MIS 2015 : 15-18 june 2015Mathematics in Savoie 2015

Evolution Equations : Long time behavior and Control

Program and Abstracts

Scientific committee

DIDIER BRESCH, Laboratoire de Mathématiques, Université Savoie Mont-Blanc, FranceJEAN-MICHEL CORON, Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, ParisGILLES LEBEAU, Laboratoire Jean Dieudonné, Université NiceENRIQUE ZUAZUA, Basque Center for Applied Mathematics, BCAM, Bilbao, Espagne

Organising comittee

KAÏS AMMARI, UR Analysis and Control of PDE, University of Monastir, TunisiaSTÉPHANE GERBI, Laboratoire de Mathématiques, Université Savoie Mont-Blanc, France

Local organising comittee

CLAUDIA BILLAT, Laboratoire de Mathématiques, Université Savoie Mont-Blanc, FranceSTÉPHANE GERBI, Laboratoire de Mathématiques, Université Savoie Mont-Blanc, FranceNADINE MARI, Laboratoire de Mathématiques, Université Savoie Mont-Blanc, FranceYVAN MASSON, Laboratoire de Mathématiques, Université Savoie Mont-Blanc, France

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Sponsors and partnership

The organisers wish to thank the sponsors and the partner listed below.

Sponsors

— Université Savoie Mont-Blanc— CNRS— Université Joseph Fourier, Grenoble— INP Grenoble— Région Rhône-Alpes,— GDRI LEM2I, Laboratoire Euro-Maghrébin de Mathématiques et leurs Interactions— GDR MACS, Modélisation, Analyse et Conduite des Systèmes dynamiques— GDRE CONEDP, Control of Partial Differential Equations,— GDR AEDP, Analyse des Equations aux Dérivées Partielles— Maimosine, Maison de la Modélisation et de la Simulation, Nanosciences et Environnement— PERSYVAL-lab, Labex PERvasive SYstems and ALgorithms

Partnership

SMF, Société Mathématique de France

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Université Savoie Mont-BlancAmphitéâtre du Pole Montagne, Le Bourget du Lac

Summer School Program

Monday 15 June 2015

9h00-9h15 - Welcome9h15-10h45 - Emmanuel Trélat : Control and stabilization of nonlinear PDE’s: several toolsand applications10h45-11h15 - Coffee break11h15-12h45 - Farid Ammar Khodja : Controllability of parabolic systems12h45-14h00 - Lunch

Session Chairman : Gilles Lebeau14h15-15h00 - Thierry Gallay : Orbital stability of periodic waves for the nonlinear Schrödingerequation15h00-15h45 - Arnaud Münch : Approximations of null controls using Least-square type methods15h45-16h00 - Coffee break and Posters presentation

Session Chairman : Emmanuel Trélat16h00-16h45 - Armen Shirikyan : Controllability implies ergodicity16h45-17h30 - Taoufik Hmidi : On the doubly connected V-states for the generalized quasi-geostrophic equations

Tuesday 16 June 2015

9h00-10h30 - Emmanuel Trélat : Control and stabilization of nonlinear PDE’s: severaltools and applications10h30-11h00 - Coffee break and Posters presentation11h00-12h30 - Farid Ammar Khodja : Controllability of parabolic systems12h30-14h00- Lunch

Session Chairman : Arnaud Münch14h15-15h00 - Reinhard Racke : Instability of coupled systems with delay15h00-15h45 - Maher Zerzeri : Rate of decay of some dissipative systems15h45-16h00 - Coffee break and Posters presentation

Session Chairman : Serge Nicaise16h00-16h45 - Lionel Rosier : Null controllability of the heat equation by the flatness approach16h45-17h30 - Cristina Pignotti : Stability results for a class of second-order evolution equationswith intermittent delay

Wednesday 17 June 2015

9h00-10h30 - Emmanuel Trélat : Control and stabilization of nonlinear PDE’s: severaltools and applications10h30-11h00 - Coffee break11h00-12h30 - Farid Ammar Khodja : Controllability of parabolic systems12h30-14h00- Lunch14h00-17h00 - Social Event: guided visit of Chambery city center19h30 - Conference Dinner at the Restaurant “Le Savoyard”, Chambéry

Thursday 18 June 2015

9h00-10h30 - Emmanuel Trélat : Control and stabilization of nonlinear PDE’s: severaltools and applications10h30-11h00 - Coffee break and Posters presentation11h00-12h30 - Farid Ammar Khodja : Controllability of parabolic systems12h30-14h00- Lunch

Session Chairman : Farid Ammar Khodja14h15-15h00 - Serge Nicaise : Stabilization and asymptotic behavior of the telegraph equation15h00-15h45 - Carlos Castro : Null controllability of a system of thermoelastic plates15h45-16h00 - Coffee break and Posters presentation16h00-16h10 - Closing

Courses

Farid Ammar KhodjaLaboratoire de Mathématiques, Université de Franche-Comté, Besançon, FRANCE

“Controllability of parabolic systems”

We give in this talk some recent controllability results of linear hyperbolic systems and we will apply themto solve some nonlinear control problems.

