The aim of this discussion is to give a broad view of the links between fractional differential equations (FDEs) or fractional partial differential equations (FPDEs) and so-called diffusive representations (DR). Many aspects will be investigated: theory and numerics, continuous time and discrete time, linear and nonlinear equations, causal and anti-causal operators, optimal diffusive representations, fractional Laplacian.

Many applications will be given, in acoustics, continuum mechanics, electromagnetism, identification, ..

Matignon, Denis

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 To cite this document: Matignon, Denis Fractional equations and diffusive systems: an overview. (In Press: 2013) In: IFAC workshop on Control of Systems Modeled by Partial Differential Equations, CPDE'13, 25-27 Sept 2013, Paris, France. Open Archive Toulouse Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.  This is an author-deposited version published in: http://oatao.univ-toulouse.fr/  Eprints ID: 8974 Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr  Fractional equations and diffusive systems:an overviewDenis Matignon ∗∗University of Toulouse, ISAE; 10, av. E. Belin; BP 54032. 31055Toulouse Cedex 4, France. denis.matignon@isae.frAbstract: The aim of this discussion is to give a broad view of the links between fractionaldifferential equations (FDEs) or fractional partial differential equations (FPDEs) and so-called diffusive representations (DR). Many aspects will be investigated: theory and numerics,continuous time and discrete time, linear and nonlinear equations, causal and anti-causaloperators, optimal diffusive representations, fractional Laplacian. Many applications will begiven, in acoustics, continuum mechanics, electromagnetism, identification, ...Keywords: fractional equations, diffusive systems, pseudo-differential operators, hereditarymechanics, stability, numerical methods.1. INTRODUCTIONFractional differential systems have become quite popularin the recent decades, giving rise to a wide literature, bothon the theoretical and on the applied sides; monographs,and special issues of international journals are now devotedto this active research field.The aim of this discussion paper is two fold: first explainto what extent diffusive representations can be helful forfractional equations, second give many fields of applicationwhere this technique has proved useful.1.1 Fractional integral and derivativesThese causal linear operators can be defined in many ways,see e.g. Matignon [2009a].Fractional integral Let β ∈ (0, 1), and set hβ(t) :=1Γ(β) tβ−1 for t > 0 only; then, hβ ∈ L1loc(R+). For anyT > 0, let u ∈ L2(0, T ), and define Iβu := hβ ⋆ u or, moreexplicitely:Iβu(t) =∫ t01Γ(β)τβ−1 u(t− τ) dτ .This is the Riemann-Liouville fractional integral of orderβ ∈ (0, 1) of u: it is causal, and belongs to L2(0, T ) also.In terms of causal Laplace transform, Hβ(s) = s−β inℜe(s) > 0; hence, the interpretation of the fractionalintegral is a causal low-pass filter, with a gain of −6βdB per octave.Fractional derivative The fractional derivative is theinverse of the fractional integral, but some technicalitiesare to be found in this case. Let α ∈ (0, 1), and for anyT > 0, let u ∈ H1(0, T ), (that is u ∈ L2(0, T ), u has aweak derivative say u˙ which does belong to L2(0, T )), anddefine Dαu = I1−αDu := h1−α ⋆ u˙ or, more explicitely:Dαu(t) =∫ t01Γ(1− α)τ−α u˙(t− τ) dτ .This is the fractional derivative of order α ∈ (0, 1) of u:it is causal, and belongs to L2(0, T ). In terms of causalLaplace transform, H˜α(s) = s+α in ℜe(s) > 0; hence, theinterpretation of the fractional derivative is a causal high-pass filter, with a gain of +6α dB per octave.Open question Both these operators present the draw-back of not being differential operators, not easily givingrise to a semigroup; moreover, a hereditary behaviour canbe foreseen, with a long-memory decay. Therefore, we tryto make a link between these fractional operators, andordinary differential equations (ODEs), or just convolutionby families of decaying exponentials: we derive functionalidentities for the kernel hβ and its Laplace transform Hβ ,in the time-domain and the frequency-domain; these arebeing used to define input-output representations, andeven state-space realizations.1.2 Diffusive representationsWe refer to Staffans [1994] and Montseny [1998] for