Abstract
The approach of Lyapunov functions is one of the most efficient ones for the investigation of the stability of stochastic systems, in particular, of singular stochastic systems. The main objective of the paper is the analysis of the stability of stochastic perturbed singular systems by using Lyapunov techniques under the assumption that the initial conditions are consistent. The uniform exponential stability in mean square and the practical uniform exponential stability in mean square of solutions of stochastic perturbed singular systems based on Lyapunov techniques are investigated. Moreover, we study the problem of stability and stabilization of some classes of stochastic singular systems. Finally, an illustrative example is given to illustrate the effectiveness of the proposed results.
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References
Barbata, A., Zasadzinski, M., Harouna, S.: \(\text{ H}^\infty \) observer design for a class of nonlinear singular stochastic systems. IFAC-PapersOnLine 50, 3885–3892 (2017)
Ben Hamed, B., Ellouze, I., Hammami, M.A.: Practical uniform stability of nonlinear differential delay equations. Mediterr. J. Math. 8, 603–616 (2011)
BenAbdallah, A., Ellouze, I., Hammami, M.A.: Practical stability of nonlinear time-varying cascade systems. J. Dyn. Control Syst. 15, 45–62 (2009)
Berger, T.: Bohl exponent for time-varying linear differential algebraic equations. Int. J. Control 10, 1433–1451 (2012)
Boukass, E.K., Lam, X.: On stability and stabilizability of singular stochastic systems with delays. J. Optim. Theory Appl. 127, 249–262 (2005)
Campbell, S.L.: Singular Systems of Differential Equations. Pitman, Marshfield (1980)
Campbell, S.L., Meyer, C.D., Rose, N.J.: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31, 411–425 (1976)
Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems. Springer International Publishing, New York (2016)
Caraballo, T., Garrido-Atienza, M.J., Real, J.: The exponential behaviour of nonlinear stochastic functional equations of second order in time. Stoch. Dyn. 3, 169–186 (2003)
Caraballo, T., Garrido-Atienza, M.J., Real, J.: Stochastic stabilization of differential systems with general decay rate. Syst. Control Lett. 48, 397–406 (2003)
Caraballo, T., Hammami, M.A., Mchiri, L.: On the practical global uniform asymptotic stability of stochastic differential equations. Stoch. Int. J. Probab. Stoch. Process. 88, 45–56 (2016)
Cong, N.D., The, N.T.: Stochastic differential algebraic equation of index 1. Vietnam J. Math. 38, 117–131 (2010)
Dai, L.: Singular Control Systems. Springer, New York (1989)
Debeljkovic, D.L., Dragutin, L.J., Jovanović, M.B., Drakulić, V.: Singular system theory in chemical engineering theory: stability in the sense of Lyapunov: a survey. Hemijska Industrija 55, 260–272 (2001)
Jiang, X., Tian, S., Zhang, T., Zhang, W.: Stability and stabilization of nonlinear discrete-time stochastic systems. Int. J. Robust Nonlinear Control 29, 6419–6437 (2019)
Khasminskii, R.: Stochastic stability of differential equations. In: Board, A. (ed.) Stochastic Modelling and Applied Probability, 2nd edn. Springer, Cham (2012)
Luo, S., Deng, F., Chen, W.: Stability analysis and synthesis for linear impulsive stochastic systems. Int. J. Robust Nonlinear Control 28, 1–14 (2018)
Mao, X.: Stochastic stabilization and destabilization. Syst. Control Lett. 23, 279–290 (1994)
Mao, X.: Stochastic Differential Equations and Applications. Ellis Horwood, Chichester (1997)
Ownes, D.H., Lj, D.: Consistency and Liapunov stability of linear descriptor systems: a geometric analysis. IMA J. Math. Control Inf. 2, 139–151 (1985)
Pham, Q.C., Tabareau, N., Slotine, J.E.: A contraction theory approach to stochastic incremental stability. IEEE Trans. Autom. Control 54, 1285–1290 (2009)
Rosenbrock, H.H.: Structural properties of linear dynamical systems. Int. J. Control 20, 191–202 (1974)
Sathananthan, S., Knap, M.J., Strong, A., Keel, L.H.: Robust stability and stabilization of a class of nonlinear discrete time stochastic systems: an LMI approach. Appl. Math. Comput. 219, 1988–1997 (2012)
Scheutzow, M.: Stabilization and destabilization by noise in the plane. Stoch. Anal. Appl. 1, 97–113 (1993)
Winkler, R.: Stochastic differential algebraic equations of index 1 and applications in circuit simulation. J. Comput. Appl. Math. 163, 435–463 (2004)
Yan, Z., Zhang, W.: Finite-time stability and stabilization of Itô-type stochastic singular systems. Abstract Appl. Anal. 1–10 (2014)
Zhang, Q., Xing, S.: Stability analysis and optimal control of stochastic singular systems. Optim. Lett. 8, 1905–1920 (2014)
Zhang, W., Zhao, Y., Sheng, L.: Some remarks on stability of stochastic singular systems with state-dependent noise. Automatica 51, 273–277 (2015)
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T. Caraballo: The research of Tomás Caraballo has been partially supported by the Spanish Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the Project PGC2018-096540-I00, and by Junta de Andalucía (Consejería de Economía y Conocimiento) under Project US-1254251.
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Caraballo, T., Ezzine, F. & Hammami, M.A. On the Exponential Stability of Stochastic Perturbed Singular Systems in Mean Square. Appl Math Optim 84, 2923–2945 (2021). https://doi.org/10.1007/s00245-020-09734-8
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DOI: https://doi.org/10.1007/s00245-020-09734-8
Keywords
- Stochastic perturbed singular systems
- Consistent initial conditions
- Lyapunov techniques
- Itô formula
- Brownian motion
- Nontrivial solution
- Practical exponential stability in mean square
- Stabilization