Asymptotic Behavior of Solutions for Linear Implicit Difference Equations with Index 1
Main Article Content
Abstract
In this paper, we deal with the asymptotic behavior of solutions of constant coefficient linear implicit difference equations with index 1. Supposing that all solutions of the original implicit equation are bounded (resp. tend to zero as tends to infinity), we provide sufficient conditions imposed on the perturbations so that all solutions of the perturbed equations remain bounded (resp. tend to zero as tends to infinity).
Key words: Implicit Difference Equations, IDEs, SDEs, Matrix Pencil, Kronecker Index
References
[1] P. K. ANH, D. S. HOANG, Stability of a class of singular difference equations, Inter. J. Difference Equ., 1, 181-193 (2006).
[2] P. K. Anh, N. H. Du, L. C. Loi, Singular difference equations: an overview, Vietnam J. Math. 35, pp. 339- 372 (2007).
[3] P. K. Anh, N. H. Du, L. C. Loi, Connections between implicit difference equations and differential-algebraic equations, Acta Math. Vietnam. 29 (2004), pp. 23-39.
[4] P. K. Anh, H. T. N. Yen, On the solvability of initial-value problems for nonlinear implicit difference equations, Adv. Difference Eqns. 3 (2004), pp. 195-200.
[5] S. L. Campbell, Singular systems of differential equations I, Pitman Advanced Publishing Program, 1982.
[6] S. L. Campbell, Singular Systems of Differential Equations II, Pitman, London, 1982.
[7] L. Dai, Singular control systems, Lecture Notes in Control and Information Sciences 118, Springer-Verlag 1989.
[8] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations, Analysis and Numerical Solution, European Math. Soc. Publ. House, 2006.
[9] L. C. Loi, Linear Implicit Nonautonomous Difference Equations, Ph.D. dissertation, Hanoi, Vietnam National Univ., 2004.
[10] R. M rz, Numerical methods for differential-algebraic equations, Acta Numerica, 1 (1992), pp 141-198.
[11] B. Rodjanadid, N. V. Sanh, N. T. Ha, N. H. Du, Stability radii for implicit difference equations, Asian-European Journal of Mathematics, Vol. 2, No. 1 (2009) pp. 95-115.
[12] N. H. Du, V. H. Linh, On the robust stability of implicit linear systems containing a small parameter in the leading term, IMA J. Math. Control Inform, 23 (2006) 67-84.
[13] V. H. Linh, N. N. Tuan, Asymptotic integration of linear differential-algebraic equations, Electronic Journal of Qualitative Theory of Differential Equations, 2014, No. 12, 1-17.
[14] R.P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
[15] T. Berger. Robustness of stability of time-varying index-1 DAEs. Preprint TUIlmenau, Germany, 2013.
[2] P. K. Anh, N. H. Du, L. C. Loi, Singular difference equations: an overview, Vietnam J. Math. 35, pp. 339- 372 (2007).
[3] P. K. Anh, N. H. Du, L. C. Loi, Connections between implicit difference equations and differential-algebraic equations, Acta Math. Vietnam. 29 (2004), pp. 23-39.
[4] P. K. Anh, H. T. N. Yen, On the solvability of initial-value problems for nonlinear implicit difference equations, Adv. Difference Eqns. 3 (2004), pp. 195-200.
[5] S. L. Campbell, Singular systems of differential equations I, Pitman Advanced Publishing Program, 1982.
[6] S. L. Campbell, Singular Systems of Differential Equations II, Pitman, London, 1982.
[7] L. Dai, Singular control systems, Lecture Notes in Control and Information Sciences 118, Springer-Verlag 1989.
[8] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations, Analysis and Numerical Solution, European Math. Soc. Publ. House, 2006.
[9] L. C. Loi, Linear Implicit Nonautonomous Difference Equations, Ph.D. dissertation, Hanoi, Vietnam National Univ., 2004.
[10] R. M rz, Numerical methods for differential-algebraic equations, Acta Numerica, 1 (1992), pp 141-198.
[11] B. Rodjanadid, N. V. Sanh, N. T. Ha, N. H. Du, Stability radii for implicit difference equations, Asian-European Journal of Mathematics, Vol. 2, No. 1 (2009) pp. 95-115.
[12] N. H. Du, V. H. Linh, On the robust stability of implicit linear systems containing a small parameter in the leading term, IMA J. Math. Control Inform, 23 (2006) 67-84.
[13] V. H. Linh, N. N. Tuan, Asymptotic integration of linear differential-algebraic equations, Electronic Journal of Qualitative Theory of Differential Equations, 2014, No. 12, 1-17.
[14] R.P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000.
[15] T. Berger. Robustness of stability of time-varying index-1 DAEs. Preprint TUIlmenau, Germany, 2013.