A Remark on a Theorem of Lyapunov

Abstract

Consider the linear ordinary differential equation 1 where x ∊ Eⁿ, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:Theorem (Lyapunov). The following three statements are equivalent:(I) The spectrum σ(A) of A lies in the negative half plane.(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies 2 where ∥ ∥ denotes the Euclidean norm.(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there exist q₁,q₂>0 such that 3 satisfying 4 where I is the identity matrix.