When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?

In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))⊂z∈C;ℜz≤−α for all t≥t0 and some α>0 is sufficient for uniform asymptotic stability of the linear time-varying system x˙=A(t)x. We prove that this class contains as a proper subset the matrix functions with the values in the special orthogonal group SO(n).