anintroduction.First diffusive representations With specific weightµβ(ξ) :=sin(β pi)pi ξ−β , one has hβ(t) =∫∞0µβ(ξ) e−ξ t dξ,which helps reformulate the fractional integral as the fol-lowing input-output representation :y(t) =∫∞0µβ(ξ) [eξ ⋆ u](t) dξ ,with eξ(t) := e−ξ t, and [eξ ⋆ u](t) =∫ t0e−ξ (t−τ) u(τ) dτ .Then, the following infinite-dimensional dynamical systemcan be seen as a state-space realization of the fractionalintergral of order β:∂tϕ(ξ, t) =−ξ ϕ(ξ, t) + u(t), ϕ(ξ, 0) = 0 , (1)y(t) =∫∞0µβ(ξ)ϕ(ξ, t) dξ . (2)The well-posedness condition∫∞0µβ(ξ)1+ξ dξ < +∞ holds.Moreover, the energy balance can be checked, ∀T > 0 :∫ T0u(t) y(t) dt = Eφ(T ) +∫ T0∫ +∞0ξ µβ(ξ)φ(ξ, t)2 dξ dt ,with storage function Eφ(T ) :=12∫∞0φ(ξ, T )2 µβ(ξ) dξ.Extended diffusive representations A careful computa-tion shows that the following input-output representationholds:y˜(t) =∫∞0µ1−α(ξ) [u− ξ eξ ⋆ u](t) dξ .The following infinite-dimensional dynamical system canbe seen as a state-space realization of the fractional deriva-tive of order α:∂tϕ˜(ξ, t) =−ξ ϕ˜(ξ, t) + u(t), ϕ˜(ξ, 0) = 0 , (3)y˜(t) =∫∞0µ1−α(ξ) [u(t)− ξ ϕ˜(ξ, t)] dξ . (4)The well-posedness condition∫∞0µ1−α(ξ)1+ξ dξ < +∞ holds.Moreover, the energy balance can be checked, ∀T > 0 :∫ T0u(t) y˜(t) dt = E˜φ˜(T )+∫ T0∫ +∞0µ1−α(ξ) (u−ξ φ˜)2 dξ dt ,with storage function E˜φ˜(T ) := 12∫∞0φ˜(ξ, T )2 ξ µ1−α(ξ) dξ.Conclusion on DR The main interests or advantages ofthe diffusive realizations are:• the existence of an associated semigroup,• the dissipativity of the realization, whenever theoperator is positive,• the possibility of deriving numerical schemes withoutheredity, hence memory-saving algorithms.2. GENERAL ASPECTS2.1 Equivalent formulation of fractionally damped equationsFirst an infinite-dimensional state-space realization of thefractional derivative or integral input-ouput relation isbeing used. Thus, another way of considering the problemconsists in interpreting the fractionally damped equationas a coupled problem between a conservative classical sys-tem (either linear or non-linear, either ODE or PDE) anda diffusion equation: this naturally introduces a global en-ergy (either quadratic or non-quadratic) in the augmentedstate-space, and leads, at least formally, to the decay ofthe global energy, see e.g. Staffans [1994] first, Montsenyet al. [2000], Haddar et al. [2004] for a nonlinear FDE,Haddar et al. [2008] for a linear FPDE.2.2 Stability issuesOn FDEs, there is an if and only if condition Matignon[1998], and some related results in Bonnet et al. [2000].With DR at hand, we only have an if condition, whichdoes not impose constant coefficients, so the range ofapplication is larger; but still if the formal computation ofthe global energy balance is quite easy, the mathematicalproof remains quite involved.The main difficulty or drawback of the diffusive realiza-tions is the lack of compactness proprety,• which makes the canonical embeddings of dense sub-spaces not compact, and forbids the use of manyexistence theorems where it is required,• which does not allow the use of LaSalle’s invari-ance principle for asymptotic stability analysis, whichwould require a precompactness prorerty of the tra-jectories.Hence, we have to resort to Arendt-Batty stability theoremto conclude on asymptotic stability of the coupled system,after a careful study of the spectrum of the generator ofthe associated semigroup, see Matignon et al. [2005] for anFDE and Matignon [2006] for an FPDE.2.3 Numerical methodsFor sure, discretizing the continuous diffusive representa-tions in the ξ-domain is possible, but it must be done in acareful way, so as to ensure an equivalent energy balanceat the discrete level, see e.g. Haddar et al. [2008]; alsothe weights can be optimized w.r.t a criterion, see e.g.Deu¨ et al. [2010]. More involved numerical methods canbe found in Diethelm [2008], Haddar et al. [2010].2.4 Optimal controlStill very few papers are concerned with the question ofoptimal control of fractional differential systems, ad hocfinite-dimensional approximations of fractional derivativesare used in the first place, and classical optimal controlmethods are being applied in the second place; no proof ofconvergence of the process is provided.A first reason for that could be that optimal controlof infinite-dimensional systems is a quite involved andtechnical field, but a second one lies in the very natureof