## File: exemple2006.tex

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R.}, title = {Harmonic Analysis and Rational Approximation; Their R�les in Signals, Control and Dynamical Systems}, publisher = {Springer Verlag}, year = {2006}, volume = {327}, series = {Lecture Notes in Control and Information Sciences}, isbn = {3-540-30922-5} } @PhdThesis{th-david, author = {Avanessoff, David}, title = {Lin�arisation dynamique des syst�mes non lin�aires et param�trage de l'ensemble des solutions}, school = {Univ. de Nice - Sophia Antipolis}, year = 2006, month = jun, } @InProceedings{ref1, author = { Charlot, Gr�goire}, title = {Stability of Nonlinear switched systems in the plane}, booktitle = {44th IEEE Conf. }, year = 2005, pages={3285--3290}, address = {Seville, Spain}, month = dec } @InProceedings{ref2, author = {Boscain, Ugo}, title = {Stability of Non}, year = 2006, pages={3285--3290}, address = {Seville, Spain}, month = dec } @InProceedings{ref3, author = {Sigalotti, Mario}, title = {Stability of Nonlinear switched systems in the plane}, booktitle = {44th IEEE Conf. }, year = 2005, pages={3285--3290}, address = {Seville, Spain}, month = dec } \end{filecontents+} \maketitle \nocite{*} \begin{module}{composition}{en-tete}{} \begin{catperso}{Team Leader} \pers{Laurent}{Baratchart}{Scientist}{Inria}[DR Inria] \end{catperso} % \begin{catperso}{Deputy Team Leader} \pers{Jean-Baptiste}{Pomet}{Scientist}{Inria}[CR Inria] \end{catperso} \begin{catperso}{Administrative Assistant} \pers{France}{Limouzis}{Assistant}{Inria}[AI Inria, partial time in the team] \end{catperso} \begin{catperso}{Staff Member} \pers{Jos�}{Grimm}{Scientist}{Inria}[CR Inria] \pers{Juliette}{Leblond}{Scientist}{Inria}[DR Inria (since September, CR INRIA b efore)] \pers{Martine}{Olivi}{Scientist}{Inria}[CR Inria] \pers{Fabien}{Seyfert}{Scientist}{Inria}[CR Inria] \end{catperso} \end{module} \begin{module}{presentation}{presentation}{} \begin{moreinfo} The Apics Team is a Project Team since January 2005. \end{moreinfo} The Team develops constructive methods for modeling, identification and control of dynamical systems. \subsubsection*{Research Themes} \begin{itemize} \item Meromorphic approximation in the complex domain. \item Inverse potential problems in 3-D and analysis of harmonic fields. \item Control and structure analysis of non-linear systems. \end{itemize} \subsubsection*{International and industrial partners} \begin{itemize} \item Industrial collaborations with Alcatel-Alenia-Space. \item Exchanges with UST (Villeneuve d'Asq). \item The project is involved in a NATO Collaborative Linkage Grant (with. \end{itemize} \end{module} \begin{module}{fondements}{identif}{Identification and deconvolution} Let us first introduce the subject of Identification in some generality. Let us turn to work of the Apics Team\footnote{and of the former MIAOU-project} can be partly recast from the data. We shall explain in more detail the above steps in the sub-paragraphs to come. \subsubsection{Analytic approximation of incomplete boundary data} \label{dida-mero} \begin{participants} \pers{Laurent}{Baratchart}, \pers{Jos�}{Grimm}, \pers{Juliette}{Leblond}, \pers{Jean-Paul}{Marmorat}[CMA, �cole des Mines], \pers{Jonathan}{Partington}, \pers{Fabien}{Seyfert} \end{participants} \begin{motscle} meromorphic approximation, frequency-domain identification, extremal problems, {alg�bre �l�mentaire $(\max,+)$} \end{motscle} A prototypical Problem is: {\sl ($P$)~~Let $p \geq 1$, $N \geq 0$, $K$ be an arc of the unit circle $T$, $f \in L^p(K)$, $\psi \in L^p(T \setminus K)$ and $M>0$; find a function $g \in H^p + R_N$ such that $\|g - \psi\|_{L^p(T \setminus K)} \leq M$ and such that $g - f$ is of minimal norm in $L^p(K)$ under this constraint.} In order to impose pointwise constraints in the frequency domain one may wish to express the gauge constraint on $T\setminus K$ in a more subtle manner, depending on the frequency: {\sl ($P'$)~~Let $p \geq 1$, $N \geq 0$, $K$ be an arc of the unit circle $T$, $f \in L^p (K)$, $\psi \in L^p(T \setminus K)$ and $M \in L^p(T \setminus K)$; find a function $g \in H^p + R_N$ such that $|g - \psi|\leq M$ a.e.