One application of Lyapunov’s matrix equation

Abstract

This article considers application of Lyapunov's matrix equation to investigation of the sign definiteness of forms in the spaces R '~ or their octants.Suppose that some quadratic form V(,c) = < x, B~->. where x E R '~ and B r = B is some n x n matrix, is to be investigated for sign definiteness. We choose an arbitrary quadratic form U(x) =< z, C~ > that is sign definite in R ". which we henceforth call the "'standard," and we choose an auxiliary system of linear equations with constant coefficients .r' = Ax.(1).4 T = A, and the elements ay of this matrix are subject to determination in such a way thatFrom (2) we proceed to Lyapunov's matrix equation, which, in our case, takes the formIn relation (3) (2) and (3), we have the system of inhomogeneons linear equations Here. if/i = max~{p~} < 0, the form V(x) is negative definite in R '~. and ifp = min{p,} > 0, the form V(x) is positive definite.Necessity. Let the form V(x) be negative definite. As we noted above, in this case the matrix equation (3) has a unique solution for the matrix .4 and identity (2) holds, where the functions V(x) and I,(~) = U(x) with reversed signs. As a result, the zero solution of system (1) which, obviously, has a unique solution in the case under consideration. In place of auxiliary system (1). consider the system z' = .41z.

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

“…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: The following theorems hold, which are slight generalizations of the results of our paper [3]. …”

“…The method just examined for investigating the positive-and negative-definiteness of quadratic forms in a cone can be extended to homogeneous polynomials of degree three. Indeed, using Euler's theorem on homogeneous functions, we represent a form of degree three V(xl,..., xn) as The main results of this paper were reported at the International Mathematical Congress on "Lyapunov Readings" [4] in September of 1992 (in the city of Khar'kov).…”

“…Indeed, the eigenvalues of the triangular matrix A are its diagonal entries ajj; it therefore follows from (6) that the zero solution of the system (1) is exponentially unstable. When inequalities (5) hold, the cone K(al,..., a,,) is a positively invariant set [3] for the system (1), on which U(x) is a positive-definite function. From this it is easy to conclude that the quadratic form V(x) is positive-definite in' the cone K(ab..., an).…”

“…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.…”