fractional operators. They are causal, but highly non-local in time (with a weakly integrable singularity at theorigin); hence their adjoint becomes necessarily anti-causaland still non-local in time. Thus, one can easily imaginethat the complexity of the theory for forward fractionaldynamical systems becomes even more intricate whenthe coupled equations of the adjoint systems are derived;because we will be left with coupled forward and backwardfractional dynamics in order to solve the optimal controlproblem: at first glance, it seems very unlikely that Riccatiequations (if any) could be either analysed or even solved(not to speak of adequate numerical schemes for these) insuch a complicated setting.In order to overcome this intrinsic difficulty, in Matignon[2010] we propose to use the equivalent diffusive repre-sentations of fractional systems, and to work on it, as forinfinite dimensional systems of integer order.3. APPLICATIONS3.1 Viscoelasticity in continuum mechanicsA large class of such systems, often found in mechanics, isbased on causal pseudo-differential time-operators, some-times with a long-memory behaviour: classical examplesare fractional integrals and derivatives. Pseudo-differentialoperators are hereditary: the whole past of the physicalstate is involved in the dynamic expression of the sys-tem evolution. This generally introduces major technicaldifficulties. Moreover, from the thermodynamical pointof view, consistency of the model is a difficult questionin most cases. In e.g. Deu¨ et al. [2010], the fractionalZener model is first translated into its equivqlent diffusiverealization, then discretized using a coupled Newmark-diffusive scheme, with optimal choice of the weights. AnFPDE in dimension 2 will be treated using the sameapproach, and the classical finite element method (FEM)in space.3.2 Visco-thermal losses in musical acousticsThe famousWebster-Lokshin model which describes acous-tic waves traveling in a duct with viscothermal losses atthe lateral walls is a wave equation with spatially-varyingcoefficients, which involves fractional-order integrals andderivatives with respect to time: it has been presented inLokshin and Rok [1978] and Polak [1991], and studied ine.g. He´lie et al. [2006a] and Haddar et al. [2008] thanks toDR.3.3 Identification in signal processingOnce an equivalent diffusive model has been found, it isof interest to find a low order finite dimensional approx-imation: to this end, optimal formulation is preferred inpractise: a first worked-out example can be found in Garciaet al. [1998], and a whole family of fractional and diffusivesystems of increasing complexity has been explored inHe´lie et al. [2006b].3.4 Matched impedance of a beam in control of PDEsFor the formulation of second order systems in time, it canbe necessary to deal with vector-valued X, hence positivematrix-valued PDOs of diffusive type; not only diagonalmatrices of scalar positive PDOs are useful in practice,as an example, in order to solve the impedance matchingproblem for the Euler–Bernoulli beam, an impedance ma-trix of the form[a ∂+αt 11 b ∂−αt], was used with α = 12 , seeMontseny et al. [1997]: a necessary and sufficient conditionfor the positivity of this operator is ab > 1(cosαpi/2)2 . Itsequivalent DR should be useful to build an absorbingfeedback, which realizes an impedance matching, but sofar, the theoretical question is still open, even thoughnumerical simulations prove efficient.3.5 Polarized media in electomagnetismIn polarized media, a hierarchy of models can be found,see e.g. Petropoulos [2005]. First Debye ou Maxwell : 11+τs ,then Cole–Cole : 11+(τs)α , then Davidson–Cole :1(1+τs)γ ,and last Havriliak–Negami : 1(1+(τs)α)γ . All of these are ofdiffusive type, meaning that there exists a specific positivemeasure µ, satisfying the well-posedness condition, whichcan be used to represent the system as diffusive.4. CONCLUSIONStill many things are to be done in this very active field ofresearch: study existence for nonlinear systems, both FDEsand FPDEs; study asymptotic stability in the nonlinearcase; go to the MIMO case for Euler-Bernoulli matchedimpedance matrix; do some more progress on optimalnumerical methods; develop the ideas, both theoreticaland numerical, for the optimal control of such systems;address new family of examples, such as those in elec-tromagnetism;together with computer science developpe-ment, define some distributed systems strategy to simulatethese types of distributed parameter systems.REFERENCESC. Bonnet and J. R. Partington, Coprime factorizations and stability of fractionaldifferential systems, Systems Control Lett., 41 (2000), pp. 167174.J.