\ on $T \setminus K$ and such that $g - f$ is of minimal norm in $L^p(K)$ under this constraint.} Deeply linked with Problem $(P)$, is the following completion Problem: {\sl ($P''$)~~Let $p \geq 1$, $N \geq 0$, $K$ an arc of the unit circle $T$, $f \in L^p(K)$, $\psi \in L^p(T \setminus K)$ and $M >0$; find a function $h \in L^p(T \setminus K)$ such that $\|h - \psi\|_{L^p(T\setminus K)} \leq M$, and such that the distance to $H^p + R_N$ of the concatenated function $f\vee h$ is minimal in $L^p(T)$ under this constraint.} A version of this problem where the constraint depends on the frequency is: {\sl ($P'''$)~~Let $p \geq 1$, $N \geq 0$, $K$ an arc the unit circle $T$, $f \in L^p(K)$, $\psi \in L^p(T \setminus K)$ and $M \in L^p(T \setminus K)$; find a function $h \in L^p(T \setminus K)$ such that $|h - \psi|\leq M$ a.e.\ on $T \setminus K$, and such that the distance to $H^p + R_N$ of the concatenated function $f\vee h$ is minimal in $L^p(T)$ under this constraint.} Let us mention that Problem $(P'')$ reduces to Problem $(P)$ that in turn reduces, although implicitly, to an extremal Problem without constraint, (i.e., a Problem of type $(P)$ where $K=T$) that is denoted conventionally by $(P_0)$. In the case where $p=\infty$, Problems $(P')$ and $(P''')$ can viewed as special cases of $(P)$ and $(P'')$ respectively, but if $p<\infty$ the situation is different. where the constraint on the approximant is expressed in terms of its real and imaginary parts while the criterion takes only its real part into account: {\sl Let $p \geq 1$, $K$ be an arc of the unit circle $T$, $f \in L^p(K)$, $\psi \in L^p(T \setminus K)$, and $\alpha, \beta, M>0$; find a function $g \in H^p$ such that $\alpha \, \|\mbox{\rm Re} ({g - \psi})\|_{L^p(T \setminus K)} + \beta \, \|\mbox{\rm Im} ({g - \psi})\|_{L^p(T \setminus K)} \leq M$ and such that $\mbox{\rm Re} (g - f)$ is of minimal norm in $L^p(K)$ under this constraint.} see sections \moduleref{APICS}{domaine}{dom-fissures} and \moduleref{APICS}{resultats}{fissures}, where data and physical prior information concern real (or imaginary) parts of analytic functions. This allows one to \begin{enumerate} \item extend the index theorem to the case $2\leq p\leq\infty$ \item study asymptotic errors with \item characterize the asymptotic (cf. section \ref{didactique-poles}). \end{enumerate} In connection with the second and third items above,see section \ref{AHH}. \subsubsection{Scalar rational approximation} \label{didactique-approx-rat-scal} \begin{participants} \pers{Laurent}{Baratchart}, \pers{Martine}{Olivi}, \pers{Edward}{Saff}, \pers{Herbert}{Stahl}[TFH Berlin], \pers{Maxim}{Yattselev} \end{participants} \begin{motscle} rational approximation, critical point, orthogonal polynomials \end{motscle} Rational approximation is the second step mentioned in section~\moduleref{APICS}{fondements}{identif}. The Problem can be stated as: {\sl Let $1\leq p\leq\infty$, $f\in H^p$ and $n$ an integer; find a rational function without poles in the unit disk, and of degree at most $n$ that is nearest possible to $f$ in $H^p$.} In this way we are led to consider minimizing a criterion of the form: \label{crit} \left\|f - \frac{p_m}{q_n} \right\|_{L^2(d \mu)} where, by definition, $\|g\|_{L^2(d \mu)}^2=\frac{1}{2\pi} \int_{-\pi}^{\pi}|g(e^{i\theta})|^2 d\mu(\theta),$ $\min \left\||f| - \left|\frac{p_n}{q_n}\right| \right\|_{L^p(T)}.