Viscoelastic materials and viscothermal losses: theory and numerics - - PowerPoint PPT Presentation

Viscoelastic materials and viscothermal losses: theory and numerics with applications to mechanical engineering and musical acoustics

• D. Matignon

denis.matignon@isae.fr

ISAE, DMIA & University of Toulouse

• E. Zuazua’s CIMI Excellence Chair on

" Control, PDEs, Numerics and Applications” — Wednesday, April 9th, 2014 — Toulouse, France.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 1 / 96

Outline

1

An introduction with examples Viscoelastic materials Viscothermal losses Fractional integrals and derivatives

2

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model

3

Stability of coupled diffusive models An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model

4

Non-linear models A damped pendulum The brassy effect

5

Conclusion

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 2 / 96

An introduction with examples

Outline

1

An introduction with examples Viscoelastic materials Viscothermal losses Fractional integrals and derivatives

2

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model

3

Stability of coupled diffusive models An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model

4

Non-linear models A damped pendulum The brassy effect

5

Conclusion

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 3 / 96

An introduction with examples Viscoelastic materials

Passive damping using viscoelastic materials

Constrained Layer Damping Example of application (Ariane V)

❚❤❡ ▲▼❙❙❈ ■♥tr♦❞✉❝t✐♦♥ ❱✐s❝♦❡❧❛st✐❝ ♠❛t❡r✐❛❧s ❙❛♥❞✇✐❝❤ str✉❝t✉r❡s ❘❡❞✉❝❡❞ ♦r❞❡r ♠♦❞❡❧s ■❋❙ ❈♦♥❝❧✉s✐♦♥

❈♦♥t❡①t ♦❢ ♠② t❤❡s✐s

❈♦♠♣♦s✐t❡ ♣r♦♣❡❧❧❡r ❢♦r s✉❜♠❛r✐♥❡ ❝❧❛ss ✷✶✷❆

✭P❛✉❧ ❡t ❛❧✱ ❚❤②ss❡♥❑r✉♣♣ t❡❝❤❢♦r✉♠ ✶✱ ✷✵✶✶✮✳

Viscoelastic material Elastic structure

❆♣♣❧✐❝❛t✐♦♥s ♦❢ ✈✐s❝♦❡❧❛st✐❝ ♠❛t❡r✐❛❧s✳

▲✉❝✐❡ ❘♦✉❧❡❛✉ ▼♦❞❡❧✐♥❣ ♦❢ s❛♥❞✇✐❝❤ str✉❝t✉r❡s ✇✐t❤ ✈✐s❝♦❡❧❛st✐❝ ❞❛♠♣✐♥❣ ✶✼✴✵✹✴✷✵✶✸ ✶✵✴✹✹

How to characterize the viscoelastic behavior and simulate the transient dynamic response of complex damped structures ?

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 4 / 96

An introduction with examples Viscoelastic materials

Modeling of viscoelastic materials

Existing viscoelastic models The complex modulus G∗(ω) = G′(ω) + iG′′(ω) is used to fit experimental data for a wide variety of materials over a large frequency range

❚❤❡ ▲▼❙❙❈ ■♥tr♦❞✉❝t✐♦♥ ❱✐s❝♦❡❧❛st✐❝ ♠❛t❡r✐❛❧s ❙❛♥❞✇✐❝❤ str✉❝t✉r❡s ❘❡❞✉❝❡❞ ♦r❞❡r ♠♦❞❡❧s ■❋❙ ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧✐♥❣ ♦❢ ✈✐s❝♦❡❧❛st✐❝ ♠❛t❡r✐❛❧s

❊①✐st✐♥❣ ✈✐s❝♦❡❧❛st✐❝ ♠♦❞❡❧s

• ❡♥❡r❛❧✐③❡❞ ▼❛①✇❡❧❧ ♠♦❞❡❧
• ∗(ω) = ●✵

 ✶ +

❏

• ❥=✶

✐ωτ❥γ❥ ✐ωτ❥ + ✶  

• ❍▼ ✭●♦❧❧❛✲❍✉❣❤❡s✲▼❝❚❛✈✐s❤✮ ♠♦❞❡❧ ✭●♦❧❧❛ ❡t ❛❧✱ ✶✾✾✸✮
• ∗(ω) = ●✵

 ✶ +

❏

• ❥=✶

ˆ α❥ −ω✷ + ✷✐ω ˆ ξ❥ ˆ ω❥ −ω✷ + ✷✐ω ˆ ξ❥ ˆ ω❥ + ˆ ω❥

✷

  ❆❉❋ ✭❆♥❡❧❛st✐❝ ❉✐s♣❧❛❝❡♠❡♥t ✜❡❧❞✮ ♠♦❞❡❧ ✭▲❡s✐❡✉tr❡ ✶✾✾✷✮

• ∗(ω) = ●✵

 ✶ +

❏

• ❥=✶

∆❥ ω✷ + ✐ωΩ❥ ω✷ + Ω✷

❥

  ❋r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♠♦❞❡❧

• ∗(ω) = ●✵ + ●∞(✐ωτ)α

✶ + (✐ωτ)α

▲✉❝✐❡ ❘♦✉❧❡❛✉ ▼♦❞❡❧✐♥❣ ♦❢ s❛♥❞✇✐❝❤ str✉❝t✉r❡s ✇✐t❤ ✈✐s❝♦❡❧❛st✐❝ ❞❛♠♣✐♥❣ ✶✼✴✵✹✴✷✵✶✸ ✶✽✴✹✹

Fractional derivative model has only 4 parameters (Go < G∞, τ > 0, 0 < α < 1), and it proves consistent with the framework of thermodynamics [Lion (1997)].

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 5 / 96

An introduction with examples Viscoelastic materials

Identification of viscoelastic constitutive law

DMA Metravib in a shear mode config. Measurments of the modulus amplitude and loss angle for Deltane 350 (Paulstra)

Environmental Thermometer Samples Grips chamber

❚❤❡ ▲▼❙❙❈ ■♥tr♦❞✉❝t✐♦♥ ❱✐s❝♦❡❧❛st✐❝ ♠❛t❡r✐❛❧s ❙❛♥❞✇✐❝❤ str✉❝t✉r❡s ❘❡❞✉❝❡❞ ♦r❞❡r ♠♦❞❡❧s ■❋❙ ❈♦♥❝❧✉s✐♦♥

▼♦❞❡❧✐♥❣ ♦❢ ✈✐s❝♦❡❧❛st✐❝ ♠❛t❡r✐❛❧s

■❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❢r❛❝t✐♦♥❛❧ ❞❡r✐✈❛t✐✈❡ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs

▲❡❛st sq✉❛r❡ ♠❡t❤♦❞

10 10

1

10

2

10

3

10

4

10

5

10

6

10

6

10

7

10

8

10

9

Frequency [Hz] Storage modulus [Pa] 10 10

1

10

2

10

3

10

4

10

5

10

6

0.2 0.4 0.6 0.8 1 Frequency [Hz] Phase angle φ [oC] Experimental Model Experimental Model

■❞❡♥t✐✜❡❞ ♣❛r❛♠❡t❡rs

• ∗(ω) = ●✵ + ●∞(✐ωτ)α

✶ + (✐ωτ)α ❘❡❧❛①❡❞ ♠♦❞✉❧✉s

• ✵ = ✶.✷✾ ▼P❛

❯♥r❡❧❛①❡❞ ♠♦❞✉❧✉s

• ∞ = ✵.✼✷ ●P❛

❘❡❧❛①❛t✐♦♥ t✐♠❡ τ = ✵.✷✹µs ❖r❞❡r ♦❢ t❤❡ ❞❡r✐✈❛t✐♦♥ α = ✵.✺

▲✉❝✐❡ ❘♦✉❧❡❛✉ ▼♦❞❡❧✐♥❣ ♦❢ s❛♥❞✇✐❝❤ str✉❝t✉r❡s ✇✐t❤ ✈✐s❝♦❡❧❛st✐❝ ❞❛♠♣✐♥❣ ✶✼✴✵✹✴✷✵✶✸ ✶✾✴✹✹

Assumptions : (i) isotropic material and (ii) Poisson’s ratio frequency independent E∗(ω) ∝ G∗(ω)

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 6 / 96

An introduction with examples Viscoelastic materials

Techniques for time-domain simulation

Resolution methods are based on :

1

either time discretization of the fractional dynamics [Grunwald-Letnikov, 1867], [Lubich, 1986], [Diethelm et al., 2005], [Galucio & Deü, 2004]...

