On quadratic Liapunov functionals for linear difference equations

This paper studies some stability properties of vector linear difference equations of the form x(t) = ∑kj=1 Ajx(t − rj), 0 < r1 < r2 < … < rk < ∞. This is done through the direct method of Liapunov, by means of a quadratic functional defined in an appropriate L2 space. The derivative of this functional along the equation, nevertheless, is defined in a finite dimensional euclidean space. The positive definiteness of certain matrices involved, together with the bounded real lemma of system theory, furnishes useful criteria to determine stability properties of the given equation. In particular, a criterion is given to determine stability with respect to the delays ri.