Emmanuel TrélatLaboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris, FRANCE“Control and stabilization of nonlinear PDE’s : several tools and applications”

In this course I will describe several existing mathematical techniques and approaches used in order tocontrol and/or stabilize linear and nonlinear PDE’s. Examples will involve e.g. nonlinear heat, wave, fluidequations.

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Invited speakers

Carlos CastroDepartamento de Matemática e Informática, Universidad Politécnica de Madrid, SPAIN

“Null controllability of a system of thermoelastic plates”

We present a new strategy for the control of coupled systems of PDE’s. The main idea is to write the solutionsand controls of the system in series form where each term satisfies a new control problem, still coupled, butthat can be written in cascade form. This means that the coupling term appears only in one of the equationsand the controllability can be deduced from suitable observability inequalities for the uncoupled equations.The control for the fully coupled system is then obtained combining the controllability of this reducedsystem with the convergence of the series. We apply this technique to the linear system of thermoelasticplates when we consider two different controls supported in some, possibly different, open sets. This workis in collaboration with L. de Teresa.

Thierry GallayInstitut Joseph Fourier, Université Grenoble I, Grenoble, FRANCE

“Orbital stability of periodic waves for the nonlinear Schrödinger equation”

We study the cubic defocusing NLS equation in one space dimension, which admits a six-parameter familyof (quasi-)periodic travelling waves. Using the conservation of the charge, the momentum, and the energy,one can prove that these periodic waves are orbitally stable within the class of solutions having the sameperiodicity properties as the wave itself. In this talk, I shall present some recent work in collaboration withDmitry Pelinovsky, which shows that cnoidal waves or arbitrary amplitude are orbitally stable with respectto "subharmonic perturbations" (these are perturbations whose period is an integer multiple of the period ofthe original wave.) The proof relies on the existence of an additional conserved quantity, but otherwise doesnot use the fact that the NLS equation is completely integrable.

Taoufik HmidiIRMAR, Université de Rennes I, FRANCE

“On the doubly connected V-states for the generalized quasi-geostrophic equations”

We shall discuss some results on the existence of the V-states for the generalized quasi-geostrophic equa-tions. They are special patches rotating uniformly around their centers of mass . In the first part, we reviewsome recent results on the simply connected case. In the second part, we focus on the existence of the V-states with only one hole. All these structures are constructed by using the bifurcation theory. This is a jointwork with Z. Hassainia.

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Arnaud MünchLaboratoire de Mathématiques, Université Blaise Pascal, Clermont-Ferrand, FRANCE

“Approximations of null controls using Least-square type methods”

We are concerned in this talk with the approximation of null controls for PDEs. We mainly focus on the heatequation and on the Navier-Stokes system. We explore a least-squares type formulation of the controllabilityproblem and show that it allows the construction of strong convergent sequences. The approach consistsfirst in introducing a class of functions satisfying a priori the boundary conditions in space and time -in particular the controllability requirement at the final time - then finding among this class one elementsatisfying the main equation of the PDE. The second step is accomplished by setting up an error (quadratic)functional, defined for all feasible functions, which measures how far those are from being a solution of theunderlying equation. Assuming controllability properties on the PDE, we show the strong convergence ofany minimizing sequence then illustrate the method with some numerical experiments. We also emphasizethat the method applies for direct and inverse problems. Joint works with Pablo Pedregal (UniversidadCastilla-La-Mancha, Spain).

Serge NicaiseLAMAV, Université de Valenciennes, FRANCE

“Stabilization and asymptotic behavior of the telegraph equation”

We analyze the stability of different models of the telegraph equation set in a real interval. They correspondto the coupling between a first order hyperbolic system and a first order differential equations of parabolictype. We show that some models have an exponential decay rate, while other ones are only polynomiallystable. In that last case we show that the obtained polynomial decay is optimal.

