Abstract:This paper develops sharp Hautus-type criteria, stochastic counterparts of the classical Popov-Belevitch-Hautus test, for exact controllability and stabilizability of backwardstructured stochastic linear systems. The main finding is that the stochastic Hautus obstruction is not a left eigenvector, as in deterministic linear systems, nor an arbitrary symmetric eigenmatrix, but a positive semidefinite eigenmatrix of a Lyapunov-type operator. We prove that exact controllability is equivalent to the absence of such nonzero positive semidefinite eigenmatrices that are orthogonal to the control directions. This cone restriction is sharp: excluding all symmetric eigenmatrices with the same orthogonality property is sufficient but not necessary. We further show that stabilizability is characterized by the same cone-restricted Hautus condition imposed only on the nonstable spectral part of the Lyapunov-type operator. Thus the stochastic Hautus theory developed here is governed by a simultaneous spectral restriction and cone restriction. In addition to these criteria, we provide finite-rank and Gramian characterizations underlying exact controllability, establish the corresponding controllability decomposition, and show that exact controllability implies stabilizability.
From: Jingrui Sun [view email]
[v1]
Wed, 27 May 2026 09:57:08 UTC (23 KB)
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