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We survey the problem of deciding the stability or stabilizability ofuncertain linear systems whose region of uncertainty is a polytope. Thisnatural setting has applications in many fields of applied science, fromControl Theory to Systems Engineering to Biology. We focus on the algorithmicdecidability of this property when one is given a particular polytope. Thissetting gives rise to several different algorithmic questions, depending on thenature of time (discrete/continuous), the property asked(stability/stabilizability), or the type of uncertainty (fixed/switching).Several of these questions have been answered in the literature in the lastthirty years. We point out the ones that have remained open, and we answer allof them, except one which we raise as an open question. In all the cases, theresults are negative in the sense that the questions are NP-hard. As abyproduct, we obtain complexity results for several other matrix problems inSystems and Control.

Polytopic uncertainty for linear systems: New and old complexity results