Cristina PignottiUniversità degli Studi dell’Aquila, L’Aquila, ITALY

“Stability results for a class of second-order evolution equations with intermittent delay”

We consider second ?order evolution equations in an abstract setting with intermittently delayed/not ?delayeddamping. We give sufficient conditions for asymptotic and exponential stability. In particular, under suitableconditions, we can consider unbounded damping operators. Some concrete examples are also illustrated.Joint work with S. Nicaise.

Reinhard RackeDepartment of Mathematics and Statistics, University of Konstanz, GERMANY

“Instability of coupled systems with delay”

We consider linear initial-boundary value problems that are a coupling like second-order thermoelasticity,or the thermoelastic plate equation or its generalization, the alpha-beta-system. Now, there is a delay termgiven in part of the coupled system, and we demonstrate that the expected inherent damping will not prevent

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the system from not being stable ; indeed, the systems will shown to be ill-posed : a sequence of boundedinitial data may lead to exploding solutions (at any fixed time).

Lionel RosierCentre Automatique et Systèmes, Ecole des Mines de Paris, FRANCE

“ Null controllability of the heat equation by the flatness approach”

We revisit the null controllability of the heat equation by using the flatness approach, which provides expli-citly the trajectory and the control as series in some Gevrey class. This is a joint work with Philippe Martinand Pierre Rouchon.

Armen ShirikyanDépartement de Mathématiques, Université de Cergy-Pontoise, FRANCE

“Controllability implies ergodicity”

In this talk, we discuss the interconnection between controllability properties of a dynamical system andlarge-time asymptotics of trajectories for the associated stochastic system. We begin with a result on thefinite-dimensional case which applies to differential equations on a smooth Riemannian manifold. We showhow the approximate controllability to a given point and solid controllability imply the uniqueness of astationary measure and exponential mixing in the total variation distance. We next turn to problems in infinitedimension and formulate a sufficient condition (in terms of controllability properties) for the exponentialmixing in the Kantorovich-Wasserstein distance. This result applies, for instance, to the 2D Navier-Stokessystem driven by a random force acting on the boundary. Finally, we formulate some open problems oncontrollability properties of the Navier-Stokes system, which would have interesting applications in theergodic theory of the associated random flow.

Maher ZerzeriLAGA, Université Paris XIII, Villetaneuse, FRANCE

“Rate of decay of some dissipative systems”

In this talk, we show that the fastest decay rate for some dissipative systems is given by the supremum of thereal part of the spectrum of the associated in infinitesimal generator, if the corresponding operator satisfiedsome spectral gap condition. We give also some applications to illustrate our setting.

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Posters

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Evolution Equations : long time behavior and control, 15-18 June 2015, Universit de Savoie, Le Bourget du

Lac, France.

Inverse source and coeffcient problems associated to theGinzburg-Landau equation

Bianca-Elena ARAMA

”Alexandru Ioan Cuza” University,Iasi, Romania

e-mail: bianca.gavrilei@yahoo.com

We consider the controlled Ginzburg-Landau equation with an internal distributed control in asub-domain. The complex Ginzburg-Landau equation describes the evolution of a complex-valuedfield y = y(x, t) by

yt(t, x) = −Ay(t, x) +Ry(t, x) + C (y) y + f(t, x)y = 0

y(x, τ) = y0(x)

Ω× (0, T )∂Ω× (0, T )

Ω,

where −Ay(t, x) = (a+ ib)∆y and C (y) y = −(α+ iβ) |y|2 y.Moreover, f(t, x) satisfies

|ft(t, x)| ≤ C |f(τ, x)| , (1)

for almost all (t, x) ∈ [0, T ]× Ω.Here a, b, R, α and β are some positive real numbers.The fundamental technique approached in this paper is estimating Carleman type inequalities

for the adjoint linearized system. We renew the computations made by Rosier and Zhang in [?],and obtain explicit coefficients in the Carleman estimates, with respect to T , where [0, T ] is themaximum interval of time we consider. We also obtain sharp Carleman inequalities.

Our inverse source problem consists of determining f(t, x) by using over-determining datay|[0,T ]×ω, where ω is an arbitrary sub-domain.

Our inverse coefficient problem consists of determining R by using over-determining datay|[0,T ]×ω, where ω is an arbitrary sub-domain.

References

[1] Imanuvilov, OY, Yamamoto, M. Lipschitz stability in inverse parabolic problems by the Carle-man estimate, Inverse problems, 1998.

[2] Lefter CG, Lorenzi A. Approximate controllability for an integro-differential control problem,Applicable Analysis. 2008.

[3] Rosier, L., Zhang, BY. Null controllability of the complex Ginzburg-Landau equation, Annalesde L’Institut Henri Poincare (2009).

