Gérard Montseny

Fractional integro-differential boundary control of the Euler-Bernoulli beam

Absorbing boundary conditions are generally associated to long-range memory behaviors. In the case of the Euler-Bernoulli beam, they are naturally based on Abel-Volterra operators of order 1/2. Diffusive realizations of them are introduced and used for the construction of an original and efficient boundary dynamic feedback control.

Simple approach to approximation and dynamical realization of pseudodifferential time operators such as fractional ones

We present a simple approach to the dynamical realization of a wide class of non-rational transfer functions, involving a natural and efficient method with rigorous theoretical background: the so-called diffusive representation. It includes the construction of finite-dimensional approximations for state-space realizations of a wide class of such operators.

Markovian diffusive representation of 1/f/sup /spl alpha// noises and application to fractional stochastic differential models

This paper is devoted to linear stochastic differential systems with fractional noise (or fractional Brownian motion) input. On the basis of a convenient Markovian description of such noises, elaborated from a diffusive representation of fractional integrators previously introduced in a deterministic context, the fractional differential system is equivalently transformed into a standard (but infinite-dimensional) one, with white-noise input. Finite dimensional approximations may easily be obtained from classical discretization schemes. With this equivalent representation, the correlation function of processes described by linear fractional stochastic differential systems may be expressed from the solution of standard differential systems, which generalizes, in some way, the well-known differential Lyapunov equation, which appears when computing the covariance matrix associated with standard linear stochastic systems.

Diffusive Representation for Operators Involving Delays

The theory of diffusive representation (DR) is essentially devoted to state-space realizations of integral operators of complex nature encountered in many concrete or theoretical situations. This approach has allowed to construct efficient solutions of non trivial problems in various fields (see [13], [9], [10]). Under their standard form, these state realizations are diffusive, which straightforwardly leads to cheap numerical approximations as well as dissipative properties useful for analysis or control purposes. This diffusive nature however imposes a restriction: the so-realized operators are pseudodifferential, which excludes in particular delay operators and so any operator involving delays.

Diffusive Realization of the Impedance Operator on Circular Boundary for 2D Wave Equation

A new formulation of the perfectly matched operator on circular boundary for 2D wave equation is introduced. It is based on the “diffusive representation”, useful for a wide class of causal operators and which enables exact and easily approximable time-local realisations of dissipative nature.