Explicit Solutions and Stability of Linear Differential Equations with multiple Delays

We give an explicit formula for the general solution of a one dimensional linear delay differential equation with multiple delays, which are integer multiples of the smallest delay. For an equation of this class with two delays, we derive two equations with single delays, whose stability is sufficient for the stability of the equation with two delays. This presents a new approach to the study of the stability of such systems. This approach avoids requirement of the knowledge of the location of the characteristic roots of the equation with multiple delays which are generally more difficult to determine, compared to the location of the characteristic roots of equations with a single delay.

• Felix Che Shu

• Delay Differential Equation
• Explicit Solution
• Exponential Stability
• Lyapunov Exponents
• Multiple Delays.

<p>[1] J. K. Hale and S. M. V. Lunel, Theory of Functional Differential
Equations, Berlin: Springer, 1993.
[2] U. K¨uchler and B. Mensch, ”Langevins stochastic differential equation
extended by a time delayed term,” Stochastics and stochastic reports, vol.
40, 1992, pp. 23–42.
[3] F. C. Shu, ”Explicit solutions of irreducible linear systems of delay
differential equations of dimension 2,” Applied Mathematics E-Notes,
vol. 11, 2011, pp. 261–273.
[4] F. C. Shu, ”Stability of a 2 dimensional irreducible linear system of delay
differential equations,” Applied Mathematics E-Notes, vol. 12, 2012, pp.
36–43.</p>