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ON THE GLOBAL EXISTENCE OF SOLUTIONS FOR SYSTEM

OF THE HEAT EQUATIONS AND THE CORRESPONDING

DAMPED WAVE SYSTEM

MOHAMED BERBICHE

We consider the Cauchy problem for a strongly coupled semi-linear heat equa-tions with some kind of nonlinearity in multi-dimensional space RN . We see undersome conditions on the exponents and on the dimension, that the time-global ex-istence, uniqueness of solutions for small data and their asymptotic behaviors areobtained. This observation will be applied to the corresponding system of thedamped wave equations in low dimensional space.

References

1. M. Escobedo, M. A. Herrero, Boundedness and blow-up for a semilinear reaction-diffusion

equation, J. Diff. Eqs. 89 (1991), 176–202.2. H. Fujita,On the blowing up of solutions of the Cauchy problem for ut = ∆u+up, J. Fac. Sci.

Univ. Tokyo, Sect. I13 (1966) 109–124.

3. K. Marcati, P. Nishihara, The Lp − Lq estimates of solutions to one-dimensional dampedwave equations and their application to compressible flow through porous media, J. Differential

Equations 191 (2003) 445—469.4. T. Hosono, T. Ogawa,Large time behavior and Lp − Lqestimate of solutions of 2-dimensional

nonlinear damped wave equations, J. Differential Equations 203 (2004) 82–118.5. G. Todorova, B. Yordanov,Critical exponent for a nonlinear wave equation with damping, J.

Differential Equations (174) (2001) 464–489.

Departement de Mathematiques Faculte des Sciences, Universite de Khenchela, Route

de Constantine BP1252, El-Houria 40004, Khenchela AlgerieE-mail address: berbichemed@yahoo.fr

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A mathematical model for induction hardening

including nonlinear magnetic field and controlled

Joule heating

Marian Slodicka ∗and Jaroslav Chovan †

Department of Mathematical Analysis, Ghent University, Galglaan 2,9000 Ghent, Belgium

Abstract

We investigate a mathematical model for the electromagnetic “in-duction heating” of a continuous medium at rest. We assume that themagnetic induction field B is a nonlinear function of the magnetic fieldH and that the electric conductivity σ is temperature dependent. Thecoupling between the equations describing electro-magnetic field andthe heat equation is provided through the nonlinear function σ on theone hand and through the Joule heating term on the other hand. Weprove an existence of a weak solution to this coupled nonlinear systemof Maxwell-heat equations with the truncated quadratic Joule heatingterm.

∗marian.slodicka@ugent.be†jaroslav.chovan@ugent.be

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Spatial-structured optimal control of pollution by nitrogen fertilizers

Eloïse Comtea

aMIA, Université de La Rochelle, Avenue Michel Crépeau, 17042-La Rochelle, France

Abstract

We present a problem of optimal stock pollution control based on a pdes model for the hydrology.

The main goal is to find an optimal nitrogen fertilizer policy for agricultural crops, taking into account de-pollution costs. Our viewpoint is to use a realist model for the transport of the pollutant in the underground.The spatial structure of this problem is the key point. Pollution and pumping are not necessarily localizedat the same place. Moreover, their exist around pumping wells green areas without nitrogen pollution. Wechoose to fine-tuned model pollutant transport as a convection-diffusion-reaction equation to analyse theoptimal pollution policy and to see how it differs from the one obtained in usual models.

As a first toy-model, we consider a zone of agricultural fields on which nitrogen fertilizers are injected anda well where water is drawn, separated from fields by an untreated green zone. The goal is to control theemissions p(t, x) in time and space to optimize the crop benefit minus the treatment water cost, knowingthat pollution is positively correlated to agricultural production.To this aim, we consider the following general problem in (0, T )× (0, L) :

maxp(t,x)∈L2(0,T ;L2([0,L1]))

T∫0

(L1∫0

f (p (t, x)) dx

)e−ρt dt−

T∫0

D (c (t, L)) e−ρt dt

p(t, x) < p

R∂c(t,x)∂t = S∂2c(t,x)

∂x2 − v∂c(t,x)∂x − g (c (t, x)) + p(t, x)χ(x)[0,L1] + γ

c(0, x) = c0(x), ∂c(0,t)∂x = 0, ∂c(L,t)

∂x = 0

where c0 ∈ H1(0, L).Here, f(p(t, x)) denotes the private benefit derived from the emissions p(t, x), D (c (t, L)) is the environ-mental damage caused by the pollution stock c(t, L) at the well. Moreover, L1 denotes the end of the cropfields.We show the existence and the uniqueness of the solution for the optimal control problem. We study so-lutions of the dual problem, knowing they are necessary conditions satisfied by the general solution. Theanalysis of the (space-structured) steady state given by two ordinary differential equation of order 2 cou-pled with the Lagrangian variable behavior, shows that the stationary solution exists. Studying the linearizedmodel with Fourier transform, we prove the existence of one stable variety of dimension one and of oneinstable variety of dimension two.

