We derive absolute observation stability and instability results for controlled evolutionary inequalities which are based on frequency-domain characteristics of the linear part of the inequalities.The uncertainty parts of the inequalities (nonlinearities which represent external forces and constitutive laws) are described by certain local and integral quadratic constraints. Other terms in the considered evolutionary inequalities represent contact-type properties of a mechanical system with dry friction. The absolute stability criteria with respect to a class of observation operators (or measurement operators) gives the opportunity to prove the weak convergence of arbitrary solutions of inequalities to their stationary sets. In particular the obtained results can be used for the investigation of continuum type memories in neural networks. In such a way we can introduce a new type of neurons with hysteretic nonlinearities which are described by evolutionary variational inequalities.