Dochain, Denis

HASTIR, Anthony

WINKIN, Joseph

English

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RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHEAuthor(s) - Auteur(s) :Publication date - Date de publication :Permanent link - Permalien :Rights / License - Licence de droit d’auteur :Bibliothèque Universitaire Moretus PlantinInstitutional Repository - Research PortalDépôt Institutionnel - Portail de la Rechercheresearchportal.unamur.beUniversity of NamurLocal Exponential Stability of Nonlinear Distributed Parameter SystemsHASTIR, Anthony; WINKIN, Joseph; Dochain, DenisPublished in:39th Benelux Meeting on Systems and ControlPublication date:2020Document VersionPeer reviewed versionLink to publicationCitation for pulished version (HARVARD):HASTIR, A, WINKIN, J & Dochain, D 2020, Local Exponential Stability of Nonlinear Distributed ParameterSyst ms: Application to a Nonisothermal Tubular Reactor. in 39th Benelux Meeting on Systems and Control.General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Download date: 02. Jul. 2022Local exponential stability of nonlinear distributed parameter systems: Application to anonisothermal tubular reactorAnthony Hastir?, Joseph J. Winkin? and Denis Dochain†? Namur Institute for Complex Systems (naXys) and Department of Mathematics,University of Namur, Rempart de la vierge 8, B-5000 Namur, Belgium† Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM),Université Catholique de Louvain, Avenue Georges Lemaitre 4-6, B-1348 Louvain-La-Neuve, Belgiumanthony.hastir@unamur.be,joseph.winkin@unamur.be,denis.dochain@uclouvain.beDeducing exponential stability of an equilibrium of a non-linear distributed parameter, i.e. infinite-dimensional, sys-tem on the basis of the stability of a linear approximationof it is in general quite challenging. Some of the existingtheories, see e.g. [4, 1], rely on the Fréchet differentiabilityof the nonlinear semigroup generated by the nonlinear oper-ator dynamics. However, checking Fréchet differentiabilityfor nonlinear operators defined on an infinite-dimensionalspace is difficult or even impossible if these are unbounded.In many cases, the general theory cannot be applied and acase-by-case study has to be performed by working directlyon the semigroup instead of its generator.The approach that is proposed here is based on anadapted concept of Fréchet differentiability which takesdifferent spaces and norms into account. This is calledthe (Y,X)−Fréchet differentiability, where X is the state(Hilbert) space and Y is an auxiliary space chosen to han-dle more easily norm-inequalities when working in infinite-dimensions (typically L∞ or Sobolev spaces (H p, p ∈ N0),which are all multiplicative algebras). The systems thatare considered here are governed by the following abstractODE: ẋ(t) = Ax(t)+N(x(t)), x(0) = x0, (1)where A : D(A)⊂ X → X and N : D(N)⊂ X → X are linearand nonlinear operators, respectively. Let xe ∈D(A)∩D(N)be an equilibrium of (1).Definition : Let (Y,‖ · ‖Y ) be an infinite-dimensional (pos-sibly Banach) space such that D(A)∩D(N) ⊂ Y ⊆ X. Theoperator N is called (Y,X)−Fréchet differentiable at xe ifthere exists a bounded linear operator dN(xe) : X → X suchthat for all h∈D(A)∩D(N),N(xe+h)−N(xe)= dN(xe)h+R(xe,h) where lim‖h‖Y→0 ‖R(xe,h)‖X/‖h‖X = 0.This allows more easily checkable adapted Fréchet differen-tiability conditions, provided that local exponential stabilityof the equilibrium of (1) holds in a weaker sense, see [3].Based on this new concept, our approach to deduce expo-nential stability of the equilibrium of (1) can be summura-zied as in Figure 1. Let (S(t))t≥0 be the nonlinear semigroupgenerated by the operator A+N on X . The standard conceptof Fréchet differentiability is needed for (S(t))t≥0 on Y , with(Txe(t))t≥0 as Fréchet derivative, the linear semigroup gen-erated by the Gâteaux derivative A+dN(xe) of A+N. Aftershowing that (Txe(t))t≥0 satisfies some Lyapunov-type sta-bility condition and that it is exponentially stable on X , thenew concept of (Y,X)−Fréchet differentiability pops up tomake the connection between Y and X , in order to deduceTarget Local (exp.) stability/instability of (S(t))t≥0 at xeon XGo to Y• (S(t))t≥0 is Y−Fréchet differentiable at xewith (Txe (t))t≥0 as Fréchet derivative• (Txe (t))t≥0 is "Lyapunov stable" on YGo back to X• (S(t))t≥0 is (Y,X)−Fréchet differentiable atxe with (Txe (t))t≥0 as Fréchet derivative• (Txe (t))t≥0 is (exp.) stable/unstable on XLocal means ‖x0‖Y < δFigure 1: Illustration of the new theoretical framework.local exponential stability of the equilibrium of (1) on X ,see [3]. Local means here that the Y−norm of the initialcondition has to be small instead of its X−norm.Our theoretical approach is illustrated on a nonisother-mal axial dispersion tubular reactor which exhibits differ-ent numbers of equilibria depending on the diffusion coeffi-cients: see [2] for the existence and multiplicity of the equi-libria. Our main result states that, in the case of only oneequilibrium profile, the latter is locally exponentially stablefor the nonlinear system governing the dynamics. Moreover,in the case where the reactor exhibits three equilibria, localexponential bistability is established, that is, the pattern ”sta-ble – unstable – stable” is highlighted, see [3].AcknowledgmentThis research was conducted with the financial support ofF.R.S-FNRS. Anthony Hastir is a FNRS Research Fellowunder the grant FC 29535. The scientific responsibility restswith its authors.References[1] R. Al Jamal and K. Morris, 2018. ”Linearized stabil-ity of partial differential equations with application to stabi-lization of the kuramotosivashinsky equation”, SIAM Jour-nal on Control and Optimization 56 (1), 120 – 147[2] A. Hastir, F. Lamoline, J.J. Winkin and D. Dochain,2019. ”Analysis of the existence of equilibrium profilesin nonisothermal axial dispersion tubular reactors”, IEEETransactions on Automatic Control, to appear[3] A. Hastir, J.J. Winkin and D. Dochain, 2019. ”Expo-nential stability of nonlinear infinite-dimensional systems:application to nonisothermal axial dispersion tubular reac-tors”, submitted[4] N. Kato, 1995. ”A principle of linearized stability fornonlinear evolution equations”, Transactions of the Ameri-can Mathematical Society 347 (8), 2851 – 2868.

Local Exponential Stability of Nonlinear Distributed Parameter Systems:Application to a Nonisothermal Tubular Reactor

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Local Exponential Stability of Nonlinear Distributed Parameter Systems:Application to a Nonisothermal Tubular Reactor

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