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.
By using Lyapunov functions, we obtain, for the first time, necessary and sufficient conditions for the exponential stability of some nonlinear systems of differential and difference equations.In the present paper, we obtain necessary and sufficient conditions for the global exponential stability and exponential instability for one class of nonlinear systems of differential and difference equations. We investigate the exponential stability of perturbed systems with nonlinear first approximation.Note that the problem of the exponential stability of nonlinear systems is connected with the work by Grujic [1], in which sufficient conditions were obtained for the global exponential stability of one nonlinear complicated system with nonlinear isolated subsystems, which are assumed to be either exponentially stable or exponentially unstable.
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