An introduction with examples Viscothermal losses A linear example : Webster-Lokshin model The 1-D wave equation with fractional order damping in time � � � � ∂ t w − 1 η z ∂ 1 / 2 + ε z ∂ − 1 / 2 ∂ 2 r 2 tt w + ∂ z z ∂ z w = 0 t t r 2 z is nothing but the coupling between : a hyperbolic PDE : Webster horn equation 1 a parabolic PDE, thanks to memory variables and diffusive 2 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 ? applied mathematics : energy balance , 1 automatic control : root locus in the Laplace plane , 2 signal processing : a spectrogram . 3 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 ∂ t 2 u ( t , x ) + YI . ∂ 4 pour 0 < x < L , ∂ x 4 u ( t , x ) = 0 , densité du matériau ρ : module de Young du matériau avec : Y , I = wh 3 moment géométrique de la barre : 12 avec les conditions aux limites suivantes : pas de moment appliqué aux extrémités : ∂ 2 ∂ 2 x u ( t , 0 ) = 0 , x u ( t , L ) = 0 ; pas de force en x = L , et une force f ( t ) en x = 0 : ∂ 3 ∂ 3 x u ( t , 0 ) = f ( t ) , x u ( 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 : � L E ( t ) = 1 ρ S ( ∂ t u ) 2 + YI ( ∂ 2 x u ) 2 d x ; 2 0 quand f = 0 , on trouve, après quelques calculs : d E d t = 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 ∂ x 4 u ( t , x ) + Y I . ∂ 4 ∂ 4 ∂ t 2 u ( t , x ) + ε ∂ ∂ tu ( t , x ) + η ∂ ∂ x 4 u ( t , x ) = 0 ∂ t � ε > 0 amortissement fluide , : avec amortissement structurel . η > 0 : Le bilan d’énergie sera : � L � L d E ( ∂ t u ) 2 dx − η x u ) 2 dx ≤ 0 . ( ∂ t ∂ 2 d t = − ε 0 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) Γ( β ) t β − 1 for t > 0 only ; for any Let β ∈ ( 0 , 1 ) , and set h β ( t ) := 1 T > 0 , let v ∈ L 2 ( 0 , T ) , and define I β v := h β ⋆ v , i.e. : � t 1 Γ( β ) τ β − 1 v ( t − τ ) d τ . I β v ( t ) = ( h β ⋆ v )( t ) = 0 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) Γ( β ) t β − 1 for t > 0 only ; for any Let β ∈ ( 0 , 1 ) , and set h β ( t ) := 1 T > 0 , let v ∈ L 2 ( 0 , T ) , and define I β v := h β ⋆ v , i.e. : � t 1 Γ( β ) τ β − 1 v ( t − τ ) d τ . I β v ( t ) = ( h β ⋆ v )( t ) = 0 The Riemann-Liouville fractional integral of order β . Let α ∈ ( 0 , 1 ) , and for any T > 0 , let v ∈ H 1 ( 0 , T ) , and define D α v = D I 1 − α v or, more explicitly : � t D α v ( t ) = d d t ( h 1 − α ⋆ v )( t ) = d 1 Γ( 1 − α ) τ − α v ( t − τ ) d τ . d t 0 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, h β ( f ) = ( 2 i π f ) − β , � a causal low-pass filter, with a gain of − 6 β d B per octave ; it is passive, since h β ( f )) ∝ cos ( β π 2 ) | f | − β > 0 . ℜ e ( � 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, h β ( f ) = ( 2 i π f ) − β , � a causal low-pass filter, with a gain of − 6 β d B per octave ; it is passive, since h β ( f )) ∝ cos ( β π 2 ) | f | − β > 0 . ℜ e ( � in the Laplace domain, H β ( s ) = s − β , in ℜ e ( s ) > 0 ; it is passive, since for − π 2 < arg ( s ) < π 2 , ℜ e ( H β ( 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 d th 1 − α ( f ) = ( 2 i π f ) α , a causal high-pass filter, with a gain of + 6 α d B per octave ; it is passive, since � d th 1 − α ( f )) ∝ cos ( απ d 2 ) | f | + α > 0 . ℜ e ( 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 d th 1 − α ( f ) = ( 2 i π f ) α , a causal high-pass filter, with a gain of + 6 α d B per octave ; it is passive, since � d th 1 − α ( f )) ∝ cos ( απ d 2 ) | f | + α > 0 . ℜ e ( in the Laplace domain, H α ( s ) = s α , � in ℜ e ( s ) > 0 ; it is passive, since for − π 2 < arg ( s ) < π 2 , ℜ e ( H α ( 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 An introduction with examples 1 Viscoelastic materials Viscothermal losses Fractional integrals and derivatives Stability of fractional models : FDEs and FPDEs 2 Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model Stability of coupled diffusive models 3 An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model Non-linear models 4 A damped pendulum The brassy effect Conclusion 5 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 q � � a k D k α y ( t ) = b l D l α u ( t ) , k = 0 l = 0 It is a causal pseudo-differential system, the symbol of which is, by Laplace transf. in some right-half plane C + a := ℜ e ( s ) > a : � � l = q � � b l σ l � Q ( σ ) � H ( s ) = Q ( s α ) � with � l = 0 . � k = p P ( s α ) � � � � a k σ k P ( σ ) � k = 0 D. Matignon (ISAE) 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) q ≤ p ⇐ ⇒ BIBO stability | 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 t → + ∞ . as D. Matignon (ISAE) 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 N m n � � r nm E ⋆ m h ( t ) = α ( λ n , t ) , n = 1 m = 1 with R ( σ ) = � N � m n m = 1 r nm ( σ − λ n ) − m ; n = 1 where E ⋆ m α ( λ, t ) denotes a Mittag-Leffler function (a hypergeometric special function), the Laplace transform of which is ( s α − λ ) − m . Remark : for α = 1 , it reduces to the well-known causal m ! t m − 1 exp ( λ t ) , for t ≥ 0 . polynomial–exponential E ⋆ m 1 ( λ, t ) := 1 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 31 / 96
Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory N. & S. Stability condition : an illustration Stability of E α ( λ t α ) with LT s α − 1 ( s α − λ ) − 1 , as a fct. of arg ( λ ) . Laplace plane : s σ -plane D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 32 / 96
Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Mittag-Leffler functions in C (I) t �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = 0 Real part Imaginary part D. Matignon (ISAE) 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 �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = π/ 8 Real part Imaginary part D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 34 / 96
Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Mittag-Leffler functions in C (III) t �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = π/ 4 Real part Imaginary part D. Matignon (ISAE) 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 �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = 3 π/ 8 Real part Imaginary part D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 36 / 96
Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Mittag-Leffler functions in C (V) t �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = π/ 2 Real part Imaginary part D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 37 / 96
Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Mittag-Leffler functions in C (VI) t �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = 3 π/ 4 Real part Imaginary part D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 38 / 96
Stability of fractional models : FDEs and FPDEs Fractional Differential Equations : Theory Mittag-Leffler functions in C (VII) t �→ E α ( λ t α ) for α = 1 2 and arg ( λ ) = π Real part Imaginary part D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 39 / 96