-F. Deu¨ and D. Matignon, Simulation of fractionally damped mechanical systemsby means of a Newmark-diffusive scheme, Computers and Mathematics withApplications, vol. 59 (2010), pp. 1745–1753.K. Diethelm, An investigation of some nonclassical methods for the numericalapproximation of Caputo-type fractional derivatives, Numer. Algor., 47 (2008),361-390.G. Garcia and J. Bernussou (1998), Identification of the dynamics of a lead acidbattery by a diffusive model, ESAIM Proc., 5 (1998), pp. 87–98.H. Haddar and D. Matignon, Well-posedness of non-linear conservative systemswhen coupled with diffusive systems, in IFAC conf. on Nonlinear ControlSystems (NOLCOS), vol. 1, Stuttgart, Germany, sept 2004, Theoretical and numerical analysis of the Webster-Lokshin model,Research Rep. RR-6558, Institut National de la Recherche en Informatiqueet Automatique (INRIA), 2008.H. Haddar, J. R. Li, and D. Matignon, Efficient solution of a wave equation withfractional-order dissipative terms, Journal of Computational and AppliedMathematics, vol. 234 (2010), pp. 2003–2010.T. He´lie and D. Matignon, Diffusive representations for the analysis and simu-lation of flared acoustic pipes with visco-thermal losses, Math. Models Meth.Appl. Sci., 16 (2006), pp. 503–536., Representations with poles and cuts for the time-domain simulation offractional systems and irrational transfer functions, Signal Processing (SP),86 (2006), pp. 2516–2528.Y. Le Gorrec, and D. Matignon, Diffusive systems coupled to an oscillator:a Hamiltonian formulation. In IFAC conf. on Lagrangian and HamiltonianMethods and Nonlinear Control (LHMNLC’12), August 29–31, 2012, Berti-noro, Italy. (invited session).Lokshin, A. A. and V. E. Rok. Fundamental solutions of the wave equation withretarded time. Dokl. Akad. Nauk SSSR 239, 1305–1308, 1978. (in Russian).Matignon, D. (1998), Stability properties for generalized fractional differentialsystems, ESAIM Proc., 5 (1998), pp. 145–158., (2006). Asymptotic stability of the Webster-Lokshin model, in Mathe-matical Theory of Networks and Systems (MTNS), Kyoto, Japan, jul 2006,pp. 11 p. CD–Rom. (invited session)., (2009). An introduction to fractional calculus, vol. 1 of Scaling, Fractalsand Wavelets of Digital Signal and Image Processing Series, ISTE–Wiley,2009, ch. 7, pp. 237–278., (2009). Diffusive representations for fractional Laplacian: systemstheory framework and numerical issues. Physica Scripta, vol. T136, 014009(6 p.)., (2010). Optimal control of fractional systems: a diffusive formulation, in19th International Symposium on Mathematical Theory of Networks andSystems - MTNS 2010, 2010, pp. 3 p. CD–Rom. (invited session).D. Matignon and C. Prieur, Asymptotic stability of linear conservative systemswhen coupled with diffusive systems, ESAIM Control Optim. Calc. Var., 11(2005), pp. 487–507.D. Matignon and H. Zwart, Standard diffusive systems as well-posed linearsystems. International Journal of Control, (2012). To appear.Montseny, G. (1998), Diffusive representation of pseudo-differential time-operators, ESAIM Proc., 5 (1998), pp. 159–175.Montseny, G., J. Audounet and D. Matignon (1997). Fractional integrodifferentialboundary control of the Euler–Bernoulli beam. In: Conference on Decisionand Control. ieee-css, siam. San Diego, California. pp. 4973–4978.Montseny, G., J. Audounet and D. Matignon (2000). Diffusive representation forpseudo-differentially damped non-linear systems. In: Nonlinear Control in theYear 2000 (A. Isidori, F. Lamnabhi-Lagarrigue and W. Respondek, Eds.).Vol. 2. cnrs, ncn. Springer Verlag. pp. 163–182.Polack, J.D., (1991). Time domain solution of Kirchhoff’s equation for soundpropagation in viscothermal gases: a diffusion process. J. Acoustique, 4,47–67.Petropoulos, P., (2005). On the time-domain response of Cole-Cole dielectrics,IEEE Trans. on Antennas and Propagation, 53, pp. 3741–3746.Staffans, O. J. (1994). Well-posedness and stabilizability of a viscoelasticequation in energy space. Trans. Amer. Math. Soc. 345(2), 527–575.

Fractional equations and diffusive systems: an overview

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