On a variation of the Gershgorin circle theorem with applications to stability theory

The conclusions of the Gershgorin circle theorem are significantly extended for the special case of real matrices. The extended result is applied to the problem of the absolute stability of both continuous- and discrete-time Lur´e systems containing sector nonlinearities. Specifically necessary conditions for the existence of common unic Lyapunov functions are presented in the form of constraints upon the root locus of the linear, time-invariant component.