Representation of solutions of systems of linear differential equations with multiple delays and nonpermutable variable coefficients
Abstract
Solutions of nonhomogeneous systems of linear differential equations with multiple constant delays are explicitly stated without a commutativity assumption on the matrix coefficients. In comparison to recent results, the new formulas are not inductively built, but depend on a sum of noncommutative products in the case of constant coefficients, or on a sum of iterated integrals in the case of time-dependent coefficients. This approach shall be more suitable for applications.
Representation of a solution of a Cauchy problem for a system of higher order delay differential equations is also given.
Keyword : Laplace transform, multiple delays, matrix polynomial, nonhomogeneous equation, noncommutative product
How to Cite
Pospíšil, M. (2020). Representation of solutions of systems of linear differential equations with multiple delays and nonpermutable variable coefficients. Mathematical Modelling and Analysis, 25(2), 303-322. https://doi.org/10.3846/mma.2020.11194
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
J. Diblík, M. Fečkan and M. Pospíšil. Representation of a solution of the Cauchy problem for an oscillating system with multiple delays and pairwise permutable matrices. Abstr. Appl. Anal., 2013(ID 931493):10 pages, 2013. https://doi.org/10.1155/2013/931493
J. Diblík, M. Fečkan and M. Pospíšil. Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices. Ukr. Math. J., 65(1):64–76, 2013. https://doi.org/10.1007/s11253-013-0765-y
J.K. Hale and S.M. Verduyn Lunel. Introduction to Functional Differential Equations. Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-46124342-7
D.Ya. Khusainov and J. Diblík. Representation of solutions of linear discrete systems with constant coefficients and pure delay. Adv. Differ. Equ., 2006(Article ID 80825):13 pages, 2006. https://doi.org/10.1155/ADE/2006/80825
D.Ya. Khusainov, J. Diblík, M. Růžičková and J. Lukáčková. Representation of a solution of the Cauchy problem for an oscillating system with pure delay. Nonlinear Oscil., 11(2):276–285, 2008. https://doi.org/10.1007/s11072-008-0030-8
D.Ya. Khusainov and G. V. Shuklin. Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Zilina Math. Ser., 17:101–108, 2003.
N.N. Lebedev. Special Functions and their Applications. Prentice Hall, Inc., USA, 1965.
Ch. Liang, J. Wang and D. O’Regan. Controllability of nonlinear delay oscillating systems. Electron. J. Qual. Theory Differ. Equ., 47:1–18, 2017. https://doi.org/10.14232/ejqtde.2017.1.47
Ch. Liang, W. Wei and J. Wang. Stability of delay differential equations via delayed matrix sine and cosine of polynomial degrees. Adv. Differ. Equ., 2017(131):17 pages, 2017. https://doi.org/10.1186/s13662-017-1188-0
N.I. Mahmudov. Representation of solutions of discrete linear delay systems with non permutable matrices. Appl. Math. Lett., 85:8–14, 2018. https://doi.org/10.1016/j.aml.2018.05.015
M. Medved’ and M. Pospíšil. Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices. Nonlinear Anal.-Theory Methods Appl., 75(7):3348–3363, 2012. https://doi.org/10.1016/j.na.2011.12.031
M. Medved’ and M. Pospíšil. Representation and stability of solutions of systems of difference equations with multiple delays and linear parts defined by pairwise permutable matrices. Commun. Appl. Anal., 17(1):21–46, 2013.
