On delay-dependent global asymptotic stability for pendulum-like systems (Q1039378)

This paper deals with global asymptotic stability of the trivial solution of the delayed nonlinear pendulum-like system \[ x'(t) = Px(t) + P_hx(t-h) + R\varphi(S^Tx(t)), \] where \(h>0\) is a given constant, \(P, P_h, R\) and \(S\) are real constant matrices of dimensions \((n+1)\times(n+1)\), \((n+1)\times(n+1)\), \((n+1)\times m\) and \((n+1)\times m\) respectively, and the mapping \(\varphi\) satisfies \(\varphi_i(\sigma) = \varphi_i(\sigma_i)\) with \(\sigma = (\sigma_1, \sigma_2, \dots, \sigma_m)\). With an appropriate linear transformation, the above system can be written in a canonical form. Then global asymptotic stability and robust global asymptotic stability are obtained under the requirement of existence of a positive number and some (at least 12) matrices satisfying certain matrix inequalities.