Simultaneous stabilization of linear systems is a fundamental issue in system and control theory, and is of theoretical significance as well as practical interest. Some open problems have been proposed in the literature as indications of the potential complexity of simultaneous stabilization questions. These problems are simply-stated but are very hard to solve. The system and control community will benefit from the resolutions to these difficult problems. The first portion of this dissertation is devoted to some problems associated with simultaneous stabilization of linear systems. This part goes as follows: Firstly, researches on simultaneous stabilization are reviewed and some open problems are introduced. Secondly, two open problems, namely Belgian chocolate problem and French champagne problem are considered. For the Belgian chocolate problem, based on the recent development in automated inequality-type theorem proving, the exact of bounds which guarantee the existence of stabilizing controllers with fixed order no more than four are determined. Meanwhile, some numerical examples are worked out and improve the relevant results in the literature. In addition, two conjectures concerning this problem are formulated. For the French champagne problem, the explicit bounds which guarantee the existence of stabilizing controllers with fixed order no more than three are presented. Some numerical examples improve the relevant results in the literature. Thirdly, according to the results of complex analysis presented by Nehari, a condition proposed by Blondel is analyzed, which concerns the simultaneous stabilization of three linear systems. It is shown that there is an error in the sufficiency part of its proof. In the second portion of this dissertation, some high-degree global optimization problems concerning a class of triangle functions are considered. Based on a theoretical analysis and the algorithms as well as functions in the generic program BOTTEMA, an open problem posed by Yang which concerns the global optimization of a 15-th degree triangle function is solved completely. Meanwhile, the degree of triangle functions in solvable cases of similar problems can be improved by our method.