Best to do some searches about this but here's a good site:
http://www.engr.udayton.edu/faculty/rkashani/mee527/nyq_stab/nyq_stability.htm
The Nyquist criterion is used to analyze a system's stability using frequency response methods. The RouthHurwitzCriterion? ( http://www-me.mit.edu/Lectures/RLocus/2.12-rationale.html) examines the characteristic polynomial.
A linear system is stable if and only if all the poles of the system TransferFunction have negative real parts. ControllabilityAndObservability for a system can be determined from its TransferFunction.