Diffusive representation of pseudo-differential time-operators

Abstract

The concept of ''diffusive representation'' was previously introduced in the aim of transforming non standard convolutive causal operators such as fractional integrodifferential ones, into infinite-dimension dissipative classical input-output dynamic systems. The existence of an explicit dissipative semigroup makes then possible the use of classical tools of functional and numerical analysis of PDE, energy methods, control, filtering, etc., generally ill-fitted to the standard convolutive formulations, namely when long memory dynamics are present. The aim of the paper is to present in a synthetic statement, first the general frame of diffusive representations, and secondly some of the essential characteristics and properties of this new tool. Simple and concrete examples are given. More significant applications (essentially in the fractional context) will be found in the referenced papers.