Parameter-dependent Lyapunov functions for polytopes of polynomials

The authors show that certain types of polytope of polynomials have parameter-dependent Lyapunov functions. The functions are quadratic ones with coefficients being just the Hermite matrix whose positive definiteness ensures Hurwitz stability of polynomials. It is demonstrated that the polytopes of polynomials have corresponding polytopes of Lyapunov functions and that thereby stability of the polytopes comes from that of their extreme polynomials. The results obtained lead to an alternative proof for some known results, including weak Kharitonov´s theorem, via the Lyapunov route and would possibly provide some tool for searching links between the Lyapunov approach and established frequency domain results on stability of systems with structured uncertainties