I need to find a controller C(s) that can stabilize a system G(s) with the following transfer function:
G(s) = \$s^2/(s^2-1)\$
The overall system response Y(s) should be the convolution of the system transfer function G(s) and some control system C(s). In the frequency domain convolution becomes multiplication.
Y(s) = G(s) * C(s)
The criteria for stability is that the overall response Y(s) has a phase of less than 360 degrees for all points where the system has a gain greater than 1.
There is more than one answer here, because there a many different C(s) that could produce stable Y(s).
One trivial answer would be a control system that is the inverse of the system. That is C(s) = alpha / G(s) would lead to stability for all alpha less than 1.
It's quite easy mathematically, if not altogether desirable practically, to cancel the double zero at the origin. Then all that's needed is to introduce a numerator term that, essentially, overwhelms the -1 in the denominator and produces a stable 2nd order response when the loop is closed.