Keywords: Optimal pollution control, Partial differential equations, Parabolic partial differential equations

Email address: eloise.comte@univ-lr.fr (Eloïse Comte)

Preprint submitted to Elsevier 30 avril 2015

Lossless error estimates for the stationary phasemethod and applications to the free Schrodinger

equation

Florent Dewez∗

joint work with Felix Ali Mehmeti†

Abstract. In an old result due to A. Erdelyi, asymptotic expansions of oscillatory inte-grals are given with explicit error estimates for one integration variable [4]. In addition,this method covers the case of L1 singularities of the amplitude. We slightly improvetheses results and apply them to obtain asymptotic expansions of the solution of the freeSchrodinger equation in one space variable, with certain initial conditions having singularFourier transform. We obtain information on the localisation and the motion of the wavepackets, as well as optimal L∞-time decay estimates in certain regions in space-time. Inparticular we construct initial conditions such that the L∞-norm decays slower as com-pared with the classical case [5]. Our results [1],[2] represent a refinement of existingresults [3].

Keywords. Asymptotic expansion, stationary phase method, error estimate, Schrodingerequation, L∞-time decay, singular frequency, space-time cone.

References

[1] F. Ali Mehmeti, F. Dewez, Explicit error estimates for the stationary phase methodI: The influence of amplitudes singularities. arXiv:1412.5789v1 [math.AP] (2014).

[2] F. Ali Mehmeti, F. Dewez, Explicit error estimates for the stationary phase methodII: Interaction of amplitude singularities with stationay points. arXiv:1412.5792v1[math.AP] (2014).

[3] T. Cazenave, J. Xie, L. Zhang, A note on decay rates for Schrodinger’s equation.Proceedings of the American Mathematical Society, 138 (2010) no. 1, 199-207.

[4] A. Erdelyi, Asymptotics expansions. Dover Publications, New York, 1956.

[5] M. Reed, B. Simon, Methods of modern mathematical physics II : Fourier Analy-sis, Self-Adjointness. Academics press, San Diego New York Boston London SydneyTokyo Toronto, 1975.

∗Universite Lille 1, Laboratoire Paul Painleve, CNRS U.M.R 8524, 59655 Villeneuve d’Ascq Cedex,France. Email: florent.dewez@math.univ-lille1.fr†Universite de Valenciennes et du Hainaut-Cambresis, LAMAV, FR CNRS 2956, Le Mont Houy, 59313

Valenciennes Cedex 9, France. Email: felix.ali-mehmeti@univ-valenciennes.fr

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On stability and instability of small oscillations ofa pendulum with a capillary viscous fluid

O. A. Dudik

Crimean Engineering and Pedagogical University, Simferopol, Crimea

dudik tnu@mail.ru

We begin from the complete mathematical statement of the hydrodynamical problem con-sisting of a pendulum with a cavity partially filled by a capillary viscous fluid. Then we considerthis problem as a system of operator-differential equations in Hilbert spaces. We prove thatthe investigated problem is reduced to the Cauchy problem for the abstract parabolic equationwith operator coefficient that is a generator of an analytic semigroup. This allows us to obtainthe theorem on strong solvability of the problem.

Further, we study normal oscillations of the problem (solutions depending on t as exp(−λt))under the conditions of static stability with respect to the linear approximation. If the operatorof the potential energy is positive definite then its normal oscillations are damped, becauseReλ > 0. As λ = Reλ + iImλ, then Reλ is the damping rate, and Imλ 6= 0 is the frequencyof these oscillations. We prove results about the spectrum of the problem and prove that thesystem of root elements forms the basis. We found that eigenvalues λ are located in the left-hand half-plane on the real axis and eigenvalues can transit from the right-hand half-plane tothe left-hand one only through the origin of the complex plane.

Also we prove that if the static stability assumptions is not satisfied, then the inversionof Lagrange’s theorem on stability is valid. Let the potential energy of the studied hydrome-chanical system does’t have minimum in the equilibrium state (the static instability) and theminimum eigenvalue λmin of the operator of the potential energy is negative. Then the problemhas at least one negative eigenvalue λmin < 0; it corresponds to the normal oscillations of thesystem, depending on t as exp(−λmint) = exp(|λmin|t), i.e. exponentially increasing with time.In other words the investigated hydromechanical system is dynamically unstable.