2

• r diffusive representations [Monsteny, 1998], [Heleschewitz, 2000],

[Hélie & M., 2006], [Deü & M., 2010] For large-scale systems, the first method proves memory consuming, whereas the second has no hereditary behavior ; therefore, it seems more appropriate. See e.g. the works in collaboration with J.-F. Deü (CNAM, Paris) : Simulation of fractionally damped mechanical systems by means of a Newmark-diffusive scheme. Computer and Mathematics with Applications, 2010, vol. 5 (1745-1753). Time-domain finite element analysis of viscoelastic sandwich structures using fractional derivative models. Composite Structures, 2014.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 7 / 96

An introduction with examples Viscothermal losses

Trumpet-like instrument (I)

Decomposition into elementary subsystems. Transfer functions of interest : Reflection between p+

0 and p− 0 .

Transmission between p+

0 and p4.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 8 / 96

An introduction with examples Viscothermal losses

Trumpet-like instrument (II)

Time-domain representation Frequency-domain rep. Real-time simulations in Pure-Data environment on optimized models (with P ≤ 10) for each quadripole Qk : bounded frequency range, logarithmic scale & relative error : see [Hélie and M. (2006)].

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 9 / 96

An introduction with examples Viscothermal losses

A 1-D wave equation with viscothermal losses

The transfer function, in ℜe(s) > 0 : H(s) = 2Γ(s/c) s/c + Γ(s/c) e(s/c−Γ(s/c))L with Γ(s) =

• s2 + η s

3 2 + 1, and η > 0

is involved in the description of pressure waves in a duct of length L, with visco-thermal losses at the walls : a boundary layer phenomenon. η ∝ √lv + (γ − 1)√lh R ; R : radius of the duct, c ≃ 340 m.s−1 : sound speed, γ := Cp/Cv ≃ 1.4 : thermodynamic coefficient, lv ≃ 4.10−8 m : characteristic length of viscous effects, lh ≃ 6.10−8 m : characteristic length of thermal effects.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 10 / 96

An introduction with examples Viscothermal losses

A linear example : Webster-Lokshin model

The 1-D wave equation with fractional order damping in time ∂2

ttw +

• ηz ∂1/2

t

+ εz ∂−1/2

t

• ∂tw − 1

r2

z

∂z

• r2

z ∂zw

• = 0

is nothing but the coupling between :

1

a hyperbolic PDE : Webster horn equation

2

a parabolic PDE, thanks to memory variables and diffusive representations of the fractional integrals and derivatives. = ⇒ Some references on this acoustic model : [Kirchhoff (1868)], [Lokshin and Rok (1978)] (cited by [Dautray and Lions (1985)]), [Bruneau, Herzog, Kergomard & Polack (1991)], [Polack (1991)].

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 11 / 96

An introduction with examples Classical damping model for PDEs

Classical damping models for beams

Let us concentrate for a minute on classical damping models, e.g. in the case of the Euler-Bernoulli beam !

Let us examine : the undamped case : conservative, with fluid damping : dissipative, with structural damping : dissipative, with combined fluid & structural dampings : dissipative. How to understand these features ?

1

applied mathematics : energy balance,

2

automatic control : root locus in the Laplace plane,

3

signal processing : a spectrogram.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 12 / 96

An introduction with examples Classical damping model for PDEs

Modèle de barres dû à Euler et Bernoulli

⇒ Analysons un modèle standard de barre d’Euler-Bernoulli. ρ S . ∂2 ∂t2 u(t, x) + YI . ∂4 ∂x4 u(t, x) = 0, pour 0 < x < L , avec    ρ : densité du matériau Y : module de Young du matériau I = wh3

12

: moment géométrique de la barre , avec les conditions aux limites suivantes : pas de moment appliqué aux extrémités : ∂2

xu(t, 0) = 0,

∂2

xu(t, L) = 0;

pas de force en x = L, et une force f(t) en x = 0 : ∂3

xu(t, 0) = f(t),

∂3

xu(t, L) = 0.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 13 / 96

An introduction with examples Classical damping model for PDEs

Le modèle d’Euler- Bernoulli est conservatif

En définissant l’énergie mécanique totale selon : E(t) = 1 2 L ρS (∂tu)2 + YI (∂2

xu)2 dx ;

quand f = 0, on trouve, après quelques calculs : dE dt = 0 . Ce modèle est conservatif, ce qui a des conséquences spectrales et temporelles très fortes.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 14 / 96

An introduction with examples Classical damping model for PDEs

Analyse du modèle sans amortissement

ε = 0, η = 0 Lieu des pôles dans le plan de Laplace Spectrogramme

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 15 / 96

An introduction with examples Classical damping model for PDEs

Modèle de barres amorties

On choisit a priori de structurer l’amortissement selon : ρ S. ∂2 ∂t2 u(t, x) + ε ∂ ∂tu(t, x) + η ∂ ∂t ∂4 ∂x4 u(t, x) + Y I. ∂4 ∂x4 u(t, x) = 0 avec ε > 0 : amortissement fluide, η > 0 : amortissement structurel. Le bilan d’énergie sera : dE dt = −ε L (∂tu)2 dx − η L (∂t∂2

xu)2 dx ≤ 0 .

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 16 / 96

An introduction with examples Classical damping model for PDEs

Modèle avec amortissement de Rayleigh

(G) (C) (D) Rayleigh type dampings : spectrograms of ∂2

t u(t, L).

(G) : a = 4.10−2 and b = 3.10−9 (SI), sounds like a metallic bar, (C) : a = 2.10−2 and b = 5.10−8 (SI), sounds like a glass bar, (D) : a = 1.10−2 and b = 5.10−7 (SI) sounds like a wooden bar.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 17 / 96

An introduction with examples Classical damping model for PDEs

Analyse du modèle avec amortissement fluide

ε > 0, η = 0 Lieu des pôles dans le plan de Laplace Spectrogramme

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 18 / 96

An introduction with examples Classical damping model for PDEs

Analyse du modèle avec amortissement structurel

ε = 0, η > 0 Lieu des pôles dans le plan de Laplace Spectrogramme

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 19 / 96

An introduction with examples Classical damping model for PDEs

Analyse du modèle avec amortissement structurel

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 20 / 96

An introduction with examples Classical damping model for PDEs

Analyse du modèle avec amortissement structurel

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 21 / 96

An introduction with examples Classical damping model for PDEs

Analyse du modèle avec amortissements combinés : fluide et structurel

ε > 0, η > 0 Lieu des pôles dans le plan de Laplace Spectrogramme

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 22 / 96

An introduction with examples Fractional integrals and derivatives

Fractional damping models

Let us go back to non-classical damping models, with fractional derivatives !