$ $\left\||f|^2 - \left|\frac{p_n}{q_n}\right|^2 \right\|_{L^\infty(T)} <\varepsilon,$ \paragraph{OK} \label{didactique-approx-rat-mat} If one introduces now as a new variable the rational matrix $R$ defined by $R=\left(\begin{array}{cc} L & H \\ 0 & I_m \end{array} \right)^{-1}$ and if $T$ stands for the first block-row, \label{defLL} \|T\|_{\Lambda}^2={\bf Tr}\left\{\frac{1}{2\pi} \int_{0}^{2\pi}T(e^{i\theta})\, d\Lambda(\theta)\,T^*(e^{i\theta})\right\}, \end{module} \begin{module}{fondements}{nl}{Structure and control of non-linear systems} In order to control a system, one generally relies on a model. \subsubsection{Feedback control and optimal control} \label{nl-stab} Stabilization by continuous state feedback---or output feedback, that is, \subsubsection{Transformations and equivalences of non-linear systems and models} \label{nl-trans} \paragraph{Dynamic linearization.} The problem of dynamic linearization,. \paragraph{Topological Equivalence} In what precedes, we have not taken into account the degree of \emph{smoothness} of the transformations under consideration.\end{module} \begin{module}{domaine}{chapeau}{Introduction} The botton line of the team's activity is twofold, \end{module} \begin{module}{domaine}{dom-fissures}{Geometric inverse problems for the Laplacian} Localizing cracks, pointwise sources or occlusions in a two-dimensional. \end{module} \begin{module}{domaine}{resonn}{Identification and design of resonant systems} One of the best training ground for the research of the team in function theory is the identification and design of physical systems. \begin{figure} \begin{center} \includegraphics{miaou_transf} \end{center} \caption{Transducer model.} \label{trans} \end{figure} \begin{figure} \begin{center} \includegraphics{miaou_coup} \end{center} \caption{Configuration of the filter} \label{filtrescnes} \end{figure}\end{module} \begin{module}{domaine}{spatial}{Spatial mechanics} The use of satellites in telecommunication networks motivates. \end{module} \begin{module}{domaine}{optique}{Non-linear optics} The increased capacity of numerical channels in information technology is a major industrial challenge. \end{module} \begin{module}{domaine}{plat}{Transformations and equivalence of non-linear systems} The works presented in module~\ref{nl-trans} lie upstream. \end{module} \begin{module}{logiciels}{logi-tralics}{The Tralics software} \label{RARL2} \begin{participant} \pers{Jos�}{Grimm}[\corresp] \end{participant} The development of a \LaTeX\ to XML translator, named Tralics was continued. \end{module} \begin{module}{logiciels}{RARL2}{The RARL2 software} \begin{participants} \pers{Jean-Paul}{Marmorat}, \pers{Martine}{Olivi}[\corresp] \end{participants} \label{didactique-poles} RARL2 (R�alisation interne et Approximation Rationnelle L2) is a software for rational approximation (see module \ref{didactique-approx-rat-mat}). Its web page is \htmladdnormallink{\url{http://www-sop.inria.fr/miaou/RARL2/rarl2.html}} {http://www-sop.inria.fr/miaou/RARL2/rarl2.html}. It is germane to the arl2 function of hyperion \end{module} \begin{module}{logiciels}{RGC}{The RGC software} The identification of filters modeled see section~\moduleref{APICS}{resultats}{Couplages-Algebrique}. \end{module} \begin{module}{logiciels}{PRESTO-HF}{PRESTO-HF} \begin{participant} \pers{Fabien}{Seyfert} \end{participant} PRESTO-HF: a toolbox dedicated to lowpass parameter identification for hyperfrequency filters \htmladdnormallink{\url{http://www-sop.inria.fr/miaou/Fabien.Seyfert/Presto_web_page/presto_pres.html}} {http://www-sop.inria.fr/miaou/Fabien.Seyfert/Presto_web_page/presto_pres.html} The miaou' should be replaced by apics' here. \end{module} \begin{module}{logiciels}{logi-endymion}{The Endymion software} \label{endymion} \begin{participant} \pers{Jos�}{Grimm}[\corresp] \end{participant} We have started the development of Endymion, a software licensed under the CeCILL license version two, see \href{http://www.cecill.info}{http://www.cecill.info/}. \end{module} \begin{module}[A]{resultats}{RAjose}{Tools for producing the Activity Report (this document)} The great novelty in the RAWEB2002 (Scientific Annex to the Annual Activity Report of Inria), was the use of XML as intermediate language, and the possibility of bypassing \LaTeX. for the example we get \verb+${\#119987 _y=lim_{x\#8594 0}sin^2{(x)}}$+. \end{module} \begin{module}[A]{resultats}{tralics}{Tralics: