Fractional Derivatives and Diffusive Representations: Semigroup formulation, Stability issues and Engineering Applications
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(14) . Matignon, Denis Fractional Derivatives and Diffusive Representations: Semigroup formulation, Stability issues and Engineering Applications. (2017) In: Control of Distributed Parameter Systems (CDPS 2017), 3 July 2017 - 7 July 2017 (Bordeaux, France).. .
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(24) Fractional Derivatives and Diffusive Representations: Semigroup formulation, Stability issues and Engineering Applications. Denis MATIGNON Universit´e de Toulouse & ISAE-Supa´ero, 10, av. Edouard Belin. 31400 Toulouse. France [email protected]. Abstract Viscoelastic materials are often characterized by a completely monotone kernel : this gives rise to dynamical systems involving a convolution term. These systems can be treated in a quite general framework, but still, an efficient way of tackling these convolution terms is to transform them into so-called diffusive representations. The idea is to add an extra memory variable to the original system, which helps suppress the convolution term : it amounts to a kind of a realization, in the sense of systems theory. In the linear case, the analysis of such an augmented system can be performed within the framework of evolution semigroups. Eventhough some Lyapunov functional is to be found for the augmented system, LaSalle’s invariance principle can not be applied to it, since a lack of compactness is to be found in the equivalent model : hence, for the proof of asymptotic stability property, we resort to Arendt-Batty theorem. In the second part of the talk, we will focus on the following worked-out examples, stemming from diverse engineering applications: • the Webster-Lokshin wave equation in musical acoustics, with a fractional derivative in time. Some numerical schemes will be presented to illustrate the behaviour of these viscous systems; • a sandwich structure composed of elastic and viscoelastic materials, described by the Zener model, including the frequency-dependence of the mechanical properties of a viscoelastic material; • the linearised Euler equations with impedance boundary conditions which arise in aeronautics, and makes use both of fractional derivatives and time delays: so-called parabolic-hyperbolic realization will be used to study this system theoretically, and to perfom numerical simulations; • the Burgers-Lokshin equation, a non-linear fractional PDE arising in the modelling of brass instruments, whence the famous brassy effect.. Collaborators: Part of the talk is joint work with Ghislain Haine, from ISAE-Supa´ero, and with Florian Monteghetti, Ph.D. student co-advised with E. Piot (Onera, France).. 1.
(25) References [1] De¨u , J.-F., and Matignon, D. (2010). Simulation of fractionally damped mechanical systems by means of a Newmark-diffusive scheme. Computers and Mathematics with Applications, vol. 5(5), pp. 1745–1753. [2] Haddar, H., Li, J.R., and Matignon, D. (2010). Efficient solution of a wave equation with fractional-order dissipative terms. Journal of Computational and Applied Mathematics, vol. 2(6), pp. 2003–2010. [3] H´elie, T., and Matignon, D. (2006). Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses. Mathematical Models and Methods in Applied Sciences, vol. 16(4), pp. 503–536. [4] H´elie, T., and Matignon, D. (2006). Representations with poles and cuts for the timedomain simulation of fractional systems and irrational transfer functions. Signal Processing, vol. 86(10), pp. 2516–2528. [5] Kergomard, J., Debut, V., and Matignon, D. (2006). Resonance modes in a onedimensional medium with two purely resistive boundaries: Calculation methods, orthogonality, and completeness. The Journal of the Acoustical Society of America, vol. 119(3), pp. 1356–1367. [6] Lombard, B. and Matignon, D. (2016). Diffusive approximation of a time-fractional Burger’s equation in nonlinear acoustics. SIAM Journal on Applied Mathematics, vol. 76(5), pp. 1765–1791. [7] Matignon, D. (1998). Stability properties for generalized fractional differential systems. In Fractional Differential Systems: Models, Methods and Applications (FDS’98), vol. 5, pp. 145–158. SMAI, Paris. [8] Matignon, D. (2009). An introduction to fractional calculus. In P. Abry, P. Gonc¸alv`es, and J. Levy-Vehel (eds.), Scaling, Fractals and Wavelets, vol. 1, Digital signal and image processing series, pp. 237–277. ISTE - Wiley. [9] Matignon, D. and Prieur, C. (2005). Asymptotic stability of linear conservative systems when coupled with diffusive systems. ESAIM: Control, Optimisation and Calculus of Variations, vol. 11(3), pp. 487–507. [10] Matignon, D. and Prieur, C. (2014). Asymptotic stability of Webster-Lokshin equation. Mathematical Control and Related Fields (MCRF), vol. 4(4), pp. 481–500. [11] Monteghetti, F., Haine, G., and Matignon, D. (2017). Stability of Linear Fractional Differential Equations with Delays: a coupled Parabolic-Hyperbolic PDEs formulation, Open Invited Track: Time-Delay Systems and PDEs, in Proc. 20th World Congress of the International Federation of Automatic Control, (IFAC), Toulouse, France. [12] Monteghetti, F., Matignon, D., Piot, E., and Pascal, L. (2016). Design of broadband timedomain impedance boundary conditions using the oscillatory-diffusive representation of acoustical models. The Journal of the Acoustical Society of America, vol. 140(3), pp. 1663–1674. [13] Monteghetti, F., Matignon, D., Piot, E., and Pascal, L. (2017). Asymptotic stability of the linearised Euler equations with long-memory impedance boundary condition. in Proc. 13th International Conference on Mathematical and Numerical Aspects of Wave Propagation (WAVES 2017) Minneapolis, USA..
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