Stability of fractional models : FDEs and FPDEs Example 1 : a 1-dof oscillator Influence of fractional order α with u 0 = 0 , v 0 = 1 . Example : u ( t ) + 0 . 5 ( d α u )( t ) + u ( t ) = 0 ¨ 1 1 0.8 0.6 0 u α 0.4 − 1 0 0.2 1 5 − 0 10 1 0.5 0.5 0 15 0 − 0.5 1 t − 1 0 20 v α u 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)] D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 40 / 96
Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model The Lokshin model is non standard 1 t ∂ t w + η 2 ∂ t w − ∂ 2 L et ∂ 2 t w + 2 η ∂ 2 x w = 0 , t > 0 , x ∈ ] 0 , 1 [ with init. cond. w ( t = 0 ) = 0 and ∂ t w ( 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. D. Matignon (ISAE) 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 1 t ∂ t w + η 2 ∂ t w − ∂ 2 L et ∂ 2 t w + 2 η ∂ 2 x w = 0 , t > 0 , x ∈ ] 0 , 1 [ with init. cond. w ( t = 0 ) = 0 and ∂ t w ( 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 x w ) dampings. D. Matignon (ISAE) 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 1 t ∂ t w + η 2 ∂ t w − ∂ 2 L et ∂ 2 t w + 2 η ∂ 2 x w = 0 , t > 0 , x ∈ ] 0 , 1 [ with init. cond. w ( t = 0 ) = 0 and ∂ t w ( 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 x w ) dampings. the spatial modes are no more orthogonal, due to the boundary conditions. D. Matignon (ISAE) 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 1 t ∂ t w + η 2 ∂ t w − ∂ 2 L et ∂ 2 t w + 2 η ∂ 2 x w = 0 , t > 0 , x ∈ ] 0 , 1 [ with init. cond. w ( t = 0 ) = 0 and ∂ t w ( 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 x w ) 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 ! D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 41 / 96
Stability of fractional models : FDEs and FPDEs Example 2 : Lokshin model Analytic solution of a fractional PDE 3 t + η 2 ∂ 1 L et ( ∂ 2 t ) w − ∂ 2 t + 2 η ∂ 2 x w = 0 , t > 0 , x ∈ ] 0 , 1 [ with init. cond. w ( t = 0 ) = 0 and ∂ t w ( t = 0 ) = 0 , and dynamical boundary conditions of absorbing type ( a 0 b 0 > 0 , a 1 b 1 > 0 ) and controlled at x = 0 : 1 1 [ a 0 ( ∂ t + η ∂ t ) + b 0 ∂ − x )] w ( t , x = 0 ) = a 0 ( ∂ t + η ∂ 2 t ) u ( t ) 2 1 [ a 1 ( ∂ t + η ∂ t ) + b 1 ∂ x )] w ( t , x = 1 ) = 0 2 with output : y ( t ) := w ( t , x = 1 ) . Then y = h ⋆ u , with � � + ∞ � c η 2 ( σ η + 2 ( σ η − ∈ L 1 ( R + ) ∩ C ∞ ( R + ) h ( t ) = E 1 n , t ) − E 1 n , t ) n n = −∞ are the roots of σ 2 + ησ = s 0 n := − α 0 + 2 i π f 0 where the σ η ± n . n ⇒ BIBO stability comes from | arg ( σ η ± n ) | > π 4 , ∀ n ∈ Z . = D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 42 / 96
Stability of coupled diffusive models Outline An introduction with examples 1 Viscoelastic materials Viscothermal losses Fractional integrals and derivatives Stability of fractional models : FDEs and FPDEs 2 Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model Stability of coupled diffusive models 3 An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model Non-linear models 4 A damped pendulum The brassy effect Conclusion 5 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 43 / 96
Stability of coupled diffusive models An introduction to diffusive representations Some useful identities (1) For δ > 1 , consider the numerical identity : � ∞ π d x δ 1 + x δ = δ ) . sin ( π 0 δ in it, and get : 1 Let s ∈ R + ∗ , substitute x = ( ξ s ) � ∞ sin ( π δ ) 1 1 1 s + ξ d ξ = . π ξ 1 − 1 s 1 − 1 0 δ δ An analytic continuation from R + ∗ to C \ R − leads to the first functional identity with 1 − 1 δ := β ∈ ( 0 , 1 ) : C \ R − H β : → C � ∞ s + ξ d ξ = 1 1 �→ µ β ( ξ ) s β , s 0 sin ( β π ) ξ − β . with density µ β ( ξ ) = π D. Matignon (ISAE) 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 : R + h β : → R � ∞ 1 Γ( β ) t β − 1 . µ β ( ξ ) e − ξ t d ξ = �→ t 0 So to speak, fractional kernels h β are nothing but a superposition of decaying exponential, with an appropriate weight µ β . 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. D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 45 / 96
Stability of coupled diffusive models An introduction to diffusive representations Input-output representation Let v and y = I β v be the input and output of the causal fractional integral of order β ∈ ( 0 , 1 ) : � ∞ y ( t ) = µ β ( ξ ) [ e ξ ⋆ v ]( t ) d ξ , 0 � t with e ξ ( t ) := e − ξ t 1 t ≥ 0 , and [ e ξ ⋆ v ]( t ) = 0 e − ξ ( t − τ ) v ( τ ) d τ . Now for fractional derivative of order α ∈ ( 0 , 1 ) , let v the input, and y = D α v = D [ I 1 − α v ] the output. � ⇒ a careful computation shows that : = � ∞ � y ( t ) = µ 1 − α ( ξ ) [ v − ξ e ξ ⋆ v ] ( t ) d ξ ; 0 which makes use both of v and e ξ ⋆ v . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 46 / 96
Stability of coupled diffusive models An introduction to diffusive representations State-space representation (1) : setting Let ϕ ( ξ, . ) := [ e ξ ⋆ v ]( t ) be the state, parametrized by ξ > 0 . (1) ∂ t ϕ ( ξ, t ) = − ξ ϕ ( ξ, t ) + v ( t ) , ϕ ( ξ, 0 ) = 0 , � ∞ (2) y ( t ) = µ β ( ξ ) ϕ ( ξ, t ) d ξ ; 0 and (3) ∂ t � ϕ ( ξ, t ) = − ξ � ϕ ( ξ, t ) + v ( t ) , � ϕ ( ξ, 0 ) = 0 , � ∞ (4) µ 1 − α ( ξ ) [ v ( t ) − ξ � � y ( t ) = ϕ ( ξ, t )] d ξ . 0 are state-space representations for I β and D α , respectively. Note : functional spaces must be specified for these representations to make sense ; more precisely : � ∞ µ β ( ξ ) | ϕ | 2 d ξ < ∞} , ϕ belongs to H β := { ϕ s . t . 0 � ∞ ϕ | 2 ξ d ξ < ∞} ; ϕ belongs to � � H α := { � ϕ s . t . µ 1 − α ( ξ ) | � 0 see e.g. [Haddar and M. (2008)], [M. and Zwart (2004)]. D. Matignon (ISAE) 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 E φ ( T ) := 1 φ ( ξ, T ) 2 µ β ( ξ ) d ξ , the energy 2 0 balance holds ∀ T > 0 for fractional integrals : � T � T � + ∞ ξ µ β ( ξ ) φ ( ξ, t ) 2 d ξ d t . v ( t ) y ( t ) d t = E φ ( T ) + 0 0 0 D. Matignon (ISAE) 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 E φ ( T ) := 1 φ ( ξ, T ) 2 µ β ( ξ ) d ξ , the energy 2 0 balance holds ∀ T > 0 for fractional integrals : � T � T � + ∞ ξ µ β ( ξ ) φ ( ξ, t ) 2 d ξ d t . v ( t ) y ( t ) d t = E φ ( T ) + 0 0 0 � ∞ φ ( T ) := 1 φ ( ξ, T ) 2 ξ µ 1 − α ( ξ ) d ξ , the with storage function � � E e 2 0 energy balance holds ∀ T > 0 for fractional derivatives : � T � T � + ∞ φ ) 2 d ξ d t . µ 1 − α ( ξ ) ( v − ξ � y ( t ) d t = � v ( t ) � φ ( T ) + E e 0 0 0 D. Matignon (ISAE) 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) : 1 Let with ξ > 0 H RC ( s ) = s + ξ , ℜ e ( s ) + ξ ⇒ ℜ e ( H RC ( s )) = ≥ 0 , ℜ e ( s ) ≥ 0 for | s + ξ | 2 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 49 / 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) : 1 Let with ξ > 0 H RC ( s ) = s + ξ , ℜ e ( s ) + ξ ⇒ ℜ e ( H RC ( s )) = ≥ 0 , ℜ e ( s ) ≥ 0 for | s + ξ | 2 a GL-circuit is passive and high-pass ( + 6 dB/oct) : s Let with ξ > 0 H GL ( s ) = s + ξ , | s | 2 + ℜ e ( s ) ξ ⇒ ℜ e ( H GL ( s )) = ≥ 0 , ℜ e ( s ) ≥ 0 for | s + ξ | 2 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 49 / 96