Let us examine : the time-domain definitions, the Fourier-domain definitions, the Laplace-domain definitions.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 23 / 96

An introduction with examples Fractional integrals and derivatives

Definitions (time domain)

Let β ∈ (0, 1), and set hβ(t) :=

1 Γ(β) tβ−1 for t > 0 only ; for any

T > 0, let v ∈ L2(0, T), and define Iβv := hβ ⋆ v, i.e. : Iβv(t) = (hβ ⋆ v)(t) = t 1 Γ(β) τ β−1 v(t − τ) dτ . The Riemann-Liouville fractional integral of order β.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 24 / 96

An introduction with examples Fractional integrals and derivatives

Definitions (time domain)

Let β ∈ (0, 1), and set hβ(t) :=

1 Γ(β) tβ−1 for t > 0 only ; for any

T > 0, let v ∈ L2(0, T), and define Iβv := hβ ⋆ v, i.e. : Iβv(t) = (hβ ⋆ v)(t) = t 1 Γ(β) τ β−1 v(t − τ) dτ . The Riemann-Liouville fractional integral of order β. Let α ∈ (0, 1), and for any T > 0, let v ∈ H1(0, T), and define Dαv = D I1−αv or, more explicitly : Dαv(t) = d dt(h1−α ⋆ v)(t) = d dt t 1 Γ(1 − α) τ −α v(t − τ) dτ . The Riemann-Liouville fractional derivative of order α.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 24 / 96

An introduction with examples Fractional integrals and derivatives

Definitions (frequency domain-1)

Fractional integral of order β ∈ (0, 1). in Fourier space,

• hβ(f) = (2iπ f)−β ,

a causal low-pass filter, with a gain of −6 β dB per octave ; it is passive, since ℜe( hβ(f)) ∝ cos(β π 2 ) |f|−β > 0 .

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 25 / 96

An introduction with examples Fractional integrals and derivatives

Definitions (frequency domain-1)

Fractional integral of order β ∈ (0, 1). in Fourier space,

• hβ(f) = (2iπ f)−β ,

a causal low-pass filter, with a gain of −6 β dB per octave ; it is passive, since ℜe( hβ(f)) ∝ cos(β π 2 ) |f|−β > 0 . in the Laplace domain, Hβ(s) = s−β, in ℜe(s) > 0 ; it is passive, since for −π

2 < arg(s) < π 2 ,

ℜe(Hβ(s)) = |s|−β cos(β arg(s)) > 0 .

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 25 / 96

An introduction with examples Fractional integrals and derivatives

Definitions (frequency domain-2)

Fractional derivative of order α ∈ (0, 1). in Fourier space,

• d

dth1−α(f) = (2iπ f)α , a causal high-pass filter, with a gain of +6 α dB per octave ; it is passive, since ℜe(

• d

dth1−α(f)) ∝ cos(απ 2 ) |f|+α > 0 .

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 26 / 96

An introduction with examples Fractional integrals and derivatives

Definitions (frequency domain-2)

Fractional derivative of order α ∈ (0, 1). in Fourier space,

• d

dth1−α(f) = (2iπ f)α , a causal high-pass filter, with a gain of +6 α dB per octave ; it is passive, since ℜe(

• d

dth1−α(f)) ∝ cos(απ 2 ) |f|+α > 0 . in the Laplace domain,

• Hα(s) = sα,

in ℜe(s) > 0 ; it is passive, since for −π

2 < arg(s) < π 2 ,

ℜe(Hα(s)) = |s|α cos(α arg(s)) > 0 .

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 26 / 96

An introduction with examples Fractional integrals and derivatives

Questions

DRAWBACKS : (see §2, right now !) not differential operators, no semigroup associated to them a hereditary behaviour, with long-memory decay makes use of special functions, as fractional exponentials an effective approach, only in the case of commensurate orders SOLUTION ? To overcome these intrinsic difficulties, we propose to use the equivalent diffusive representations of fractional systems, giving rise to infinite-dimensional systems of integer order ! = ⇒ But, what are diffusive representations ? (please, wait until §3 !) see e.g. [Desch & Miller (1988)], [Staffans (1994)], [Montseny (1998)].

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 27 / 96

Stability of fractional models : FDEs and FPDEs

Outline

1

An introduction with examples Viscoelastic materials Viscothermal losses Fractional integrals and derivatives

2

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model

3

Stability of coupled diffusive models An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model

4

Non-linear models A damped pendulum The brassy effect

5

Conclusion

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 28 / 96

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Fractional Differential Equations

For 0 < α < 1, consider the input u – output y relation :

p

• k=0

ak Dkαy(t) =

q

• l=0

bl Dlαu(t), It is a causal pseudo-differential system, the symbol of which is, by Laplace transf. in some right-half plane C+

a := ℜe(s) > a :

H(s) = Q(sα) P(sα) with

• Q(σ)
• l=q
• l=0

bl σl P(σ)

• k=p
• k=0

ak σk .

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 29 / 96

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Necessary and Sufficient Stability Condition

From the input-output viewpoint, the BIBO-stability result reads as follows, y = h ⋆ u, with : Theorem (M. 1994) BIBO stability ⇐ ⇒    q ≤ p | arg λ| > α π

2 ,

∀λ ∈ C, / P(λ) = 0 In which case, h has the long-memory asymptotics : h(t) ∼ K t−1−α as t → +∞.

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 30 / 96

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Sketch of the proof

Algebraic tools can be used, since the orders are commensurate to the same α : let P and Q two coprime polynomials, and let R = Q/P the rational function, we get the structure result : Proposition h(t) =

N

• n=1

mn

• m=1

rnm E⋆m

α (λn, t),

with R(σ) = N

n=1

mn

m=1 rnm (σ − λn)−m ;

where E⋆m

α (λ, t) denotes a Mittag-Leffler function (a

hypergeometric special function), the Laplace transform of which is (sα − λ)−m. Remark : for α = 1, it reduces to the well-known causal polynomial–exponential E⋆m

1 (λ, t) := 1 m! tm−1 exp(λ t), for t ≥ 0.

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Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

• N. & S. Stability condition : an illustration

Stability of Eα(λ tα) with LT sα−1(sα − λ)−1, as a fct. of arg(λ). Laplace plane : s σ-plane

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Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (I)

t → Eα(λ tα) for α = 1

2 and arg(λ) = 0

Real part Imaginary part

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 33 / 96

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (II)

t → Eα(λ tα) for α = 1

2 and arg(λ) = π/8

Real part Imaginary part

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Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (III)

t → Eα(λ tα) for α = 1

2 and arg(λ) = π/4

Real part Imaginary part

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 35 / 96

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (IV)

t → Eα(λ tα) for α = 1

2 and arg(λ) = 3π/8

Real part Imaginary part

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Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (V)

t → Eα(λ tα) for α = 1

2 and arg(λ) = π/2

Real part Imaginary part

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Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (VI)

t → Eα(λ tα) for α = 1

2 and arg(λ) = 3π/4

Real part Imaginary part

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Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory

Mittag-Leffler functions in C (VII)

t → Eα(λ tα) for α = 1

2 and arg(λ) = π

Real part Imaginary part

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Stability of fractional models : FDEs and FPDEs Example 1 : a 1-dof oscillator

Influence of fractional order α

Example : ¨ u(t) + 0.5 (dαu)(t) + u(t) = 0 with u0 = 0, v0 = 1.

0.5 1 5 10 15 20 −1 1 t α u −1 −0.5 0.5 1 − 1 0.2 0.4 0.6 0.8 1 α u v

Displacement versus time Phase portrait Remark : a continuous behaviour between undamped (α = 0) and viscous damping (α = 1) can be observed, [Deü and M. (2010)]

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Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model

The Lokshin model is non standard

Let ∂2

t w + 2 η ∂

1 2

t ∂tw + η2 ∂tw − ∂2 xw = 0,

t > 0, x ∈ ]0, 1[ with init. cond. w(t = 0) = 0 and ∂tw(t = 0) = 0, and controlled dynamic boundary conditions at x = 0 ; the system is being observed at x = 1. the damping is modelled by a fractional derivative.