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IDENTIFICATION PROBLEMS USING PROPER ORTHOGONALDECOMPOSITION

VLADIMIR VRABE AND MICHAL GALBA

We apply model order reduction techniques to an inverse identification problem for the heatequation. In particular, we use proper orthogonal decomposition (POD) in source identificationfrom the final time data. The heat equation is discretized by finite element method and theinverse problem is tackled by the steepest descent method with POD scheme. In the article,we perform several numerical experiments to test the acceleration and stability of the proposedmethod.

DEPARTMENT OF MATHEMATICAL ANALYSIS, GHENT UNIVERSITY, GENT, BELGIUME-mail address: michal.galba@ugent.be

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Laboratory of Modeling Analysis & Computer Science (MACS).

Controllability of a semilinear damped waveequation with bilinear controls

Imad El harraki (1), Ali Boutoulout (1)∗

1 Laboratory of Modeling Analysis & Computer Science, Department of Mathematics andComputer Science, Moulay Ismail University, Faculty of Sciences Meknes, Morocco.

∗Corresponding author : boutouloutali@yahoo.fr

Abstract: This paper concerns the following initial and boundary value prob-lem for the n-dimensional wave equation:

ytt = ∆y + v1y + α(t)Byt − f(y) in Q

y(x, 0) = y0(x),∂y

∂t(x, 0) = y1(x) in Ω

y(ξ, t) = 0 on Σ.

(1)

where Ω is an open bounded subset of Rn with regular boundary ∂Ω, andQ = Ω×]0, T [, Σ = ∂Ω×]0, T [ with T > 0. B is a self-adjoint and positiveoperator with support ω. The real valued coefficients v1(x, t) and α(t) are themultiplicative controls and f ∈ C1(R). Our aim is to investigate the exactand the weak controllability of the system (??). Namely, given the initial state(y0, y1), we would like to know what states (y, yt) can be exactly or approx-imately achieved by system (??) at a time T by applying a strategy for theaforementioned multiplicative controls v1(x, t); α(t). In terms of applications,a problem like this arises, e.g., in the context of smart materials, whose prop-erties can be altered by applying various external factors such as temperature,electrical current or magnetic field.

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Evolution Equations : long time behavior and controlMIS 2015, Universite de Savoie, Le Bourget du Lac, France, 15 - 18 June 2015

Lagrangian approach extended to Constrained controllability forparabolic semi-linear systems

Touria Karite (1), Ali Boutoulout (1)∗

1 TSI Team, MACS Laboratory, Faculty of Sciences. Meknes. Morocco

∗Corresponding author : boutouloutali@yahoo.fr

Abstract:The problem of the constrained controllability for distributed parabolic semi-linear system evolving

in spatial domain Ω, consists in finding the control u with minimum cost that steers the system fromthe initial state y0 to a state between two prescribed functions. We give some definitions and propertiesconcerning this concept and then we solve the problem by characterizing the optimal control in terms ofa Lagrangian multipliers with the minimized functional.

Introduction:Models for dynamical systems usually arise from the application of physical laws such as conservation

of mass, momentum, and energy. These models typically take the form of linear or nonlinear differentialequations, where the parameters involved can usually be interpreted in terms of physical properties of thesystem. That’s why this work is devoted to study the constrained controllability of semi-linear systemswhich are intermediate between the linear and the non-linear ones. We meet these systems in physics,chemistry or biology and they are more close to the reality and more reliable to represent a problemmathematically.

This kind of controllability corresponds to real industrial preoccupations; for example, in an industrialfurnace, it may be required to maintain the temperature in a subregion of the furnace between twoprescribed temperature profiles T1 and T2 . Otherwise, the products won’t be in a good shape or with agood quality.

Key word: Distributed systems, parabolic systems, regional controllability, constrained controllability,Lagrangian approach, Uzawa algorithm.

References

[1] M. Fortin, and R. Glowinski, Augmented Lagrangian Methods: Applications to the numericalsolution of boundary-value problems, Vol 15, North-Holland, 1983.

[2] R. Tyrrell Rockafellar, Lagrange multipliers and optimality, SIAM Review, Vol. 35, No. 2,pp. 183-238, Jun., 1993.

[3] E. Zerrik and F. Ghafrani (2002), Minimum energy control subject to output constraints:Numerical approach; IEE Proc - Control Theory Appl, vol 149, 105-110.

