We study stability and robustness for a large class of linear time-varying systems under the assumption that the system possesses some kind of excitation, which is necessary for uniform attractivity of the origin, but not even boundedness of the solutions is assumed a priori . Our main statements provide strict Lyapunov functions, i.e., having a strictly negative-definite derivative, constructed based on an initial candidate whose derivative is sign-undefined. The Lyapunov function that we construct guarantees uniform global asymptotic stability and input-to-state stability with respect to bounded additive inputs. As a byproduct of our main results, we provide a Lyapunov function for a class of systems reminiscent of model-reference adaptive control with non-differentiable regressors.