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 & 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.  Google Scholar

[2]

A. BoichukJ. DiblíkD. 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.  Google Scholar

[3]

R.M. CorlessG.H. GonnetD.E.G. HareD. J. Jeffrey and D.E. Knuth, On the Lambert W Function, Adv. Comp. Math., 5 (1996), 329-359.  doi: 10.1007/BF02124750.  Google Scholar

[4]

F. R. Gantmacher, The Theory of Matrices, Volume 1, AMS Chelsea Publishing, 1998.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

J. H. Lambert, Observationes variae in mathesin puram, Acta Helvetica, physicomathematico-anatomico-botanico-medica, Band Ⅲ, (1758), 128{168. Google Scholar

[9]

E. E. Tyrtyshnikov, A Brief Introduction to Numerical Analysis, Birkhäuser Boston, Inc., Boston, MA, 1997. doi: 10.1007/978-0-8176-8136-4.  Google Scholar

Figure 1.  The curve ${\mathrm{Re}\,W_0(z)}=0$
Figure 2.  Detailed eigenvalue domains
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