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Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model

The Lokshin model is non standard

Let ∂2

t w + 2 η ∂

1 2

t ∂tw + η2 ∂tw − ∂2 xw = 0,

t > 0, x ∈ ]0, 1[ with init. cond. w(t = 0) = 0 and ∂tw(t = 0) = 0, and controlled dynamic boundary conditions at x = 0 ; the system is being observed at x = 1. the damping is modelled by a fractional derivative. there is no simple energy property, unlike in the classical cases of fluid (∂1

t w) or structural (−∂1 t ∂2 xw) dampings.

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 41 / 96

Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model

The Lokshin model is non standard

Let ∂2

t w + 2 η ∂

1 2

t ∂tw + η2 ∂tw − ∂2 xw = 0,

t > 0, x ∈ ]0, 1[ with init. cond. w(t = 0) = 0 and ∂tw(t = 0) = 0, and controlled dynamic boundary conditions at x = 0 ; the system is being observed at x = 1. the damping is modelled by a fractional derivative. there is no simple energy property, unlike in the classical cases of fluid (∂1

t w) or structural (−∂1 t ∂2 xw) dampings.

the spatial modes are no more orthogonal, due to the boundary conditions.

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 41 / 96

Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model

The Lokshin model is non standard

Let ∂2

t w + 2 η ∂

1 2

t ∂tw + η2 ∂tw − ∂2 xw = 0,

t > 0, x ∈ ]0, 1[ with init. cond. w(t = 0) = 0 and ∂tw(t = 0) = 0, and controlled dynamic boundary conditions at x = 0 ; the system is being observed at x = 1. the damping is modelled by a fractional derivative. there is no simple energy property, unlike in the classical cases of fluid (∂1

t w) or structural (−∂1 t ∂2 xw) dampings.

the spatial modes are no more orthogonal, due to the boundary conditions. But still, a closed-form solution is available, from which stability can be proved !

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Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model

Analytic solution of a fractional PDE

Let (∂2

t + 2 η ∂

3 2

t + η2 ∂1 t )w − ∂2 x w = 0,

t > 0, x ∈ ]0, 1[ with init. cond. w(t = 0) = 0 and ∂tw(t = 0) = 0, and dynamical boundary conditions of absorbing type (a0b0 > 0, a1b1 > 0) and controlled at x = 0 :    [a0 (∂t + η ∂

1 2

t ) + b0 ∂−x)] w(t, x = 0) = a0 (∂t + η ∂

1 2

t ) u(t)

[a1 (∂t + η ∂

1 2

t ) + b1 ∂x)] w(t, x = 1) = 0

with output : y(t) := w(t, x = 1). Then y = h ⋆ u, with h(t) =

+∞

• n=−∞

cη

n

• E 1

2 (ση+

n , t) − E 1

2 (ση−

n , t)

• ∈ L1(R+) ∩ C∞(R+)

where the ση±

n

are the roots of σ2 + ησ = s0

n := −α0 + 2iπf 0 n .

= ⇒ BIBO stability comes from | arg(ση±

n )| > π 4 , ∀n ∈ Z.

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Stability of coupled diffusive models

Outline

1

An introduction with examples Viscoelastic materials Viscothermal losses Fractional integrals and derivatives

2

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model

3

Stability of coupled diffusive models An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model

4

Non-linear models A damped pendulum The brassy effect

5

Conclusion

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Stability of coupled diffusive models An introduction to diffusive representations

Some useful identities (1)

For δ > 1, consider the numerical identity : ∞ dx 1 + xδ =

π δ

sin(π

δ ) .

Let s ∈ R+

∗ , substitute x = (ξ s )

1 δ in it, and get :

∞ sin(π

δ )

π 1 ξ1− 1

δ

1 s + ξ dξ = 1 s1− 1

δ

. An analytic continuation from R+∗ to C \ R− leads to the first functional identity with 1 − 1

δ := β ∈ (0, 1) :

Hβ : C \ R− → C s → ∞ µβ(ξ) 1 s + ξ dξ = 1 sβ , with density µβ(ξ) = sin(β π) π ξ−β .

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 44 / 96

Stability of coupled diffusive models An introduction to diffusive representations

Some useful identities (2)

Applying an inverse Laplace transform to both sides yields to the second functional identity : hβ : R+ → R t → ∞ µβ(ξ) e−ξ t dξ = 1 Γ(β)tβ−1 . So to speak, fractional kernels hβ are nothing but a superposition of decaying exponential, with an appropriate weight µβ. From these identities, we can easily derive both : Input-output representations, (signal processing viewpoint) State-space representations, (automatic control viewpoint) for both fractional integrals and derivatives.

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Stability of coupled diffusive models An introduction to diffusive representations

Input-output representation

Let v and y = Iβv be the input and output of the causal fractional integral of order β ∈ (0, 1) : y(t) = ∞ µβ(ξ) [eξ ⋆ v](t) dξ , with eξ(t) := e−ξ t 1t≥0, and [eξ ⋆ v](t) = t

0 e−ξ (t−τ) v(τ) dτ.

Now for fractional derivative of order α ∈ (0, 1), let v the input, and

• y = Dαv = D[I1−αv] the output.

= ⇒ a careful computation shows that :

• y(t) =

∞ µ1−α(ξ) [v − ξ eξ ⋆ v] (t) dξ ; which makes use both of v and eξ ⋆ v.

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Stability of coupled diffusive models An introduction to diffusive representations

State-space representation (1) : setting

Let ϕ(ξ, .) := [eξ ⋆ v](t) be the state, parametrized by ξ > 0. ∂tϕ(ξ, t) = −ξ ϕ(ξ, t) + v(t), ϕ(ξ, 0) = 0 , (1) y(t) = ∞ µβ(ξ) ϕ(ξ, t) dξ ; (2) and ∂t ϕ(ξ, t) = −ξ ϕ(ξ, t) + v(t), ϕ(ξ, 0) = 0 , (3)

• y(t)

= ∞ µ1−α(ξ) [v(t) − ξ ϕ(ξ, t)] dξ . (4) are state-space representations for Iβ and Dα, respectively. Note : functional spaces must be specified for these representations to make sense ; more precisely : ϕ belongs to Hβ := {ϕ s.t. ∞ µβ(ξ)|ϕ|2 dξ < ∞},

• ϕ belongs to

Hα := { ϕ s.t. ∞ µ1−α(ξ)| ϕ|2 ξ dξ < ∞} ; see e.g. [Haddar and M. (2008)], [M. and Zwart (2004)].

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 47 / 96

Stability of coupled diffusive models An introduction to diffusive representations

State-space representation (2) : energy functionals

with storage function Eφ(T) := 1 2 ∞ φ(ξ, T)2 µβ(ξ) dξ, the energy balance holds ∀T > 0 for fractional integrals : T v(t) y(t) dt = Eφ(T) + T +∞ ξ µβ(ξ) φ(ξ, t)2 dξ dt .

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 48 / 96

Stability of coupled diffusive models An introduction to diffusive representations

State-space representation (2) : energy functionals

with storage function Eφ(T) := 1 2 ∞ φ(ξ, T)2 µβ(ξ) dξ, the energy balance holds ∀T > 0 for fractional integrals : T v(t) y(t) dt = Eφ(T) + T +∞ ξ µβ(ξ) φ(ξ, t)2 dξ dt . with storage function Ee

φ(T) := 1

2 ∞

• φ(ξ, T)2 ξ µ1−α(ξ) dξ, the

energy balance holds ∀T > 0 for fractional derivatives : T v(t) y(t) dt = Ee

φ(T) +

T +∞ µ1−α(ξ) (v − ξ φ)2 dξ dt .