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Evolution Equations : long time behavior and controlMIS 2015, 15-18 June 2015, Universite de Savoie, Le Bourget du Lac, France

Strategic Sensors and Regional Boundary Gradient

Observability for Hyperbolic systems

Adil Khazari†∗, , Ali Boutoulout †

† TSI Team, MACS Laboratory, Moulay Ismail University , Meknes, Morocco∗Corresponding author : adil0974@gmail.com

Abstract

The aim of this paper is to introduce the concept of regional gradient observability on a subregionΓ of the boundary of the evolution domain Ω in connection with the strategic sensors. Then we givecharacterization of such sensors in order that regional gradient observability can be achieved. Theobtained results are applied to a two-dimensional systems and various cases of sensors are considered.We also show that, there exists a dynamical system for diffusion system is not gradient observable inthe usual sense, but it may be regionally gradient observable.

INTRODUCTION

In system theory, the gradient observability is related to the possibility of reconstruction of the gradientof the state from the knowledge of system dynamics and the output [3, 4]. The notion of regional analysiswas extended by El Jai et al. [1]. The study of this notion motivated by certain concret-real problem, inthermic, mechanic, environment [2].The purpose of this paper is to give some results related to the link between regional gradient observabilityand strategic sensors. We consider a class of linear hyperbolic system and we explore various resultsconnected with the different types of measurements, domains, and boundary conditions.

References

[1] A. El Jaı, M. Amouroux, and E. Zerrik, Regional observability of a distributed systems, int. J. ofSystems Science, vol 25. no. 2, (1994) 301–313.

[2] A. El Jaı, E. Zerrik, and M.C. Simon, Regional observability of a thermal process, Int. J. of sensorsand actuators. A, vol 40. no. 4,(1995) 518–521.

[3] E. Zerrik and H. Bourray, Gradient observability for diffustion system, Int. J. Appl. Math. Comput.Sci 13 (2003), 139–150.

[4] E. Zerrik, H. Bourray and A. El Jai, Regional flux reconstruction for parabolic systems, InternationalJournal of Systems Science, 34 (2003), 641–650.

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ON A MODEL OF A FLEXURAL PRESTRESSED SHELL

MAROJE MAROHNIC

Abstract. We derive a linearized prestressed elastic shell model from anonlinear Kirchhoff model of elastic plates. The model is given in termsof displacement and micro-rotation of the cross-sections. In additionto the standard membrane, transverse shear, and flexural terms, themodel also contains a nonstandard prestress term. The prestress is ofthe same order as flexural effects, hence the model is appropriate whenflexural effects dominate over membrane ones. We prove the existenceand uniqueness of the solutions by Lax-Milgram theorem and comparesolution with the solution of the standard shell model via numericalexamples. This is joint work with Josip Tambaca.

University of Zagreb, Department of Mathematics, Zagreb, CroatiaE-mail address: maroje.marohnic@math.hr

Date: June 4, 2015.

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Evolution Equations : long time behavior and controlMIS 2015, Universite de Savoie, Le Bourget du Lac, France, 15 - 18 June 2015

Constrained observability for distributed hyperbolic systemsvia Lagrangian approach

Abdel-ilah Saaof (1)∗, Ali Boutoulout (1)

1 TSI Team, MACS Laboratory, Faculty of Sciences. Meknes. Morocco

∗Corresponding author : abdelilah.saaof@gmail.com

Abstract:The problem of the constrained observability for distributed hyperbolic system evolving in spatial

domain Ω consists in the reconstruction of the initial conditions, in a subregion ω of Ω, provided that theinitial state must be between two prescribed functions in ω and that the initial speed must be between twoothers functions also prescribed in ω. We give some definitions and proprieties concerning this conceptand then we describe an approach, based on the Lagrangian multipliers , for solving this problem. Thisapproach leads to an algorithm for the reconstruction of the initial conditions.

Introduction:In the distributed systems analysis, one of the interesting problems is the knowledge of the initial

conditions of a such system, this is called observation problem.Constrained observability is a special case of the problems of observability, where the initial conditions

are required to be between certain thresholds. There are many reasons for introducing this concept :Firstly, the mathematical model of a real system is obtained either from the measurements, or fromapproximation techniques and is very often affected by perturbations. Consequently the solution of sucha system is only approximately known. Secondly, the observation error is smaller than in general caseand the initial conditions to be reconstructed are to be between some bounds. This kind of observabilityis encountered in various real problems, in physics, chemistry or biology.

