Lyudmila Bantsur - The method of stability analysis of nonlinear systems using Liapunov functions in the form of norms is proposed

The Jacobian matrix set of the system on some positively invariant set is considered. It is show that the autonomous global Liapunov function in the form of norm exist if and only if the Jacobian matrice are simultaneously dissipative, i.e. such a norm exit that all the matrices considered generate contractive semigroups in this norm. Some conditions for simultaneous dissipativity are discussed. The conditions obtained allow to judge the stability ``in the large''. The conditions for the existence of the Liapunov norm for the perturbed autonomous system with positively homogeneous right-hand side are obtained.