Kharitonov's theorem and the second method of lyapunov

In this paper Kharitonov's theorem for the robust stability of interval polynomials is proved using the second method of Lyapunov. The Hermite matrix is taken as the matrix of the quadratic form which is used as a Lyapunov function to prove Hurwitz stability. It is shown that if the four Hermite matrices corresponding to the four Kharitonov extreme polynomials are positive definite, the Hermite matrix of any polynomial of the polynomial family remains positive definite.