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Kadiev, Ramazan; Ponosov, Arcady

Positive invertibility of matrices and exponential stability of linear stochastic systems with delay. (English) Zbl 1511.34085

Int. J. Differ. Equ. 2022, Article ID 5549693, 13 p. (2022).
The paper is concerned with certain linear stochastic differential equations with delay. The aim is to provide results on exponential moment stability of large systems of equations. The approach is based on the regularization method which seeks an auxiliary equation that regularizes the original problem. The presentation of the approach is designed to be used more widely than the present paper requires. Stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained- the authors remark that some of the results provide fresh insights also for deterministic systems. Sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Ito equations are given. The paper concludes with some examples and a discussion of open problems.
Reviewer: Neville J. Ford (Chester)

MSC:

34K50 Stochastic functional-differential equations
34K06 Linear functional-differential equations
34K20 Stability theory of functional-differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

exponential stability; Ito equations; Lyapunov stability; regularization method; delay equations
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