Systems with memory

Distributed parameter systems with memory are encountered in viscoelasticity and diffusion processes with complex molecular structure (nonFickian diffusion). In particular, certain thermodynamics problems fall in this class. We study the fundamental mathematical properties of this class of systems, following recent results obtained in this field, which show a rich  structure in terms of functional analysis, and a large set of sequences associated with such systems, which are Riesz bases in suitable Hilbert spaces. These facts have important consequences for the controllability and the solution of input identification problems for this class of systems and furthermore they show that quantities which are strongly related in the memoryless case are in fact largely independent in the presence of memory. For example, which reference to diffusion processes, the flux in the memoryless case is identified by the concentration while in the presence of memory these quantities are independent (for large times) in spite of the fact that the flux is determined by the past history of the concentration. This kind of analysis relies on techniques from  harmonic and  functional analysis.

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