Stability inquiry of $x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})\times H(t-3)$

Question

Determine if $x(t)$ is stable, where $x(t)$ is a solution to the differential equation:

$x'''+x''+4x'+4x = e^t(H(t) - H(t-3))$

and x(t) is defined as:

$x(t) = 0.1e^{-t} + 0.1e^t - 0.1\sin(2t) + 0.1e^3 (\sin(2(t-3)) + e^{-t+3}-e^{t-3})\times H(t-3)$

How do I approach this? Do I use eigenvalues? If so, I get 2i, - 2i and 1, is that correct and what do these values imply?

Any help is greatly appreciated!

Answer 1

The exponentially growing terms can (and should) be combined to $$ 0.1·e^t·(H(t)-H(t-3)) $$ so that it is plainly visible and also true under numerical evaluation that this term only contributes on the interval $[0,3]$.

All other terms are bounded or even decaying, so that the function is stable in a signal context and represents the stability of the zero solution of the homogeneous equation.

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PS: The eigenvalues are $-1$ and $\pm 2i$.