About fractional models physical consistency: Case of implicit differentiation based fractional order models

Abstract

As recently shown, a fractional model can be viewed as a doubly infinite model: its “real state” is of infinite dimension as it is distributed, but it is distributed on an infinite domain. It is shown in the paper, that this last feature induces a physically inconsistent property: the model real state has the ability to store an infinite amount of energy. This property demonstration is based on an electrical interpretation of fractional models. As a consequence, even if fractional models permit to capture accurately the input-output dynamical behavior of many physical systems, such a property highlights a physical inconsistence of fractional models: they do not reflect the reality of macroscopic physical systems in terms of energy storage ability. This property is shown for implicit fractional models and extends previous result of the authors for explicit fractional models.