Avoidance and intersection in the complex plane, a tool for simultaneous stabilization

The authors study the following problem: `under what condition(s) is it possible to find a single controller which stabilizes k single-input-single-output (SISO) linear time-invariant plants p_i(s) (i=l,. . .,k )?´. They show that the problem admits a solution if and only if an avoidance condition in the complex plane is satisfied, and this result is used to derive a sufficient condition for k plants to be simultaneously stabilizable