Stability of Polynomial Systems via Polynomial Lyapunov Functions

The stability of a class of polynomial systems is investigated by constructing a polynomial Lyapunov function. The key technique is to convert the polynomial Lyapunov candidate and it derivative into formal quadratic forms and to test their positivity and negativity respectively. A new mathematical tool, semi-tensor product of matrices, is implemented to convert polynomials into their formal quadratic forms and vise versa, back and forth. Certain formulas are proposed for this purpose. The advantage of this approach is that the solvability of the problem can be converted into a set of algebraic conditions.