Dynamical identification of a space varying coefficient in a system with persistent memory

In this paper we solve the problem of the identification of a coefficient which appears in the equation of a distributed system with persistent memory. This system is encountered for example when modelling the interconnection of linear elastic and viscoelastic systems. The additional data used in the identification are subsumed in the input output map from the deformation to the traction on the boundary. We extend a dynamical approach to identification introduced by Belishev in the case of purely elastic (memoryless) bodies and based on a special equation due to Blagoveshchenskiı̌. So, in particular, we extend Blagoveshchenskiı̌ equation to our class of systems with persistent memory.