On the exponential stability of neutral linear systems with variable delays (Q2043924)

The authors establish a sufficient condition for the exponential stability of a system of linear neutral equations with time-varying delays \[ \dot{x}(t) - \sum_{i=1}^2 B_i \dot{x}(t -\tau_i(t)) = A_0 x(t) - \sum_{i=1}^2 A_i(t) x(t -\tau_i(t)), \] where \(x \in \mathbb{R}^n\). In the principal set of assumptions, the delays are bounded and a mandatory drift of the oldest memory is enforced (\(\dot{\tau} \leq \delta < 1\)). This stability condition is the satisfiability of a linear matrix inequality (LMI) ; the inequality leads to an explicit differential inequality \(\dot{W} \leq -2 \alpha W\) for a suitable Lyapunov-Krasovskii functional \(W\) composed of five discrete or distributed terms. This LMI depends on ten \(n \times n\) matrices and is illustrated in the paper with an example where \(n=2\), \(A_2 = B_2 = 0\) and \(A_1\) is constant but the delay \(\tau_1\) is time-varying. The authors also consider a less restrictive set of assumptions, without mandatory drift. The structure of the result is similar to the previous case; the Lyapunov-Krasovskii functional now has only three of the five terms that were used in the main case.