ECS-9531911 Pujara The main focus of this project is to investigate some necessary and sufficient conditions for an interval polytope of characteristic polynomial to contain a stable polynomial. In this project, this problem will be called the "Interval Polytope Problem". Note that a polytope is being called an interval polytope if it is generated by an uncertain characteristic polynomial with independent coefficients. The proposed investigation centers around the conjecture: if an interval polytope contains a stable polynomial, then the Kharitonov rectangle of at least one of the maximal subpolytopes contains a stable polynomial. There is a strong basis to surmise that the conjecture is true. This optimism is based on some very recent results of the PI which show that the conjecture is true for interval polytopes of dimension up to five. In addition to this, an investigation of numerical examples of interval polytopes of large order also reveals that the conjecture is true for these numerical examples. To make sure that the investigation of these numerical examples really truly reveals the truth, arbitrarily large and small bounds were considered around the coefficients of stable polynomials, ranging from six to nineteen degrees. A successful resolution of the interval polytope problem would lead to new techniques which will throw some additional light on a possible approach to solve the more difficult and important question: what are the necessary and sufficient conditions for a general polytope of polynomials to contain a stable polynomial? Two important applications of the general polytope case are also mentioned which are the main driving force behind this research area.