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 48 / 96

Stability of coupled diffusive models An introduction to diffusive representations

Finite-dimensional approximations of DR (1)

an RC-circuit is passive and low-pass (−6 dB/oct) : Let HRC(s) = 1 s + ξ , with ξ > 0 ⇒ ℜe(HRC(s)) = ℜe(s) + ξ |s + ξ|2 ≥ 0, for ℜe(s) ≥ 0

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Stability of coupled diffusive models An introduction to diffusive representations

Finite-dimensional approximations of DR (1)

an RC-circuit is passive and low-pass (−6 dB/oct) : Let HRC(s) = 1 s + ξ , with ξ > 0 ⇒ ℜe(HRC(s)) = ℜe(s) + ξ |s + ξ|2 ≥ 0, for ℜe(s) ≥ 0 a GL-circuit is passive and high-pass (+6 dB/oct) : Let HGL(s) = s s + ξ , with ξ > 0 ⇒ ℜe(HGL(s)) = |s|2 + ℜe(s) ξ |s + ξ|2 ≥ 0, for ℜe(s) ≥ 0

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Stability of coupled diffusive models An introduction to diffusive representations

Finite-dimensional approximations of DR (2)

Positive aggregation of RC-circuits is passive and low-pass (−6K dB/oct) : HK

RC(s) = K

• k=1

µk 1 s + ξk with µk > 0.

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Stability of coupled diffusive models An introduction to diffusive representations

Finite-dimensional approximations of DR (2)

Positive aggregation of RC-circuits is passive and low-pass (−6K dB/oct) : HK

RC(s) = K

• k=1

µk 1 s + ξk with µk > 0. Positive aggregation of GL-circuits is passive and high-pass (+6L dB/oct) : HL

GL(s) = L

• l=1

νl s s + ξl with νl > 0.

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Stability of coupled diffusive models An introduction to diffusive representations

Passivity : example of GL-circuit with L = 1 d.o.f.

Modulus (dB) Phase (deg)

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Stability of coupled diffusive models An introduction to diffusive representations

Passivity : example of GL-circuit with L = 2 d.o.f.

Modulus (dB) Phase (deg)

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 52 / 96

Stability of coupled diffusive models An introduction to diffusive representations

Passivity : example of GL-circuit with L = 3 d.o.f.

Modulus (dB) Phase (deg)

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Stability of coupled diffusive models An introduction to diffusive representations

Passivity : from finite to infinite d.o.f. (1)

[Oustaloup (1983)] : Systèmes asservis linéaires d’ordre fractionnaire, Masson.

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Stability of coupled diffusive models An introduction to diffusive representations

Passivity : from finite to infinite d.o.f. (2)

Recall that : For ℜe(s) > 0, ∞ µβ(ξ) 1 s + ξ dξ = 1 sβ with µβ(ξ) ∝ 1 ξβ and also For ℜe(s) > 0, ∞ να(ξ) s s + ξ dξ = sα with να(ξ) ∝ 1 ξ1−α = ⇒ Fractional integrals & derivatives are nothing but continuous positive aggregations of RC & GL circuits !

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 55 / 96

Stability of coupled diffusive models An introduction to diffusive representations

Conclusions on DR

The main advantages of the diffusive realizations are : the existence of an associated semigroup (on an augmented state-space), the dissipativity of the realization (whenever the operator is positive), the possibility of deriving numerical schemes without heredity, hence memory-saving algorithms.

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 56 / 96

Stability of coupled diffusive models Example 3 : a 1-dof oscillator

A 1-dof oscillator (1) : with FD and FI

The analysis of a 1-d.o.f oscillator, ¨ x + ˜ y + ˙ x + y + ω2 x = 0 , damped by : ˙ x = v, instantaneous w.r.t v, y(v), with memory, low-pass behaviour, (e.g. (µk)1≤k≤K or µβ) ˜ y(v), with memory, high-pass behaviour, (e.g. (νl)1≤l≤L or να) cannot be understood through the mechanical energy alone : Em := 1 2v2 + 1 2ω2 x2 , since d dtEm = v(¨ x + ω2 x) = −v2 − ˜ yv − yv ≤ ≥ 0? has no definite sign ! But still, thanks to DR...

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 57 / 96

Stability of coupled diffusive models Example 3 : a 1-dof oscillator

RECALL the two useful energy equalities !

with storage function Eφ(T) := 1 2 ∞ φ(ξ, T)2 µβ(ξ) dξ, the energy balance holds ∀T > 0 for fractional integrals : T v(t) y(t) dt = Eφ(T) + T +∞ ξ µβ(ξ) φ(ξ, t)2 dξ dt . with storage function Ee

φ(T) := 1

2 ∞

• φ(ξ, T)2 ξ µ1−α(ξ) dξ, the

energy balance holds ∀T > 0 for fractional derivatives : T v(t) y(t) dt = Ee

φ(T) +

T +∞ µ1−α(ξ) (v − ξ φ)2 dξ dt .

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 58 / 96

Stability of coupled diffusive models Example 3 : a 1-dof oscillator

A 1-dof oscillator (2) : with DR

The analysis of a 1-d.o.f oscillator, ¨ x + ˜ y + ˙ x + y + ω2 x = 0 , damped by : ˙ x = v, instantaneous w.r.t v, y(v), with memory, low-pass behaviour, (e.g. (µk)1≤k≤K or µβ) ˜ y(v), with memory, high-pass behaviour, (e.g. (νl)1≤l≤L or να) can be easily understood through the augmented energy : E := Em + Eφ + ˜ E˜

φ

• f the global system, with internal variables [x, v, φ, ˜

φ], since : d dtE = −v2 −

K

• k=1

µk ξk φ2

k − L

• l=1

νl (v − ξl ˜ φl)2 ≤ 0 . = ⇒ in finite dimension (2+K+L), asymptotic stability follows.

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 59 / 96

Stability of coupled diffusive models Example 3 : a 1-dof oscillator

A 1-dof oscillator (3) : numerical simulation

¨ ϑ + η ∂−β

t

˙ ϑ + ϑ = 0, for β = 0.75 and (ϑ0, ˙ ϑ0) = (3.5, 0). (L) (C) (R) (L) : Evolution of angle ϑ and angular velocity ˙ ϑ, (C) : section of the phase portrait in the (ϑ, ˙ ϑ)-plane, (R) : Evolution of diffusive components {φk(t) = φ(t, ξk)}1≤k≤K for K = 25.

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 60 / 96

Stability of coupled diffusive models Example 3 : a 1-dof oscillator

A 1-dof oscillator (4) : Asymptotic stability

When y = Iβv and ˜ y = Dαu, we compute, at least formally : d dtE = −v2 − ∞ µβ(ξ) ξ φ2(ξ) dξ − ∞ να(ξ) (v − ξ ˜ φ(ξ))2 dξ ≤ 0 . Is this formal computation enough to bring a rigorous proof ? Maybe not, because in infinite dimension, LaSalle’s invariance principle asks for the precompactness of the trajectories in the energy space, which is not easy to get a priori. This is the reason why we resort to the stability result by Arendt–Batty

• r Lyubich–Phong (1988) : the refined spectral analysis of the

infinitesimal generator −A of the semigroup on the Hilbert state H enables to prove the result of internal asymptotic stability ; the proof is a bit involved (for an FDE, Lax–Milgram theorem is being used). = ⇒ read all the details in [M. and Prieur, 2005].

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 61 / 96

Stability of coupled diffusive models Example 4 : Webster-Lokshin model

What about Webster-Lokshin model ?

For z ∈ (0, 1), with r(z) > 0, η(z), ε(z) ≥ 0, w(t, z) satisfies : ∂2

t w + η(z) ∂3/2 t

w + ε(z) ∂1/2

t

w − 1 r2 ∂z(r2 ∂zw) = 0 ; with static boundary conditions in z = 0 and z = 1. The model is non standard, since : there is no simple energy property, due to fractional derivatives in time,

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 62 / 96

Stability of coupled diffusive models Example 4 : Webster-Lokshin model

What about Webster-Lokshin model ?

