In the present paper the authors establish new sufficient conditions for the global asymptotic stability of linear systems operating on the unit hypercube in R-n and of linear systems with partial state saturation constraints. They also determine a set of necessary and sufficient conditions for the global asymptotic stability of second order linear systems Kith saturation constraints on both states using the above results. Systems of the type considered herein are widely used in several areas of applications, including control systems, signal processing, and artificial neural networks.