A study of positive- and negative-definiteness of homogeneous polynomials of degrees two and three in a cone using Lyapunov's second method

Abstract

We propose a solution of the problem inverse to the well-known problem of constructing Lyapunov functions for linear systems with constant coe~icients, making it possible to obtain new conditions for positive-and negative-definiteness of forms of degrees two and three in an arbitrary octant of the space R n.Consider a linear system of differential equationswhere x E R '~ and A is an upper triangular n x n matrix with letter entries. Let V(x) = (x, Bx) be a quadratic form to be investigated for positive-or negative-definiteness in a certain cone K(al,..., an) of the space R n, where as are the parameters of the cone [1] (for example, the normegative cone K~ of the space R ~ has parameters as = 1 (s = 1--,-,n)). Take any form U(x) = (x, Cx) that is positive-definite in the cone K(al,..., a~) under consideration, but not in the entire space R n. From now on we shall say that the form U(x) is standard. We require that system (1) imply the equality = v(x),from which we arrive at the Lyapunov matrix equationwhere the matrix C is to be determined. Thus we have the problem inverse to the well-known Lyapunov problem [2]. We remark that in our paper [3] we considered the case of a symmetric matrix C. The choice of an upper triangular matrix simplifies the computations and makes it possible to generalize the basic results of [3] slightly. We denote the elements of the matrices A, B, and C by a,k, b,k, and csk respectively. Equating like terms on the left-and right-hand sides of (2), we obtain a system of N = n(n + 1)/2 equations to determine the numbers ask, which decomposes into n subsystems: where c13 ~-c13 -bl3all, c23 = c23 -b13a12 -b23a22, c33 : c33;