The Lyapunov method is one of the most effective methods to analyze the stability of stochastic differential equations (SDEs). Different authors analyzed the stability of SDEs based on Lyapunov techniques when the origin can be considered as an equilibrium point. When the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions concerning a small neighborhood of the origin. The purpose is to study the asymptotic stability of a system whose solution behavior is a small ball of state space or close to it. Thus, all state trajectories are bounded and close to a sufficiently small neighborhood of the origin. In this sense, the limited boundedness of solutions of random systems, or the chance of convergence of solutions needs to be analyzed on a ball centered on the origin. This is the so called “Practical Stability”. In this article, we mainly investigate the practical uniform exponential stability in the mean square of stochastic linear time–invariant systems. In addition, we are developing the problem of stabilization of certain classes of perturbed stochastic systems. Our crucial techniques include Lyapunov techniques and generalized Gronwall inequalities. Lastly, we provide a numerical example to illustrate our theoretical findings.
| Published in | American Journal of Applied Scientific Research (Volume 9, Issue 1) |
| DOI | 10.11648/j.ajasr.20230901.12 |
| Page(s) | 14-20 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2023. Published by Science Publishing Group |
Linear Systems, Stochastic Systems, Lyapunov Techniques, Brownian Motions, Nontrivial Solution, Practical Stability, Stabilization
| [1] | B. Ben Hamed, I. Ellouze, M. A. Hammami, Practical uniform stability of nonlinear differential delay equations, Mediterranean Journal of Mathematics, 8 (2011), 603–616. |
| [2] | A. BenAbdallah, W. Hdidi, Uniting two local output controllers for linear system subject to input saturation: LMI Approach, Journal of The Franklin Institute, 15 (2018), 6969–6991. |
| [3] | M. Corless, Guaranteed rates of exponential convergence for uncertain systems, Journal of Optimization Theory and Applications, 64 (1990), 481-494. |
| [4] | T. Caraballo, On the decay rate of solutions of non- autonomous differential systems, Electronic Journal of Differential Equations, 05 (2001), 1–17. |
| [5] | T. Caraballo, F. Ezzine, M. Hammami, Partial stability analysis of stochastic differential equations with a general decay rate, Journal of Engineering Mathematics, 130 (2021), 1–17. |
| [6] | T. Caraballo, F. Ezzine, M. Hammami, On the exponential stability of stochastic perturbed singular systems in mean square, Applied Mathematics and Optimization, 84 (2021), 2923–2945. |
| [7] | X. Jiang, S. Tian, T. Zhang, W. Zhang, Stability and stabilization of nonlinear discrete-time stochastic systems, International Journal of Robust and Nonlinear Control, 29 (2019), 6419–6437. |
| [8] | R. Khasminskii, Stochastic stability of differential equations, Stochastic Modelling and Applied Probability 66, 2nd edition, 2012. |
| [9] | H. K. Khalil, Nonlinear Systems, Mac-Millan, 2nd edition, 1996. |
| [10] | X. Mao, Stochastic differential equations and applications, Ellis Horwood, Chichester, U. K, 1997. |
| [11] | X. Mao, Stochastic stabilization and destabilization, Systems & control letters, 23 (1994), 279–290. |
| [12] | B. Oksendal, Stochastic Differential Equations: an Introduction with Applications (6th ed.) Springer-Verlag, New York, 2003. |
| [13] | Q. C. Pham, N. Tabareau, J. E. Slotine, A contraction theory approach to stochastic incremental stability, IEEE Transactions on Automatic Control, 54 (2009), 1285– 1290. |
| [14] | S. Sathananthan, M. J. Knap, A. Strong, L. H. Keel, Robust stability and stabilization of a class of nonlinear discrete time stochastic systems: An LMI approach, Applied Mathematics and Computation, 219 (2012), 1988-1997. |
| [15] | M. Scheutzow, Stabilization and destabilization by noise in the plane, Stochastic Analysis and Applications, 1 (1993), 97–113. |
| [16] | E. Sontag, Input to State Stability: Basic Concepts and Results, In book: Nonlinear and optimal control theory. |
