Global stability and stabilization of polynomial systems

The problem of global stability and stabilization of polynomial systems is considered. Using semi-tensor product of matrices, an easily verifiable sufficient condition for the positivity of multi-variable polynomials is proposed. Assume a candidate of Lyapunov function is a polynomial, the above result provides a sufficient condition for the global stability of a polynomial system. The computation for the conditions is formulated and can be done by computer. Then the result is used for global stabilization of polynomial systems via polynomial state feedback. Using semi-tensor product again, the conditions become a set of linear algebraic inequalities about the parameters in controls.