Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay
The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regular...
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| Format: | Article |
| Language: | English |
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Wiley
2022-01-01
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| Series: | International Journal of Differential Equations |
| Online Access: | http://dx.doi.org/10.1155/2022/5549693 |
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| _version_ | 1849412942442790912 |
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| author | Ramazan Kadiev Arcady Ponosov |
| author_facet | Ramazan Kadiev Arcady Ponosov |
| author_sort | Ramazan Kadiev |
| collection | DOAJ |
| description | The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well. |
| format | Article |
| id | doaj-art-65f063f7fd744edfb806972e2ec53f7c |
| institution | Kabale University |
| issn | 1687-9651 |
| language | English |
| publishDate | 2022-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Differential Equations |
| spelling | doaj-art-65f063f7fd744edfb806972e2ec53f7c2025-08-20T03:34:17ZengWileyInternational Journal of Differential Equations1687-96512022-01-01202210.1155/2022/5549693Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with DelayRamazan Kadiev0Arcady Ponosov1Dagestan Research Center of the Russian Academy of Sciences & Department of MathematicsDagestan Research Center of the Russian Academy of Sciences & Department of MathematicsThe work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.http://dx.doi.org/10.1155/2022/5549693 |
| spellingShingle | Ramazan Kadiev Arcady Ponosov Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay International Journal of Differential Equations |
| title | Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay |
| title_full | Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay |
| title_fullStr | Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay |
| title_full_unstemmed | Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay |
| title_short | Positive Invertibility of Matrices and Exponential Stability of Linear Stochastic Systems with Delay |
| title_sort | positive invertibility of matrices and exponential stability of linear stochastic systems with delay |
| url | http://dx.doi.org/10.1155/2022/5549693 |
| work_keys_str_mv | AT ramazankadiev positiveinvertibilityofmatricesandexponentialstabilityoflinearstochasticsystemswithdelay AT arcadyponosov positiveinvertibilityofmatricesandexponentialstabilityoflinearstochasticsystemswithdelay |