Stability of Linear Systems with the Generalized Lipschitz Property

We consider non-autonomous multivariable linear _. systems governed by the equation u = A(t)u with the matrix A(t) satisfying the generalized Lipschitz condition kA(t) − A(τ)k ≤ a(|t − τ |) (t, τ ≥ 0), where a(t) is a positive function. Explicit sharp stability conditions are derived. In the appropriate situations our results generalize and improve the traditional freezing method. An illustrative example is presented.