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: $$V(x(t),y(t))=\frac{1}{2}y(t)^2+\frac{1}{2}x(t)^2 $$

Updated January 22, 2018 21:20 PM

Updated October 28, 2017 18:20 PM

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