# American Institute of Mathematical Sciences

January  2018, 23(1): 123-144. doi: 10.3934/dcdsb.2018008

## Asymptotic properties of delayed matrix exponential functions via Lambert function

 Brno University of Technology, CEITEC -Central European Institute of Technology, Purkyňova 656/123,612 00 Brno, Czech Republic

* Corresponding author: Z. Svoboda

Received  September 2016 Published  January 2018

In the case of first-order linear systems with single constant delay and with constant matrix, the application of the well-known "step by step" method (when ordinary differential equations with delay are solved) has recently been formalized using a special type matrix, called delayed matrix exponential. This matrix function is defined on the intervals $(k-1)τ≤q t<kτ$, $k=0,1,\dots$ (where $τ>0$ is a delay) as different matrix polynomials, and is continuous at nodes $t=kτ$. In the paper, the asymptotic properties of delayed matrix exponential are studied for $k\to∞$ and it is, e.g., proved that the sequence of values of a delayed matrix exponential at nodes is approximately represented by a geometric progression. A constant matrix has been found such that its matrix exponential is the "quotient" factor that depends on the principal branch of the Lambert function. Applications of the results obtained are given as well.

Citation: Josef Diblík, Zdeněk Svoboda. Asymptotic properties of delayed matrix exponential functions via Lambert function. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 123-144. doi: 10.3934/dcdsb.2018008
##### References:
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show all references

##### References:
 [1] N. H. Abel, Beweis eines Ausdruckes, von welchen die Binomial-Formel ein einzelner Fall ist, J. Reine Angew. Math., 1 (1826), 159-160.  doi: 10.1515/crll.1826.1.159. [2] A. Boichuk, J. Diblík, D. Khusainov and M. Růžičková, Fredholm's boundary-value problems for differential systems with a single delay, Nonlinear Anal., 72 (2010), 2251-2258.  doi: 10.1016/j.na.2009.10.025. [3] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D. J. Jeffrey and D.E. Knuth, On the Lambert W Function, Adv. Comp. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750. [4] F. R. Gantmacher, The Theory of Matrices, Volume 1, AMS Chelsea Publishing, 1998. [5] D. Ya. Khusainov and G.V. Shuklin, Linear autonomous time-delay system with permutation matrices solving, Stud. Univ. Žilina, Math. Ser., 17 (2003), 101-108. [6] D.Ya. Khusainov and G.V. Shuklin, On relative controllability in systems with pure delay, Int. Appl. Mech., 41 (2005), 210-221.  doi: 10.1007/s10778-005-0079-3. [7] V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1965-0. [8] J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physicomathematico-anatomico-botanico-medica, Band Ⅲ, (1758), 128{168. [9] E. E. Tyrtyshnikov, A Brief Introduction to Numerical Analysis, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-8136-4.
The curve ${\mathrm{Re}\,W_0(z)}=0$
Detailed eigenvalue domains
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