M. Medved’ and M. Pospíšil. Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices. J. Math. Sci., 228(3):276–289, 2018. https://doi.org/10.1007/s10958-017-3620-0
M. Medved’, M. Pospíšil and L. Škripková. On exponential stability of nonlinear fractional multidelay integro-differential equations defined by pairwise permutable matrices. Appl. Math. Comput., 227:456–468, 2014. https://doi.org/10.1016/j.amc.2013.11.012
M. Pospíšil. Representation and stability of solutions of systems of functional differential equations with multiple delays. Electron. J. Qual. Theory Differ. Equ., 54:1–30, 2012. https://doi.org/10.14232/ejqtde.2012.1.54
M. Pospíšil and F. Jaroš. On the representation of solutions of delayed differential equations via Laplace transform. Electron. J. Qual. Theory Differ. Equ., 117:1– 13, 2016. https://doi.org/10.14232/ejqtde.2016.1.117
J.L. Schiff. Laplace Transform: Theory and Applications. Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-0-387-22757-3
D. Serre. Matrices: Theory and Applications. Springer, New York, 2nd edition edition, 2010. https://doi.org/10.1007/978-1-4419-7683-3
Z. Svoboda. Representation of solutions of linear differential systems of the second order with constant delays. J. Math. Sci., 222:345–358, 2017. https://doi.org/10.1007/s10958-017-3304-9
Z. You, J. Wang, D. O’Regan and Y. Zhou. Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices. Math. Meth. Appl. Sci., 42(3):954–968, 2019. https://doi.org/10.1002/mma.5400
F. Zhang. Matrix Theory: Basic Results and Techniques. Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4614-1099-7
J. Diblík, M. Fečkan and M. Pospíšil. Representation of a solution of the Cauchy problem for an oscillating system with two delays and permutable matrices. Ukr. Math. J., 65(1):64–76, 2013. https://doi.org/10.1007/s11253-013-0765-y
J.K. Hale and S.M. Verduyn Lunel. Introduction to Functional Differential Equations. Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-46124342-7
D.Ya. Khusainov and J. Diblík. Representation of solutions of linear discrete systems with constant coefficients and pure delay. Adv. Differ. Equ., 2006(Article ID 80825):13 pages, 2006. https://doi.org/10.1155/ADE/2006/80825
D.Ya. Khusainov, J. Diblík, M. Růžičková and J. Lukáčková. Representation of a solution of the Cauchy problem for an oscillating system with pure delay. Nonlinear Oscil., 11(2):276–285, 2008. https://doi.org/10.1007/s11072-008-0030-8
D.Ya. Khusainov and G. V. Shuklin. Linear autonomous time-delay system with permutation matrices solving. Stud. Univ. Zilina Math. Ser., 17:101–108, 2003.
N.N. Lebedev. Special Functions and their Applications. Prentice Hall, Inc., USA, 1965.
Ch. Liang, J. Wang and D. O’Regan. Controllability of nonlinear delay oscillating systems. Electron. J. Qual. Theory Differ. Equ., 47:1–18, 2017. https://doi.org/10.14232/ejqtde.2017.1.47
Ch. Liang, W. Wei and J. Wang. Stability of delay differential equations via delayed matrix sine and cosine of polynomial degrees. Adv. Differ. Equ., 2017(131):17 pages, 2017. https://doi.org/10.1186/s13662-017-1188-0
N.I. Mahmudov. Representation of solutions of discrete linear delay systems with non permutable matrices. Appl. Math. Lett., 85:8–14, 2018. https://doi.org/10.1016/j.aml.2018.05.015
M. Medved’ and M. Pospíšil. Sufficient conditions for the asymptotic stability of nonlinear multidelay differential equations with linear parts defined by pairwise permutable matrices. Nonlinear Anal.-Theory Methods Appl., 75(7):3348–3363, 2012. https://doi.org/10.1016/j.na.2011.12.031
M. Medved’ and M. Pospíšil. Representation and stability of solutions of systems of difference equations with multiple delays and linear parts defined by pairwise permutable matrices. Commun. Appl. Anal., 17(1):21–46, 2013.
M. Medved’ and M. Pospíšil. Representation of solutions of systems of linear differential equations with multiple delays and linear parts given by nonpermutable matrices. J. Math. Sci., 228(3):276–289, 2018. https://doi.org/10.1007/s10958-017-3620-0
M. Medved’, M. Pospíšil and L. Škripková. On exponential stability of nonlinear fractional multidelay integro-differential equations defined by pairwise permutable matrices. Appl. Math. Comput., 227:456–468, 2014. https://doi.org/10.1016/j.amc.2013.11.012
M. Pospíšil. Representation and stability of solutions of systems of functional differential equations with multiple delays. Electron. J. Qual. Theory Differ. Equ., 54:1–30, 2012. https://doi.org/10.14232/ejqtde.2012.1.54
M. Pospíšil and F. Jaroš. On the representation of solutions of delayed differential equations via Laplace transform. Electron. J. Qual. Theory Differ. Equ., 117:1– 13, 2016. https://doi.org/10.14232/ejqtde.2016.1.117
J.L. Schiff. Laplace Transform: Theory and Applications. Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-0-387-22757-3
D. Serre. Matrices: Theory and Applications. Springer, New York, 2nd edition edition, 2010. https://doi.org/10.1007/978-1-4419-7683-3
Z. Svoboda. Representation of solutions of linear differential systems of the second order with constant delays. J. Math. Sci., 222:345–358, 2017. https://doi.org/10.1007/s10958-017-3304-9
Z. You, J. Wang, D. O’Regan and Y. Zhou. Relative controllability of delay differential systems with impulses and linear parts defined by permutable matrices. Math. Meth. Appl. Sci., 42(3):954–968, 2019. https://doi.org/10.1002/mma.5400
F. Zhang. Matrix Theory: Basic Results and Techniques. Springer-Verlag, New York, 1999. https://doi.org/10.1007/978-1-4614-1099-7