For z ∈ (0, 1), with r(z) > 0, η(z), ε(z) ≥ 0, w(t, z) satisfies : ∂2

t w + η(z) ∂3/2 t

w + ε(z) ∂1/2

t

w − 1 r2 ∂z(r2 ∂zw) = 0 ; with static boundary conditions in z = 0 and z = 1. The model is non standard, since : there is no simple energy property, due to fractional derivatives in time, the coefficients are variable with space : η → η(z), and ε → ε(z).

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 62 / 96

Stability of coupled diffusive models Example 4 : Webster-Lokshin model

What about Webster-Lokshin model ?

For z ∈ (0, 1), with r(z) > 0, η(z), ε(z) ≥ 0, w(t, z) satisfies : ∂2

t w + η(z) ∂3/2 t

w + ε(z) ∂1/2

t

w − 1 r2 ∂z(r2 ∂zw) = 0 ; with static boundary conditions in z = 0 and z = 1. The model is non standard, since : there is no simple energy property, due to fractional derivatives in time, the coefficients are variable with space : η → η(z), and ε → ε(z). But still, existence, uniqueness and asymptotic stability can be proved, using (diffusive) realization theory and involved stability theorems (Arendt-Batty), but not LaSalle’s invariance principle, since... a lack of compactness is to be found in this model !

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 62 / 96

Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Rewriting the Webster-Lokshin model

For z ∈ (0, 1), with r(z) > 0, η(z), ε(z) ≥ 0, w(t, z) satisfies : ∂2

t w + η(z) ∂3/2 t

w + ε(z) ∂1/2

t

w − 1 r2 ∂z(r2 ∂zw) = 0 ; with static boundary conditions in z = 0 and z = 1. This is equivalent to the first-order system in (p, v) : ∂tp = −r−2 ∂zv − ε ∂−1/2

t

p − η ∂1/2

t

p , ∂tv = −r2 ∂zp , p(z = 0, t) = and v(z = 1, t) = 0 . Use of standard DR for ∂−1/2

t

, and extended DR for ∂1/2

t

.

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Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Existence et uniqueness (I)

With L2

p = {p,

1

0 p2r2 dz < ∞}, L2 v = {v,

1

0 v2r−2 dz < ∞}, and

H = L2

p × L2 v × L2(0, 1; HM; ε r2 dz) × L2(0, 1;

HN; η r2 dz), the system can be put in the abstract form ∂tX + A X = 0, where : A     p v ϕ

• ϕ

    =     r−2 ∂zv + ε +∞ ϕ dM + η +∞ [p − ξ ϕ] dN r2 ∂zp ξϕ − p ξ ϕ − p     ; D(A) =        (p, v, ϕ, ϕ)T ∈ V ,

• p(0) = 0

v(1) = 0 (p − ξϕ) ∈ L2(0, 1; HM; ε r2 dz) (p − ξ ϕ) ∈ L2(0, 1; VN; η r2 dz)        . with V = H1

p × H1 v × L2(0, 1; VM; ε r2 dz) × L2(0, 1;

HN; η r2 dz).

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Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Existence et uniqueness (II)

Theorem (Haddar and M. (2008)) The operator A : D(A) ⊂ H → H is maximal monotone. The monotonicity of A comes from the energy identity : ∀X ∈ D(A), (AX, X)H = 1 ϕ2

e HMε r2 dz +

1 p − ξ ϕ2

HNη r2 dz ≥ 0 .

Corollary Hille-Yosida theorem enables to conclude to the existence and uniqueness of a strong solution to the original problem. Note : in case of dynamical boundary conditions, the Kalman-Yakubovich-Popov lemma will be used to realize the ouput impedance, which is a positive real rational function of s.

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Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Internal asymptotic stability : a difficult question

The proof is difficult. Why ? In fact, LaSalle’s invariance principle requires, in infinite dimension, the hypothesis of precompactnes of the trajectories ; but this latter hypothesis cannot be checked a priori for diffusive realizations, since a diffusion equation in an unbounded domain is hidden behind them, and the canonical embedding from H1(R) into L2(R) is not compact (Rellich theorem does not apply).

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Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Internal asymptotic stability : a difficult question

The proof is difficult. Why ? In fact, LaSalle’s invariance principle requires, in infinite dimension, the hypothesis of precompactnes of the trajectories ; but this latter hypothesis cannot be checked a priori for diffusive realizations, since a diffusion equation in an unbounded domain is hidden behind them, and the canonical embedding from H1(R) into L2(R) is not compact (Rellich theorem does not apply). The refined spectral analysis of the infinitesimal generator −A of the semigroup on the Hilbert state H enables to use the stability result by Arendt–Batty or Lyubich–Phong, et helps prove the result

• f internal asymptotic stability ; the proof is quite involved (it

makes use of the Fredholm alternative for an FPDE, since Lax–Milgram is not sufficient, like in the FDE case).

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Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Stability theorem, by Arendt & Batty, 1988

Theorem (Arendt and Batty (1988)) Let us consider the infinitesimal generator A of a bounded C0-semigroup on a reflexive Banach space. Assume that no eigenvalue

• f A lies on the imaginary axis. If σ(A) ∩ iR is countable, then the

semigroup generated by A is asymptotically stable, which means that the solutions of the differential equation x′(t) = Ax(t) tend to 0 with t → ∞. Note that, following our notations, the infinitesimal generator of the semigroup, defined by (3), is denoted by −A. Remark : in [Lyubich and Phong (1988)], the same result appeared independently.

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Stability of coupled diffusive models Example 4 : Webster-Lokshin model

Analysis of the spectrum of the generator

Sketch of the proof [M. and Prieur (2014)]. No spectrum in the right-half plane, since −A generates a contraction semigroup, it is necessarily a bounded semigroup, hence σ(−A) ∩ {s ∈ C, ℜe(s) > 0} = ∅. Spectrum on the imaginary axis ?

1

Solving for A X = 0 leads to X = 0, thanks to the boundary conditions in (3). Hence, λ = 0 is not an eigenvalue of −A ; but λ = 0 ∈ σc(−A), the continuous spectrum of −A.

2

In order to prove σ(−A) ∩ {iω, ω ∈ R, ω = 0} = ∅, we show the continuity of the resolvent (iω I + A)−1 on H : given any Y = (f, g, χ, χ) ∈ H, we seek some X = (p, v, φ, φ) ∈ D(A), such that (iω I + A) X = Y. Algebraic equation for the last two components, and a variational formulation in p, but the real part of the continuous sesquilinear form on H1

p × H1 p has no definite sign, hence we can

not apply the complex version of Lax-Milgram theorem. Therefore, we have to resort to the Fredholm alternative to conclude.

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Stability of coupled diffusive models Some numerical illustrations

Definition and Analysis of numerical schemes

Fractional derivatives are difficult to numerically approximate, and usually involve hereditary algorithms, thus turning into memory storage problems on the computer.

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Stability of coupled diffusive models Some numerical illustrations

Definition and Analysis of numerical schemes

Fractional derivatives are difficult to numerically approximate, and usually involve hereditary algorithms, thus turning into memory storage problems on the computer. Standard numerical approximations of the extended system with DR enable to define memoryless numerical schemes ; more precisely the schemes have finite memory, once the {ξk}1≤k≤K have been chosen.

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Stability of coupled diffusive models Some numerical illustrations

Definition and Analysis of numerical schemes

Fractional derivatives are difficult to numerically approximate, and usually involve hereditary algorithms, thus turning into memory storage problems on the computer. Standard numerical approximations of the extended system with DR enable to define memoryless numerical schemes ; more precisely the schemes have finite memory, once the {ξk}1≤k≤K have been chosen. The proof of convergence of the numerical schemes is based on discrete extended energy techniques, which mimick the principle

• f the extended energy for the continuous system, see e.g.

[Haddar, Li and M. (2010)].

