Application of Lyapunov functionals to studying stability of linear hyperbolic systems

The Lyapunov functional method is used to prove the stability conditions for Cauchy problems and initial-boundary value problems if the system is described by a set of linear first-order partial differential equations of the hyperbolic type. The application of the Lyapunov functional method to stability of linear hyperbolic systems with more than two equations leads to the search for functionals with diagonal matrices. The question of whether or not there exists a positive diagonal matrix G such that D^TG+GD, <0 or S^T S-G<0 does not have a simple answer. The characterization of the class of matrices D and S which have these properties is either a set of sufficient conditions or a set of necessary conditions