dynamics and reduce the effective dimension of the control space. We provide non-asymptotic guarantees for both phases and demonstrate that operating on the unstable subspace reduces sample complexity. In particular, when the number of unstable modes is much smaller than the state dimension, our analysis reveals that LTS on the unstable subspace substantially speeds up the stabilization process. Numerical experiments are provided to support this sample complexity reduction achieved by our approach.

We study the problem of learning to stabilize unknown noisy Linear Time-Invariant (LTI) systems on a single trajectory. It is well known in the literature that the learn-to-stabilize problem suffers from exponential blow-up in which the state norm blows up in the order of $\Theta(2^n)$ where $n$ is the state space dimension. This blow-up is due to the open-loop instability when exploring the $n$-dimensional state space. To address this issue, we develop a novel algorithm that decouples the unstable subspace of the LTI system from the stable subspace, based on which the algorithm only explores and stabilizes the unstable subspace, the dimension of which can be much smaller than $n$. With a new singular-value-decomposition(SVD)-based analytical framework, we prove that the system is stabilized before the state norm reaches $2^{O(k \log n)}$, where $k$ is the dimension of the unstable subspace. Critically, this bound avoids exponential blow-up in state dimension in the order of $\Theta(2^n)$ as in the previous works, and to the best of our knowledge, this is the first paper to avoid exponential blow-up in dimension for stabilizing LTI systems with noise.