### A Comparison Theorem for Stability of Linear Stochastic Implicit Difference Equations of Index-1

## Main Article Content

## Abstract

In this paper we study linear stochastic implicit difference equations (LSIDEs for short) of index-1. We give a definition of solution and introduce an index-1 concept for these equations. The mean square stability of LSIDEs is studied by using the method of solution evaluation. An example is given to illustrate the obtained results.

*Keywords:*LSIDEs, index, solution, mean square stability.

## References

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[11] R. Winkler, Stochastic differential algebraic equations of index 1 and applications in circuit simulation, Journal of Computational and Applied Mathematics 157 (2003) 477--505.

[12] O. Scheina, G. Denk, Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits, Journal of Computational and Applied Mathematics 100 (1998) 77--92.

[13] P. Kunkel, V. Mehrmann, Differential - algebraic equations, analysis and numerical solution, EMS Publishing House, Zürich, 2006.

[14] R. Lamour, R. März, C. Tischendorf, Differential-algebraic equations: a projector based analysis. Springer, Berlin, 2013.

[15] P.K. Anh, N.H. Du, L.C. Loi, Singular difference equations: an overview, Vietnam J. Math. 35 (2007), pp. 339-372.

[16] P.K. Anh, P.T. Linh, Stability of periodically switched discrete-time linear singular systems, Journal of Difference Equations and Applications 23 (2017) 1680-1693.

[17] N.H. Du, T.K. Duy and V.T. Viet, Degenrate cocycle index-1 and lyapunov exponents, Stochastics and Dynamics 7 (2) (2007) 229-245.

[18] N.H. Du, V.H. Linh, V. Mehrmann, D.D. Thuan, Stability and robust stability of linear time-invariant delay differential-algebraic equations, SIAM J. Matrix Anal. Appl. 34 (2013) 1631-1654.

[2] L. Shaikhet, Necessary and Sufficient Conditions of Asymptotic Mean Square Stability for Stochastic Linear Difference Equations, Applied Mathematics Letters 10 (3) (1997) 111-115.

[3] F. Ma, T.K. Caughey, Mean stability of stochastic difference systems, International Journal of Non-Linear Mechanics 17 (2) (1982) 69--84.

[4] Z. Yang, D. Xu, Mean square exponential stability of impulsive stochastic difference equations, Applied Mathematics Letters 20 (2007) 938--945.

[5] T. Brüll, Existence and uniqueness of solutions of linear variable coefficient discrete-time descriptor systems, Linear Algebra Appl. 431 (2009) 247-265.

[6] V.H. Linh, N.T.T. Nga and D.D. Thuan, Exponential stability and robust stability for linear time-varying singular systems of second-order difference equations, SIAM J. Matrix Anal. Appl. 39 (2018) 204-233.

[7] D.G. Luenberger, Dynamic equations in descriptor form”, IEEE Trans. Automat. Control 22 (1977) 312-322.

[8] D.D. Thuan, N.H. Son, Stochastic implicit difference equations of index-1, preprint 2019, submitted for publication.

[9] L. Brancik, E. Kolarova, Application of Stochastic Differential-Algebraic Equations in Hybrid MTL Systems Analysis, Elektronika ir Elektrotechnika 20 (2014) 41-45.

[10] N.D. Cong, N.T. The, Stochastic differential - algebraic equations of index-1, Vietnam J. Math. 38 (1) (2010), 117-131.

[11] R. Winkler, Stochastic differential algebraic equations of index 1 and applications in circuit simulation, Journal of Computational and Applied Mathematics 157 (2003) 477--505.

[12] O. Scheina, G. Denk, Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits, Journal of Computational and Applied Mathematics 100 (1998) 77--92.

[13] P. Kunkel, V. Mehrmann, Differential - algebraic equations, analysis and numerical solution, EMS Publishing House, Zürich, 2006.

[14] R. Lamour, R. März, C. Tischendorf, Differential-algebraic equations: a projector based analysis. Springer, Berlin, 2013.

[15] P.K. Anh, N.H. Du, L.C. Loi, Singular difference equations: an overview, Vietnam J. Math. 35 (2007), pp. 339-372.

[16] P.K. Anh, P.T. Linh, Stability of periodically switched discrete-time linear singular systems, Journal of Difference Equations and Applications 23 (2017) 1680-1693.

[17] N.H. Du, T.K. Duy and V.T. Viet, Degenrate cocycle index-1 and lyapunov exponents, Stochastics and Dynamics 7 (2) (2007) 229-245.

[18] N.H. Du, V.H. Linh, V. Mehrmann, D.D. Thuan, Stability and robust stability of linear time-invariant delay differential-algebraic equations, SIAM J. Matrix Anal. Appl. 34 (2013) 1631-1654.