Robust stability and controllability of stochastic differential delay equations with Markovian switching.

*(English)*Zbl 1040.93069The authors investigate the almost surely asymptotic stability for nonlinear stochastic differential delay equations with Markovian switching. Most of the existing results on stochastic differential delay equations with Markovian switching are about the moment stability, while little is known on the almost surely asymptotic stability, which is the main topic of the present publication. The results on such stability are then applied to establish a sufficient condition for the controllability of linear stochastic delay equations with Markovian switching. The robust stability is discussed in the last section by using linear matrix inequalities.

Reviewer: Yuliya S. Mishura (Kyïv)

##### MSC:

93E15 | Stochastic stability in control theory |

93D09 | Robust stability |

60J75 | Jump processes (MSC2010) |

15A39 | Linear inequalities of matrices |

93B05 | Controllability |

93D20 | Asymptotic stability in control theory |

34K50 | Stochastic functional-differential equations |

##### Keywords:

asymptotic stability; generalized Ito’s formula; controllability; robust stability; stochastic delay equations; Markovian switching; linear matrix inequalities
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\textit{C. Yuan} and \textit{X. Mao}, Automatica 40, No. 3, 343--354 (2004; Zbl 1040.93069)

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