Determine the stability of the solution $x=0$ of the equation $mx''(t)+bx'(t)+qx'(t-r)+kx(t)=0$

by AzJ   Last Updated October 29, 2017 14:20 PM

The Problem

Discuss the stability of the solution $x(t)=0$ of the equation $$mx''(t)+bx'(t)+qx'(t-r)+kx(t)=0$$ where $m>0,b\geq 0,q\geq 0,$ and $k\geq 0$ are all constants, by constructing an appropriate lyapunov functional.

Why I need help

I am having difficulty constructing an appropriate lyapunov functional needed to determine the stability as I do not have much experience with delayed differential equations.

I am also not sure that my overall approach is correct as I have not see a problem of this type (a delay differential equation of order 2) before.

Also note that lyapunov functionals, are similar but different to lyapunov functions.

My Work on the problem (so far):

First we use lyapunov functionals to determine stability (rather than use them to determine instability)as this will allow the determination of asymptomatic stability if $x=0$ is stable. If $x=0$ is not stable, then it is unstable.

In an attempt to make the problem simpler we rewrite the equation to eliminate the second derivative: let $y(t)=x'(t)$ then \begin{align} y'(t)=x''(t)&=-\frac{b}{m}x'(t)-\frac{q}{m}x'(t-r)-\frac{k}{m}x(t)\\ &=-\frac{b}{m}y(t)-\frac{q}{m}y(t-r)-\frac{k}{m}x(t) \end{align} thus we get the following system \begin{cases} x'(t)=y(t) \\ y'(t)=-\frac{1}{m}(b y(t)+q y(t-r)+k x(t)) \end{cases}

Here is my attempt at a appropriate lyapunov functional: $$V(x(t),y(t))=\frac{1}{2}y(t)^2+\frac{1}{2}x(t)^2 $$

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