Time-varying version of the lemma of Lyapunov

The lemma of Lyapunov gives the necessary and sufficient condition for the stability of a time-invariant system in terms of the existence of a positive-definite symmetric matrix. In this letter, the lemma is generalised for time-varying systems, in which it is shown that additional conditions are required for the lemma to hold. The conditions involve boundedness of certain quantities and uniform complete observability of a pair of matrixes.