| [17] | E. Sontag, Y. Wang, on characterizations of input– to–state stability with respect to compact sets, IFAC Nonlinear Control Systems Design, Tahoe City, California, USA, 1995. |
APA Style
Walid Hdidi, Faten Ezzine. (2023). On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach. American Journal of Applied Scientific Research, 9(1), 14-20. https://doi.org/10.11648/j.ajasr.20230901.12
ACS Style
Walid Hdidi; Faten Ezzine. On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach. Am. J. Appl. Sci. Res. 2023, 9(1), 14-20. doi: 10.11648/j.ajasr.20230901.12
AMA Style
Walid Hdidi, Faten Ezzine. On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach. Am J Appl Sci Res. 2023;9(1):14-20. doi: 10.11648/j.ajasr.20230901.12
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.ajasr.20230901.12,
author = {Walid Hdidi and Faten Ezzine},
title = {On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach},
journal = {American Journal of Applied Scientific Research},
volume = {9},
number = {1},
pages = {14-20},
doi = {10.11648/j.ajasr.20230901.12},
url = {https://doi.org/10.11648/j.ajasr.20230901.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajasr.20230901.12},
abstract = {The Lyapunov method is one of the most effective methods to analyze the stability of stochastic differential equations (SDEs). Different authors analyzed the stability of SDEs based on Lyapunov techniques when the origin can be considered as an equilibrium point. When the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions concerning a small neighborhood of the origin. The purpose is to study the asymptotic stability of a system whose solution behavior is a small ball of state space or close to it. Thus, all state trajectories are bounded and close to a sufficiently small neighborhood of the origin. In this sense, the limited boundedness of solutions of random systems, or the chance of convergence of solutions needs to be analyzed on a ball centered on the origin. This is the so called “Practical Stability”. In this article, we mainly investigate the practical uniform exponential stability in the mean square of stochastic linear time–invariant systems. In addition, we are developing the problem of stabilization of certain classes of perturbed stochastic systems. Our crucial techniques include Lyapunov techniques and generalized Gronwall inequalities. Lastly, we provide a numerical example to illustrate our theoretical findings.},
year = {2023}
}
TY - JOUR T1 - On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach AU - Walid Hdidi AU - Faten Ezzine Y1 - 2023/03/02 PY - 2023 N1 - https://doi.org/10.11648/j.ajasr.20230901.12 DO - 10.11648/j.ajasr.20230901.12 T2 - American Journal of Applied Scientific Research JF - American Journal of Applied Scientific Research JO - American Journal of Applied Scientific Research SP - 14 EP - 20 PB - Science Publishing Group SN - 2471-9730 UR - https://doi.org/10.11648/j.ajasr.20230901.12 AB - The Lyapunov method is one of the most effective methods to analyze the stability of stochastic differential equations (SDEs). Different authors analyzed the stability of SDEs based on Lyapunov techniques when the origin can be considered as an equilibrium point. When the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions concerning a small neighborhood of the origin. The purpose is to study the asymptotic stability of a system whose solution behavior is a small ball of state space or close to it. Thus, all state trajectories are bounded and close to a sufficiently small neighborhood of the origin. In this sense, the limited boundedness of solutions of random systems, or the chance of convergence of solutions needs to be analyzed on a ball centered on the origin. This is the so called “Practical Stability”. In this article, we mainly investigate the practical uniform exponential stability in the mean square of stochastic linear time–invariant systems. In addition, we are developing the problem of stabilization of certain classes of perturbed stochastic systems. Our crucial techniques include Lyapunov techniques and generalized Gronwall inequalities. Lastly, we provide a numerical example to illustrate our theoretical findings. VL - 9 IS - 1 ER -
Email: