Guaranteed cost control of uncertain nonlinear neutral systems via memory state feedback (Q1775699)

A neutral differential system \[ \frac{d}{dt}[x(t)-A_{2}x(t-\tau)]=A_{0}x(t)+A_{1}x(t-h)+f_{0}(t,x(t))+ f_{1}(t,x(t-h))+Bu(t),\quad t\geq 0, \] \[ x(t)=\phi(t),\quad t\leq 0 \] with a feedback control \[ u(t)=-K(x(t)+\int_{t-h}^{t}A_{1}x(s)\,ds-A_{2}x(t-\tau)) \] and with the cost function \[ J=\int_{0}^{\infty}(x^{T}(t)Qx(t)+u^{T}Su(t))\,dt \] is considered. Here \( A_{i},i=0,1,2 \) and \(B\) are known constant matrices; \( \phi\) is a continuously differentiable function; \( f_{i},i=0,1 \), are perturbations satisfying the conditions \[ \|f_{i}(t,x)\|\leq \beta_{i}\|x \|, \quad i=0,1; \] \( K \) is a control gain; \( Q,S \) are given positive-definite matrices. The objective of this article is to design a feedback controller \( u(t)\) such that the resulting closed-loop system is asymptotically stable and the value of the cost function satisfies \( J\leq J^{*},\) where \(J^{*}\) is some specified constant. Two criteria for the existence of such a controller are derived in terms of a linear matrix inequality.