A noncombinatorial necessary and sufficient ´vertex´ solution for checking robust stability of polytopes of matrices induced by interval parameters

This paper first argues that the current research by the author on the issue of developing necessary and sufficient ´vertex´ solutions for checking the stability of matrix families needs to be evaluated on its own merit without any link to the computational complexity issues such as NP-hardness because it is not proven that P ≠ NP. Then it reviews the current ´necessary and sufficient vertex solution´ presented by the author for the ´interval polytopes´, that is, polytopes of matrices induced by ´interval´ parameters. The major features of the ´interval polytopes´ in contrast to general polytopes are highlighted. Finally, and more importantly, by exploiting the special nature of ´interval polytopes´, a new improved vertex solution, which is noncombinatorial in nature, is presented in this paper for that family. This new result, which clearly offers significant computational savings, should alleviate the concerns of computational complexity issue being raised in the context of this research and establishes that at least the ´interval polytope´ stability check is in P. Future research promises to point towards even more computational reduction possibilities for the ´interval polytope´ problem.