Hautus-Type Criteria for Controllability and Stabilizability of Backward-Structured Stochastic Systems
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| Format: | Preprint |
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2026
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| author | Sun, Jingrui |
| author_facet | Sun, Jingrui |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_28242 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hautus-Type Criteria for Controllability and Stabilizability of Backward-Structured Stochastic Systems Sun, Jingrui Optimization and Control 93E03, 93B05, 93D15, 60H10 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. |
| title | Hautus-Type Criteria for Controllability and Stabilizability of Backward-Structured Stochastic Systems |
| topic | Optimization and Control 93E03, 93B05, 93D15, 60H10 |
| url | https://arxiv.org/abs/2605.28242 |