A Unified Kalman-Type Classification of Controllability for Linear Stochastic Systems
Abstract
We provide a unified algebraic classification of approximate controllability, approximate null controllability, exact null controllability, and exact controllability for finite-dimensional linear stochastic control systems with constant coefficients and multidimensional Brownian noise. The controls may enter both the drift and the diffusion terms, and the joint diffusion-control matrix may have arbitrary rank. A finite decreasing sequence of subspaces defines a common geometric obstruction. Its vanishing characterizes the two approximate properties and exact null controllability. Exact controllability to arbitrary random targets additionally requires the joint diffusion-control matrix to be surjective. The codimension of the obstruction is given by a rank difference of explicit polynomial block matrices, yielding finite tests in the original coefficients. When the joint diffusion-control matrix is surjective, we obtain a finite Kalman-type full-rank condition for exact controllability. If the obstruction vanishes, a linear state feedback with a deterministic time-dependent gain drives every initial state to zero at any prescribed positive time with finite energy and polynomial small-time cost. An equivalent orthogonal elimination algorithm computes the rank differences with polynomial storage and cost, avoiding exponentially large block matrices. A two-noise example relates the input classification to terminal errors and control energy in penalized stochastic linear-quadratic control problems.
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BibTeX
@misc{lu2026unified,
title = {{A Unified Kalman-Type Classification of Controllability for Linear Stochastic Systems}},
author = {Qi Lü and Yu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14975},
eprint = {2610.14975},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14975},
}
RIS
TY - UNPB
TI - A Unified Kalman-Type Classification of Controllability for Linear Stochastic Systems
AU - Qi Lü
AU - Yu Wang
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14975
UR - https://arxiv.org/abs/2610.14975
AB - We provide a unified algebraic classification of approximate controllability, approximate null controllability, exact null controllability, and exact controllability for finite-dimensional linear stochastic control systems with constant coefficients and multidimensional Brownian noise. The controls may enter both the drift and the diffusion terms, and the joint diffusion-control matrix may have arbitrary rank. A finite decreasing sequence of subspaces defines a common geometric obstruction. Its vanishing characterizes the two approximate properties and exact null controllability. Exact controllability to arbitrary random targets additionally requires the joint diffusion-control matrix to be surjective. The codimension of the obstruction is given by a rank difference of explicit polynomial block matrices, yielding finite tests in the original coefficients. When the joint diffusion-control matrix is surjective, we obtain a finite Kalman-type full-rank condition for exact controllability. If the obstruction vanishes, a linear state feedback with a deterministic time-dependent gain drives every initial state to zero at any prescribed positive time with finite energy and polynomial small-time cost. An equivalent orthogonal elimination algorithm computes the rank differences with polynomial storage and cost, avoiding exponentially large block matrices. A two-noise example relates the input classification to terminal errors and control energy in penalized stochastic linear-quadratic control problems.
ER -