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 ( − 6 K dB/oct) : K � 1 H K with µ k > 0 . RC ( s ) = µ k s + ξ k k = 1 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 50 / 96
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 ( − 6 K dB/oct) : K � 1 H K with µ k > 0 . RC ( s ) = µ k s + ξ k k = 1 Positive aggregation of GL-circuits is passive and high-pass ( + 6 L dB/oct) : � L s H L with ν l > 0 . GL ( s ) = ν l s + ξ l l = 1 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 50 / 96
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) D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 51 / 96
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) D. Matignon (ISAE) 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) D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 53 / 96
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. D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 54 / 96
Stability of coupled diffusive models An introduction to diffusive representations Passivity : from finite to infinite d.o.f. (2) Recall that : � ∞ s + ξ d ξ = 1 1 µ β ( ξ ) ∝ 1 For with ℜ e ( s ) > 0 , µ β ( ξ ) s β ξ β 0 and also � ∞ s 1 s + ξ d ξ = s α For with ℜ e ( s ) > 0 , ν α ( ξ ) ∝ ν α ( ξ ) ξ 1 − α 0 ⇒ Fractional integrals & derivatives are nothing but continuous = positive aggregations of RC & GL circuits ! D. Matignon (ISAE) 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. D. Matignon (ISAE) 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 + ω 2 x = 0 , ¨ x + ˜ y + ˙ damped by : x = v , instantaneous w.r.t v , ˙ y ( v ) , with memory, low-pass behaviour, (e.g. ( µ k ) 1 ≤ k ≤ K or µ β ) y ( v ) , with memory, high-pass behaviour, (e.g. ( ν l ) 1 ≤ l ≤ L or ν α ) ˜ cannot be understood through the mechanical energy alone : E m := 1 2 v 2 + 1 2 ω 2 x 2 , since d x + ω 2 x ) = v (¨ d tE m − v 2 − ˜ = yv − yv ≤ ≥ 0 ? has no definite sign ! But still, thanks to DR... D. Matignon (ISAE) 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 E φ ( T ) := 1 φ ( ξ, T ) 2 µ β ( ξ ) d ξ , the energy 2 0 balance holds ∀ T > 0 for fractional integrals : � T � T � + ∞ ξ µ β ( ξ ) φ ( ξ, t ) 2 d ξ d t . v ( t ) y ( t ) d t = E φ ( T ) + 0 0 0 � ∞ φ ( T ) := 1 φ ( ξ, T ) 2 ξ µ 1 − α ( ξ ) d ξ , the with storage function � � E e 2 0 energy balance holds ∀ T > 0 for fractional derivatives : � T � T � + ∞ φ ) 2 d ξ d t . y ( t ) d t = � µ 1 − α ( ξ ) ( v − ξ � v ( t ) � E e φ ( T ) + 0 0 0 D. Matignon (ISAE) 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 + ω 2 x = 0 , ¨ x + ˜ y + ˙ damped by : x = v , instantaneous w.r.t v , ˙ y ( v ) , with memory, low-pass behaviour, (e.g. ( µ k ) 1 ≤ k ≤ K or µ β ) y ( v ) , with memory, high-pass behaviour, (e.g. ( ν l ) 1 ≤ l ≤ L or ν α ) ˜ can be easily understood through the augmented energy : E := E m + E φ + ˜ E ˜ φ of the global system, with internal variables [ x , v , φ, ˜ φ ] , since : � K � L d d t E = − v 2 − φ l ) 2 ≤ 0 . ν l ( v − ξ l ˜ µ k ξ k φ 2 k − k = 1 l = 1 ⇒ in finite dimension (2+K+L), asymptotic stability follows. = D. Matignon (ISAE) 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 ¨ ϑ + ϑ = 0 , for β = 0 . 