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Abstract

We consider a class of elasto-plastic contact problems which can be described by parabolic evolutionary variational inequalities (EVI) in certain Lebesgue- and Sobolev-spaces. An observation (measurement), which maps from state space of the EVI in some observation space, is said to be asymptotically determining for the EVI if the asymptotic behaviour of an arbitrary solution is completely determined by the values of the observation along the trajectory of the solution. It will be shown that under certain conditions a monotone function of the observation bounds from above the free energy of the associated time-dependent state of the EVI. These conditions include properties of the frequency-domain characteristic of the linear part of the EVI, monotonicity of the material law, potential character of the contact functional, and properties of compactly and densely embedded Sobolev spaces. As a consequence we get that under such conditions the solutions of the elasto-plastic contact problem are asymptotically determined by their values in a few observation points at the body of contact, and that the fractal dimension of a compact negatively-invariant set of the EVI is finite. Note, that due to Takens (topological) embedding theorem, it is expected that one such observation point is sufficient.