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Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 69 / 96

Stability of coupled diffusive models Some numerical illustrations

Definition and Analysis of numerical schemes

Fractional derivatives are difficult to numerically approximate, and usually involve hereditary algorithms, thus turning into memory storage problems on the computer. Standard numerical approximations of the extended system with DR enable to define memoryless numerical schemes ; more precisely the schemes have finite memory, once the {ξk}1≤k≤K have been chosen. The proof of convergence of the numerical schemes is based on discrete extended energy techniques, which mimick the principle

• f the extended energy for the continuous system, see e.g.

[Haddar, Li and M. (2010)]. Some Illustrations !

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Stability of coupled diffusive models Some numerical illustrations

A linear example : Lokshin model, role of η (I)

Output signal. Wave & augmented (- -) energies For a cylinder (ηx = η), in blue η = 0.1, in red η = 0.

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Stability of coupled diffusive models Some numerical illustrations

A linear example : Lokshin model, role of η (II)

Output signal. Wave & augmented (- -) energies For a cylinder (ηx = η), in blue η = 0.2, in red η = 0.

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Stability of coupled diffusive models Some numerical illustrations

A linear example : Lokshin model, role of η (III)

Output signal. Wave & augmented (- -) energies For a cylinder (ηx = η), in blue η = 0.5, in red η = 0.

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Stability of coupled diffusive models Some numerical illustrations

Trapped modes in Webster-Lokshin model

Output signal. Wave & augmented (- -) energies. Trapped modes in a duct with two cones facing each other, in blue ε = 0.2 and η = 0.05, in red ε = η = 0.

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Non-linear models

Outline

1

An introduction with examples Viscoelastic materials Viscothermal losses Fractional integrals and derivatives

2

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model

3

Stability of coupled diffusive models An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model

4

Non-linear models A damped pendulum The brassy effect

5

Conclusion

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Non-linear models

Un résultat général

Theorem (Haddar and M. (2004)) Soit H un espace de Hilbert, soit A : D(A) ⊂ H → H un opérateur maximal monotone, et F une fonction non-linéaire F : H → H telle que le pb. d’évolution semi-linéaire : ∂tX + A X = F(X), et X(0) = X0 ∈ D(A) soit bien posé, pour t ∈ [0, Tmax), au sens de l’existence et de l’unicité de X ∈ C1([0, Tmax); H) ∩ C0([0, Tmax); D(A)), une solution forte. Alors, pour deux OPD de type diffusif et positif, l’un standard hM⋆ et l’autre étendu par dérivation hN⋆, le pb. pseudo-différentiel non-linéaire : ∂tX + hM ⋆ X + hN ⋆ X + A X = F(X), et X(0) = X0 ∈ D(A) est bien posé, pour t ∈ [0, T′max), au sens d’une unique solution forte X ∈ C1([0, T′max); H) ∩ C0([0, T′max); D(A)).

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Non-linear models

Perturbation diffusive de systèmes non-linéaires conservatifs

Remark L ’hypothèse du théorème précédent est vérifiée dans le cas où la non-linéarité est localement lipschitzienne sur H. Corollary Si le système différentiel non-linéaire de départ est conservatif, alors comme l’énergie étendue associée au système perturbé est décroissante et bornée par sa valeur initiale, le résultat d’existence locale se prolonge en un résultat d’existence globale, i.e. T′

max = +∞.

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Non-linear models A damped pendulum

A toy model : non-linear pendulum

¨ ϑ + η ∂−β

t

˙ ϑ + sin(ϑ) = 0 ; β = 0.75 et (ϑ0, ˙ ϑ0) = (3.5, 0). (G) (C) (D) (G) : Evolution of angle ϑ and angular velocity ˙ ϑ, (C) : Section of the phase portrait in the (ϑ, ˙ ϑ)-plane, (D) : Evolution of the diffusive components {φk(t) = φ(t, ξk)}1≤k≤K for K = 25.

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Non-linear models The brassy effect

The one-way Burgers-Lokshin equation

The one-way Burgers-Lokshin equation reads : ∂tu + c ∂xu + ε ∂α

t u + b ∂x(u2/2) = 0

for t > 0 , (5) with initial datum u(t = 0, x) = u0(x) at t = 0. The coefficients are : c > 0 the sound speed, ε > 0, which takes into account the specific length of both viscous and thermal effects and the radius of the duct, α ∈ (0, 1) the fractional order, b ≥ 0, or Burgers coefficient, quantifies the nonlinear effects.

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Non-linear models The brassy effect

Some references

nonlinearity (model) : Burgers model, [Sugimoto 1991], [Makarov & Ochmann 1997], [Menguy & Gilbert 2000], ... fractional derivative (model) : Lokshin model, [Lokshin & Rok 1978], [Bruneau, Herzog, Kergomard & Polack 1991], [Polack 1991], ... nonlinearity (solution) : Volterra series [Hélie & Hasler 2004], [Hélie & Smet 2008], ... fractional derivative (solution) : diffusive representations [Staffans 1994], [DM 1994], [Montseny 1998], [Haddar & DM 2008], [Hélie & DM 2006], [Blanc Chiavassa & Lombard 2013], ... See the work in collaboration with B. Lombard (LMA, Marseille) : Diffusive approximation of a fractional Burgers equation arising in nonlinear acoustics. Journal of Computational Physics, 2014.

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Non-linear models The brassy effect

Definition of the semi-discretized problem (2)

Burgers-Lokshin model is now transformed into a first-order coupled system :              ∂u ∂t + c ∂u ∂x + b ∂ ∂x u2 2

• = −ε

L

• ℓ=1

µℓ φℓ, ∂φj ∂t + γα θ2α−1

j

• c ∂u

∂x + b ∂ ∂x u2 2

• = −θ2

j φj − γα θ2α−1 j

ε L

ℓ=1 µℓ φℓ.

(6) which can be put in the following abstract form : ∂ ∂tU + ∂ ∂xF(U) = S U, (7)

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Non-linear models The brassy effect

Analysis : notations

the vector of (N + 1) unknowns is U =

• u

φ1 . . . φN T the nonlinear flux function F = (F1 F2 . . . FN+1)T is F1 = c u + b u2 2 , Fj = γα θ2α−1

j

F1, j = 2 · · · N + 1, the (N + 1) × (N + 1) diffusive matrix S is S = −          ε µ1 · · · ε µN θ2

1 + Ω µ1

· · · Ω µN . . . . . . . . . . . . Ω µ1 · · · θ2

N + Ω µN

         , with Ω = γα ε.

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Non-linear models The brassy effect

Analysis : results

1

The eigenvalues of the Jacobian matrix J = F

′ of the nonlinear

flux function F are real : c + b u, and 0 with multiplicity 2 N. Moreover, they do not depend on the coefficients of the diffusive representation ! = ⇒ the nonlinear part of the system is hyperbolic.

2

The eigenvalues of the diffusive matrix S are real and negative. Moreover, they interlace the values −θ2

j .

= ⇒ the linear part of the system is parabolic.

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Non-linear models The brassy effect

Numerical method : Strang splitting

1

Build up two independent schemes : ∂ ∂tU + ∂ ∂xF(U) = 0 = ⇒ numerical scheme Ha (8) ∂ ∂tU = S U = ⇒ numerical scheme Hb (9)

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Non-linear models The brassy effect

Numerical method : Strang splitting

1

Build up two independent schemes : ∂ ∂tU + ∂ ∂xF(U) = 0 = ⇒ numerical scheme Ha (8) ∂ ∂tU = S U = ⇒ numerical scheme Hb (9)

2

Implement the time-marching scheme between tn & tn+1 :

• U(1)

j

= Hb(∆t

2 ) Un j ,

• U(2)

j

= Ha(∆t) U(1)

j

,

• Un+1

j

= Hb(∆t

2 ) U(2) j

. (10) = ⇒ 2nd order precision in ∆t, see e.g. [LeVeque 2002].

