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A large class of visco-elastic and elasto-plastic systems, frequently encountered in physics, are based on causal pseudo-differential operators, which are hereditary: the whole past of the state is involved in the dynamic expression of the system evolution. This generally induces major technical difficulties. We consider a specific class of pseudo-differential damping operators, associated to the so-called diffusive representation which enables to built augmented state-space realizations without heredity. Dissipativity property is expressed in a straightforward and precise way. Thanks to state-space realizations, standard analysis and approximation methods as well as control-theory concepts may therefore be used. [ABSTRACT FROM AUTHOR]

The stability of non-linear fractional differential equations is studied. A sufficient stability condition on the non-linearity is given for the input-output stability, thanks to many different reformulations of the system using diffusive representations of dissipative pseudo-differential operators. The problem of asymptotic internal stability is analyzed by a more involved functional analytic method. Finally, a fractional version of the classical Hartman-Grobman theorem for hyperbolic dynamical systems of order 1 is conjectured and reformulated, based upon known necessary and sufficient stability conditions for linear fractional differential equations. [ABSTRACT FROM AUTHOR]