On the stability analysis of nonlinear systems using polynomial Lyapunov functions (Q2468034)

This paper is devoted to the method of Lyapunov functions for studying the stability of the origin for systems of ordinary differential equations \(\dot x= F(x)\). Here, \(F\) is a polynomial vector field and \(F(0)= 0\). The simplest Lyapunov function candidates are quadratic forms (homogeneous polynomials of degree 2). Unfortunately, if \(F\) is nonlinear it is not always possible to find a Lyapunov function of this type. In this case, it is natural to seek a Lyapunov function in more general polynomial form (homogeneous or not). By exploiting the properties of the Kronecker product and tensor algebra, the authors state sufficient conditions for the existence of such functions, in the form of certain linear matrix inequalities.