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Non-linear models The brassy effect

Numerical results (1)

(L) (C) (R) Gaussian pulse : snapshots of u. (L) : inviscid (ε = 0), (C) : slightly dissipative ε = 2, for α = 0.2, (R) : strongly dissipative ε = 10, for α = 0.2.

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Non-linear models The brassy effect

Numerical results (2)

(L) (C) (R) Gaussian pulse : snapshots of u, and energy. (L) : slightly dissipative ε = 2, for α = 0.5, (C) : strongly dissipative ε = 10, for α = 0.5, (R) : time evolution of the energy, with time of break t∗ = 0.002s.

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Non-linear models The brassy effect

Sismograms : b = 0

(L) (C) (R) Gaussian pulse : snapshots of u (L) : slightly dissipative ε = 1, for α = 0.5, (C) : mildly dissipative ε = 2, for α = 0.5, (R) : strongly dissipative ε = 5, for α = 0.5.

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Non-linear models The brassy effect

Sismograms : b = 1

(L) (C) (R) Gaussian pulse : snapshots of u (L) : slightly dissipative ε = 1, for α = 0.5, (C) : mildly dissipative ε = 2, for α = 0.5, (R) : strongly dissipative ε = 5, for α = 0.5.

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Conclusion

Outline

1

An introduction with examples Viscoelastic materials Viscothermal losses Fractional integrals and derivatives

2

Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model

3

Stability of coupled diffusive models An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model

4

Non-linear models A damped pendulum The brassy effect

5

Conclusion

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Conclusion

Special thanks to the following co-authors : Brigitte d’Andréa-Novel, Alain Oustaloup, Jaques Audounet, Gérard Montseny, Thomas Hélie, Houssem Haddar, Christophe Prieur, Hans Zwart, Jean Kergomard, Rémi Mignot, Jean-Francois Deü, Jing-Rebecca Li, Bruno Lombard.

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Bibliography

Some references on the Lokshin model

• A. A. LOKSHIN, Wave equation with singular retarded time, Dokl.
• Akad. Nauk SSSR, 240 (1978), pp. 43–46. (in Russian).
• A. A. LOKSHIN AND V. E. ROK, Fundamental solutions of the wave

equation with retarded time, Dokl. Akad. Nauk SSSR, 239 (1978),

• pp. 1305–1308. (in Russian).
• D. MATIGNON AND B. D’ANDRÉA-NOVEL, Spectral and

time-domain consequences of an integro-differential perturbation

• f the wave PDE, in 3rd WAVES conf., Mandelieu, France, April

1995, INRIA, SIAM, pp. 769–771.

• D. MATIGNON, J. AUDOUNET, AND G. MONTSENY, Energy decay

for wave equations with damping of fractional order, in 4th WAVES conf., Golden, Colorado, June 1998, INRIA, SIAM, pp. 638–640.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 90 / 96

Bibliography

Some references on the Webster-Lokshin model

• T. HÉLIE, D. MATIGNON,Diffusive representations for the analysis

and simulation of flared acoustic pipes with visco-thermal losses, Mathematical Models and Methods in Applied Sciences, 16 (2006), pp. 503–536.

• H. HADDAR, D. MATIGNON, Theoretical and numerical analysis of

the Webster-Lokshin model, Tech. Rep., Institut National de la Recherche en Informatique et Automatique (INRIA), 2008.

• H. HADDAR, J.R. LI, D. MATIGNON, Efficient solution of a wave

equation with fractional-order dissipative terms, Journal of Computational and Applied Mathematics, vol. 2 (2010).

• pp. 2003-2010.
• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 91 / 96

Bibliography

Some references on stability (I)

• W. ARENDT AND C. J. K. BATTY, Tauberian theorems and stability
• f one-parameter semigroups, Trans. Amer. Math. Soc., 306

(1988), pp. 837–852.

• B. D’ANDRÉA-NOVEL, F. BOUSTANY, F. CONRAD, AND B. RAO,

Control of an overhead crane : stabilization of flexibilities, Math. Control, Signals & Systems, 7 (1994), pp. 1–22.

• R. F. CURTAIN, Old and new perspectives on the positive–real

lemma in systems and control theory, Z. Angew. Math. Mech., 79 (1999), pp. 579–590.

• Z. H. LUO, B. Z. GUO, AND O. MORGUL, Stability and stabilization
• f infinite dimensional systems with applications, Communications

and Control Engineering, Springer Verlag, 1999.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 92 / 96

Bibliography

Some references on stability (II)

• Y. LYUBICH AND V. PHÓNG, Asymptotic stability of linear

differential equations on Banach spaces, Studia Mathematica, 88 (1988), pp. 37–42.

• D. MATIGNON, Stability properties for generalized fractional

differential systems, ESAIM : Proc., 5 (1998), pp. 145–158.

• D. MATIGNON AND C. PRIEUR, Asymptotic stability of linear

conservative systems when coupled with diffusive systems, ESAIM : COCV, 11 (2005), pp. 487–507.

• D. MATIGNON , Asymptotic stability of the Webster-Lokshin model,

in MTNS, Kyoto, Japan, jul 2006. (invited session).

• D. MATIGNON AND C. PRIEUR, Asymptotic stability of

Webster-Lokshin equation, MCRF, 3 (2014), to appear.

• A. RANTZER, On the Kalman–Yakubovich–Popov lemma,

Systems & Control Letters, 28 (1996), pp. 7–10.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 93 / 96

Bibliography

Some references on diffusive representations (I)

• O. J. STAFFANS, Well-posedness and stabilizability of a

viscoelastic equation in energy space, Trans. Amer. Math. Soc., 345 (1994), pp. 527–575.

• G. MONTSENY, Diffusive representation of pseudo-differential

time-operators, ESAIM : Proc., 5 (1998), pp. 159–175.

• G. MONTSENY, J. AUDOUNET, AND D. MATIGNON, Diffusive

representation for pseudo-differentially damped non-linear systems, in Nonlinear control in the year 2000, Vol. 2, vol. 259 of Lecture Notes in Control and Inform. Sci., Springer, 2001,

• pp. 163–182.
• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 94 / 96

Bibliography

Some references on diffusive representations (II)

• G. DAUPHIN, D. HELESCHEWITZ, AND D. MATIGNON, Extended

diffusive representations and application to non-standard

• scillators, in MTNS, Perpignan, France, June 2000, 10 p. (invited

session).

• D. MATIGNON AND H. ZWART, Standard diffusive systems are

well-posed linear systems, in MTNS, Leuven, Belgium, jul 2004. (invited session).

• T. HÉLIE AND D. MATIGNON, Representations with poles and cuts

for the time-domain simulation of fractional systems and irrational transfer functions, Signal Processing, 86 (2006), pp. 2516-2528.

• D. Matignon (ISAE)

Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 95 / 96

Bibliography

Some... extra references !

• G. MONTSENY, J. AUDOUNET, AND D. MATIGNON, Fractional

integrodifferential boundary control of the Euler–Bernoulli beam, in CDC, San Diego, California, dec. 1997, pp. 4973–4978. (invited session).

• H. HADDAR AND D. MATIGNON, Well-posedness of non-linear

conservative systems when coupled with diffusive systems, in NOLCOS, Stuttgart, Germany, sep. 2004, vol. 1, pp. 251–256.

• J. KERGOMARD, V. DEBUT, AND D. MATIGNON, Resonance modes

in a 1-D medium with two purely resistive boundaries : calculation methods, orthogonality and completeness, J. Acoust. Soc. Amer., 119 (2006), pp. 1356-1367.

• H. ZWART, Transfer functions for infinite-dimensional systems,

Systems Control Lett., 52 (2004), pp. 247–255.

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