The so-called "pseudo-differential operators" (PDO) generalize classical transfer functions: any such PDO can indeed be expressed by its symbol (a time-frequency representation) H(t, p). Hypothesis on the function H are weak, then the scope of PDO is general. Differential operators, convolutive operators associated to rational transfer function and fractional integrodifferential operators are basic PDO. The set of PDO is an algebra: it therefore appears as a unified framework, whose scope extends to the general context of "time-variable" operators.
The concept of "diffusive representation" (DR) is based on a particular transformation of the symbol (or kernel) into an explicit distribution (the "diffusive symbol" µ(t, x)) whose main property is to generate, from Fubini's theorem, suitable state-space realizations of differential (time-local) form. Consequently, the concept of pseudo-differential operator reveals at once to be very useful, not only from theoretical but also practical points of view. Moreover, this equivalent transformation induces a number of remarkable properties. Finally, "diffusive realizations" of causal PDOs are well-adapted to the finitedimensional approximation problem which can be tackled via standard techniques of numerical analysis. This allows to build numerical state-space realizations at the same time simple, stable, precise and cheap.
Through the central concept of symbol, the DR joins together in a same algebra, rational convolutive operators, fractional operators and a large variety of more general operators of great practical interest. The objective of this worksop is to present and discuss recent advances and open questions in the domain, in the framework of the scientific group "Pseudodifferential operators and diffusive representation in modeling, control and signal".
P. Bidan
Organizer