75 and ( ϑ 0 , ˙ ˙ ϑ 0 ) = ( 3 . 5 , 0 ) . t (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 { φ k ( t ) = φ ( t , ξ k ) } 1 ≤ k ≤ K for K = 25 . D. Matignon (ISAE) 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 = I β v and ˜ y = D α u , we compute, at least formally : � ∞ � ∞ d d t E = − v 2 − φ ( ξ )) 2 d ξ ≤ 0 . ν α ( ξ ) ( v − ξ ˜ µ β ( ξ ) ξ φ 2 ( ξ ) d ξ − 0 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 or 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]. = D. Matignon (ISAE) 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 : w − 1 r 2 ∂ z ( r 2 ∂ z w ) = 0 ; t w + η ( z ) ∂ 3 / 2 w + ε ( z ) ∂ 1 / 2 ∂ 2 t t 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, D. Matignon (ISAE) 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 : w − 1 r 2 ∂ z ( r 2 ∂ z w ) = 0 ; t w + η ( z ) ∂ 3 / 2 w + ε ( z ) ∂ 1 / 2 ∂ 2 t t 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 ) . D. Matignon (ISAE) 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 : w − 1 r 2 ∂ z ( r 2 ∂ z w ) = 0 ; t w + η ( z ) ∂ 3 / 2 w + ε ( z ) ∂ 1 / 2 ∂ 2 t t 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 ! D. Matignon (ISAE) 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 : w − 1 r 2 ∂ z ( r 2 ∂ z w ) = 0 ; t w + η ( z ) ∂ 3 / 2 w + ε ( z ) ∂ 1 / 2 ∂ 2 t t with static boundary conditions in z = 0 and z = 1 . This is equivalent to the first-order system in ( p , v ) : − r − 2 ∂ z v − ε ∂ − 1 / 2 p − η ∂ 1 / 2 ∂ t p = p , t t − r 2 ∂ z p , ∂ t v = and p ( z = 0 , t ) = v ( z = 1 , t ) = 0 . 0 Use of standard DR for ∂ − 1 / 2 , and extended DR for ∂ 1 / 2 . t t D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 63 / 96
Stability of coupled diffusive models Example 4 : Webster-Lokshin model Existence et uniqueness (I) � 1 � 1 0 p 2 r 2 dz < ∞} , L 2 0 v 2 r − 2 dz < ∞} , and With L 2 p = { p , v = { v , v × L 2 ( 0 , 1 ; H M ; ε r 2 dz ) × L 2 ( 0 , 1 ; � H N ; η r 2 dz ) , the system can H = L 2 p × L 2 be put in the abstract form ∂ t X + A X = 0 , where : � + ∞ � + ∞ r − 2 ∂ z v + ε [ p − ξ � p ϕ dM + η ϕ ] dN 0 0 r 2 ∂ z p v A = ; ϕ ξϕ − p ϕ � ξ � ϕ − p � � p ( 0 ) = 0 � � v ( 1 ) = 0 ϕ ) T ∈ V , � D ( A ) = ( p , v , ϕ, � . � ( p − ξϕ ) ∈ L 2 ( 0 , 1 ; H M ; ε r 2 dz ) � � ϕ ) ∈ L 2 ( 0 , 1 ; V N ; η r 2 dz ) ( p − ξ � v × L 2 ( 0 , 1 ; V M ; ε r 2 dz ) × L 2 ( 0 , 1 ; � H N ; η r 2 dz ) . with V = H 1 p × H 1 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 64 / 96
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 ) , � 1 � 1 H M ε r 2 dz + H N η r 2 dz ≥ 0 . � ϕ � 2 ϕ � 2 ( A X , X ) H = � p − ξ � e 0 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 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 65 / 96
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 H 1 ( R ) into L 2 ( R ) is not compact (Rellich theorem does not apply). D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 66 / 96
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 H 1 ( R ) into L 2 ( 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 of 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). D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 66 / 96
