Getting the Programs Ready

This research focuses on the dynamical system A-Pt where A and P are both matrices.

When A and P are 2 by 2 matrices:


When A and P are 3 by 3 matrices:


By Routh-Hurwitz Stability Criterion, the necessary and sufficient conditions for the related polynomials to be stable are:


In 2 by 2 cases, c1(t) is either constant or linear. a2(t) can be constant, linear or quadratic. In 3 by 3 cases, c1(t) is either linear or constant, while both c3(t) and (c1c2-c3)(t) have maximal order 3.

By solving the inequalities above, we will find intervals of t in which the system A-Pt is stable. In the past a few weeks, I wrote two Mathematica programs to that can compute the system’s stable region given a matrix A and a matrix P, one for 2 by 2 cases and the other for  3 by 3 cases.  Both programs will ask the user to enter matrices A and P entry-wise and then compute the stable region of system A-Pt for given matrices A and P. The programs involve solvers for polynomial inequalities of degree 1-3 and incorporate built-in functions to handle the union and intersection of continuous intervals. These two programs will be used as tools to find test examples.

An example of the output:


Programs (Static Web Page):    (2 by 2)   (3 by 3)

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