Key word: Distributed systems, Hyperbolic systems, Constrained observability, Regional reconstruc-tion, Lagrangian approach.

References

[1] Boutoulout A., Bourray H. and Baddi M., Constrained Observability for Parabolic Systems,Int. Journal of Math. Analysis, Vol. 5, 2011, no. 35, 1695 - 1710.

[2] Curtain R. F. and Pritchard A. J., Infinite Dimensional Linear Systems Theory, Springer-Verlag, Berlin, 1978.

[3] Curtain R.F. and Zwart H., An introduction to infinite dimentional linear systems theory,Springer Verlag, NewYork, 1995.

[4] El Jai A. and Pritchard A.J., Sensors and actuators in distributed systems analysis, Wiley,New York 1988.

[5] Fortin v. and Glowinski R., Methodes de Lagrangien Augmente. Applications a la resolutionnumerique des problemes aux limites,, (Dunod, 1982).

[6] Tyrrell Rockafellar R., Lagrange multipliers and optimality, Slam review, vol 35, (2) (1993),183-238.

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Identification of a convolution kernel in a nonlinear

wave equation

Marian Slodicka1, Lukas Seliga1

We consider a bounded domain Ω ⊂ RN , N ≥ 1 with sufficiently smooth boundary Γ.The symbol ν stands for the outer normal vector associated with Γ. We are interested indetermining of an unknown couple (u,K) obeying the following nonlinear hyperbolic problemof second order

∂ttu(x, t)−∆u(x, t) + (K ∗ u(x))(t) = f(x, t, u(x, t), ∂tu(x, t)) in Ω× (0, T ),−∇u(x, t) · ν = g(x, t), on Γ× (0, T ),

∂tu(x, 0) = v0(x) in Ω,u(x, 0) = u0(x) in Ω,

(1)

where T > 0. By K ∗ u we denote the usual convolution in time, namely (K ∗ u(x))(t) =∫ t

0K(t − s)u(x, s)ds . The missing time-convolution kernel K = K(t) will be recovered from

the following integral-type measurement

∫

Ω

u(x, t)dx = m(t), t ∈ (0, T ).

The global in time existence, uniqueness as well as the regularity of a solution are addressed.A new numerical algorithm based on Rothe’s method is designed and error estimates arederived.

1Department of Mathematical Analysis, research group of NAM2, Ghent University, Belgium, contacton presenter: lseliga@cage.ugent.be

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The linearized viscous Saint-Venant system on a tree∗

Farhat Shel †

Abstract

Feedback stabilization of a simplified model of fluid-structure interaction on atree. We study the dynamic feedback stability for some simplified model of fluid-structure interaction on a tree. We prove that, under some conditions, the energyof the solutions of the dissipative system decay exponentially to zero when the timetends to infinity. Our technique is based on a frequency domain method.

∗joint work with Kaıs Ammari and Muthusamy VANNINATHAN†UR Analysis and Control of Pde, UR 13ES64, Universite de Sfax, Institut Preparatoire aux Etudes

d’Ingenieurs de Sfax, B.P. 1172, Sfax 3018, Tunisia,farhat.shel@ipeit.rnu.tn

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THE REGULAR VORTEX PATCH PROBLEM FOR STRATIFIED EULER

EQUATIONS

TAOUFIK HMIDI AND MOHAMED ZERGUINE

Abstract. We present here the regular vortex patch problem for the stratified Euler equationsin space dimension two. We generalize Chemin’s result concerning the global persistence of theHolderian regularity of the vortex patches. Roughly speaking, we prove that if the initial densityρ0 ∈ L1 ∩ L∞ and the initial vorticity takes the form ω0 = 1Ω0 , with ∂Ω0 is a planar curve havingC1+ε− regularity, then the velocity of the stratified Euler equations is a Lipschitz function globallyin time, namely we have:

‖∇v(t)‖L∞ ≤ C0eC0t log2(1+t).

Moreover, we prove that the vorticity is split into two parts ω(t) = 1Ωt + ρ(t), where Ωt , Ψ(t,Ω0)with Ψ denotes the flow associted to the velocity v and keeps its initial regularity. The function ρis a smooth function related to the smoothing effects of density.

IRMAR, Universite de Rennes 1, Campus de Beaulieu, 35 042 Rennes cedex, FranceE-mail address: thmidi@univ-rennes1.fr

Universite Hadj Lakhdar Batna, Faculte des Sciences, Departement de Mathematiques, Lab. EDPA,05000 Batna AlgerieE-mail address: mohamed zerguine@yahoo.fr

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