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 C 0 -semigroup on a reflexive Banach space. Assume that no eigenvalue of A lies on the imaginary axis. If σ ( A ) ∩ i R is countable, then the semigroup generated by A is asymptotically stable, which means that the solutions of the differential equation x ′ ( t ) = A x ( 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. D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 67 / 96
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 , ℜ e ( s ) > 0 } = ∅ . Spectrum on the imaginary axis ? Solving for A X = 0 leads to X = 0 , thanks to the boundary 1 conditions in (3). Hence, λ = 0 is not an eigenvalue of −A ; but λ = 0 ∈ σ c ( −A ) , the continuous spectrum of −A . In order to prove σ ( −A ) ∩ { i ω, ω ∈ R , ω � = 0 } = ∅ , we show the 2 continuity of the resolvent ( i ω I + A ) − 1 on H : given any χ ) ∈ H , we seek some X = ( p , v , φ, � φ ) ∈ D ( A ) , such that Y = ( f , g , χ, � ( i ω 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 H 1 p has no definite sign, hence we can p × H 1 not apply the complex version of Lax-Milgram theorem. Therefore, we have to resort to the Fredholm alternative to conclude. D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 68 / 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. D. Matignon (ISAE) 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 { ξ k } 1 ≤ k ≤ K have been chosen. D. Matignon (ISAE) 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 { ξ k } 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 of the extended energy for the continuous system, see e.g. [Haddar, Li and M. (2010)]. D. Matignon (ISAE) 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 { ξ k } 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 of the extended energy for the continuous system, see e.g. [Haddar, Li and M. (2010)]. Some Illustrations ! D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 69 / 96
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 ( η x = η ), in blue η = 0 . 1 , in red η = 0 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 70 / 96
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 ( η x = η ), in blue η = 0 . 2 , in red η = 0 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 71 / 96
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 ( η x = η ), in blue η = 0 . 5 , in red η = 0 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 72 / 96
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 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 73 / 96
Non-linear models Outline An introduction with examples 1 Viscoelastic materials Viscothermal losses Fractional integrals and derivatives Stability of fractional models : FDEs and FPDEs 2 Fractional Differential Equations : Theory Example 1 : a 1-dof oscillator Example 2 : Lokshin model Stability of coupled diffusive models 3 An introduction to diffusive representations Example 3 : a 1-dof oscillator Example 4 : Webster-Lokshin model Non-linear models 4 A damped pendulum The brassy effect Conclusion 5 D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 74 / 96
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 : ∂ t X + A X = F ( X ) , et X ( 0 ) = X 0 ∈ D ( A ) soit bien posé, pour t ∈ [ 0 , T max ) , au sens de l’existence et de l’unicité de X ∈ C 1 ([ 0 , T max ); H ) ∩ C 0 ([ 0 , T max ); D ( A )) , une solution forte. Alors, pour deux OPD de type diffusif et positif, l’un standard h M ⋆ et l’autre étendu par dérivation � h N ⋆ , le pb. pseudo-différentiel non-linéaire : ∂ t X + h M ⋆ X + � h N ⋆ X + A X = F ( X ) , et X ( 0 ) = X 0 ∈ D ( A ) est bien posé, pour t ∈ [ 0 , T ′ max ) , au sens d’une unique solution forte X ∈ C 1 ([ 0 , T ′ max ); H ) ∩ C 0 ([ 0 , T ′ max ); D ( A )) . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 75 / 96
