Optimization approach to the robustness of linear delay systems (Q5958081)

Consider the time-delay system \[ \dot x(t)= Ax(t)+ f(t, x(t- h(t))),\tag{1} \] where \(A\in\mathbb{R}^{n\times n}\) is an asymptotically stable matrix, \(0\leq h(t)\leq\overline h\), \[ \|f(t, x(t- h(t)))\|\leq \beta\|x(t- h(t))\|,\quad \beta> 0.\tag{2} \] Theorem 1. The system (1) is uniformly asymptotically stable if the following condition holds \[ \beta< \overline\beta\equiv \sqrt{\lambda_{\min}(Q- P^2)}, \] where the positive-definite matrices \(Q\) and \(P\) satisfy the Lyapunov equation \[ A^T P+ PA= -Q, \] where \(Q= kQ_0\), \(Q_0> 0\) is an arbitrary matrix given by the designer, and \(k> 0\) is a sufficiently small scalar such that \(Q- P^2> 0\). Stability conditions are obtained by using the Lyapunov function \(V(x(t))= x^T(t) Px(t)\) and the boundedness of increase of the function \(f(t, x(t- h(t)))\) in (2).