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 = + ∞ . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 76 / 96
Non-linear models A damped pendulum A toy model : non-linear pendulum ϑ + η ∂ − β ¨ ϑ + sin ( ϑ ) = 0 ; β = 0 . 75 et ( ϑ 0 , ˙ ˙ ϑ 0 ) = ( 3 . 5 , 0 ) . t (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 { φ k ( t ) = φ ( t , ξ k ) } 1 ≤ k ≤ K for K = 25 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 77 / 96
Non-linear models The brassy effect The one-way Burgers-Lokshin equation The one-way Burgers-Lokshin equation reads : ∂ t u + c ∂ x u + ε ∂ α t u + b ∂ x ( u 2 / 2 ) = 0 for t > 0 , (5) with initial datum u ( t = 0 , x ) = u 0 ( 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. D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 78 / 96
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. D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 79 / 96
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 2 � � L ∂ u ∂ t + c ∂ u ∂ x + b ∂ = − ε µ ℓ φ ℓ , ∂ x 2 � � u 2 ℓ = 1 �� (6) ∂φ j c ∂ u ∂ x + b ∂ ∂ t + γ α θ 2 α − 1 j ∂ x 2 ε � L j φ j − γ α θ 2 α − 1 = − θ 2 ℓ = 1 µ ℓ φ ℓ . j which can be put in the following abstract form : ∂ t U + ∂ ∂ (7) ∂ x F ( U ) = S U , D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 80 / 96
Non-linear models The brassy effect Analysis : notations � � T the vector of ( N + 1 ) unknowns is U = u φ 1 . . . φ N the nonlinear flux function F = ( F 1 F 2 . . . F N + 1 ) T is F 1 = c u + b u 2 F j = γ α θ 2 α − 1 2 , F 1 , j = 2 · · · N + 1 , j the ( N + 1 ) × ( N + 1 ) diffusive matrix S is ε µ 1 · · · ε µ N 0 θ 2 0 1 + Ω µ 1 · · · Ω µ N S = − , . . . . . . . . . . . . θ 2 Ω µ 1 · · · N + Ω µ N 0 with Ω = γ α ε . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 81 / 96
Non-linear models The brassy effect Analysis : results ′ of the nonlinear The eigenvalues of the Jacobian matrix J = F 1 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. = The eigenvalues of the diffusive matrix S are real and negative. 2 Moreover, they interlace the values − θ 2 j . ⇒ the linear part of the system is parabolic. = D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 82 / 96
Non-linear models The brassy effect Numerical method : Strang splitting Build up two independent schemes : 1 ∂ t U + ∂ ∂ ⇒ numerical scheme H a (8) ∂ x F ( U ) = 0 = ∂ ⇒ numerical scheme H b (9) ∂ t U = S U = D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 83 / 96
Non-linear models The brassy effect Numerical method : Strang splitting Build up two independent schemes : 1 ∂ t U + ∂ ∂ ⇒ numerical scheme H a (8) ∂ x F ( U ) = 0 = ∂ ⇒ numerical scheme H b (9) ∂ t U = S U = Implement the time-marching scheme between t n & t n + 1 : 2 U ( 1 ) H b ( ∆ t 2 ) U n • = j , j U ( 2 ) H a (∆ t ) U ( 1 ) (10) • = , j j 2 ) U ( 2 ) U n + 1 H b ( ∆ t • = . j j ⇒ 2nd order precision in ∆ t , see e.g. [LeVeque 2002]. = D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 83 / 96
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 . D. Matignon (ISAE) Viscoelastic mater. & viscothermal losses CIMI, 09/04/2014 84 / 96
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