a Latex to XML Translator} The \textit{Tralics} software is a C++ written \LaTeX\ to XML translator \end{module} \begin{module}[B]{resultats}{fissures}{Inverse Problems for 2D and 3D elliptic operators} \subsubsection{Sources recovery in 2D and 3D} \label{AHH} The fact that 2D harmonic functions are real parts is also considered. \subsubsection{Application to EEG inverse problems} In 3D, epileptic regions in the cortex are often linked to a number of important related issues. \subsubsection{Cauchy problems in 2D and 3D} Solving Cauchy problems on an annulus can be extended. \subsubsection{More general geometries} We also started to be developed. \subsubsection{Others elliptic operators} Within the post-doctoral stay of E. Sincich, we began the University of Nice. \subsubsection{Application to magnetic dipoles recovery} The magnetic field produced by a magnetic dipole $\vec {m}$ located at a point ${\vec r}'$ is \vec B(\vec r) = \frac{\mu_0}{4\pi}\left\{ \frac{3\vec m({\vec r}')\cdot(\vec r-{\vec r}')}{|\vec r-{\vec r}'|^5} (\vec r-{\vec r}') - \frac{\vec m({\vec r}')}{|\vec r-{\vec r}'|^3} \right\}. \label{W102} $B_z(x,y,z) = \frac{\mu_0}{4\pi}\lambda_k \frac{2z^2-(x-x_k)^2-(y-y_k)^2}{[(x-x_k)^2+(y-y_k)^2+z^2]^{5/2}}$ C_z(x,y,z) = \frac{\mu_0}{4\pi^2a^2}\sum_k\lambda_k \int_{D(0,a)} \frac{2z^2-(x-\alpha-x_k)^2-(y-\beta-y_k)^2} {\left[(x-\alpha-x_k)^2+(y-\beta-y_k)^2+z^2\right]^{5/2}} d\alpha d\beta. \label{equa3} \end{module} \begin{module}[C]{resultats}{martine1}{Parametrizations of matrix-valued lossless functions} \label{Schur-realisations} The possibility to fertilize the pure algebraic LMI approach with the rich and vast topic of Schur analysis has been pointed out and deserve to be further investigated. \end{module} \begin{module}[C]{resultats}{martine2}{The mathematics of Surface Acoustic Wave filters} \end{module} \begin{module}[B]{resultats}{jul2}{Rational and Meromorphic Approximation} The results have been exploited . \end{module} \begin{module}[B]{resultats}{poles}{Behavior of poles} \label{Rpoles} This rather unexpected algorithm is currently being explored in details by E. Mina. \end{module} \begin{module}[C]{resultats}{ExtFab}{Analytic extension under pointwise constraints} Such regularity condition should greatly impinge on the numerical practice of the problem. \end{module} \begin{module}[B]{resultats}{Couplages-Algebrique}{Exhaustive determination of constrained realizations corresponding to a transfer function} \begin{eqnarray} \label{ep} p \in \CC^{r_1},\,\, E_{\sigma_1}(p)&=\{q \in \CC^{r_1}, \pi_{\sigma_1}(q)=\pi_{\sigma_1}(p)\} \\ p \in \CC^{r_2},\,\, E_{\sigma_1,\sigma_2}(p)&=\{q \in \CC^{r_1}, \pi_{\sigma_1}(q)=\pi_{\sigma_2}(p)\} \end{eqnarray} \end{module} \begin{module}[A]{resultats}{synthese}{Zolotarev problem and multi-band filter design} \label{zolo} T \end{module} \begin{module}[B]{resultats}{omux}{Frequency Approximation and OMUX design} \label{secOMUXc} This is one reason for analysing the optimization problem further. \end{module} \begin{module}[A]{resultats}{mario-optim}{On the structure of optimal trajectories} The results on the local regularity of trajectories in optimal control obtained previously have been published. \end{module} \begin{module}[A]{resultats}{alex1}{Feedback for low thrust orbital transfer} \label{secorbitet} The study concerns the control of a satellite. \end{module} \begin{module}{resultats}{flat}{Local linearization (or flatness) of control systems} a workable formulation of the question is now available. \end{module} \begin{module}{resultats}{dubbins}{Controllability for a general Dubins problem} Controllability results for systems with drift are usually obtained. \smallskip The main object of our research is given by Dubins-like systems In \footcite{ref1} we proved that $0\ne1$. This has been presented in \cite{ref2} and \refercite{ref3}. \end{module} \begin{module}{contrats}{cnes}{Contracts CNES-IRCOM-INRIA} Contracts \no 04/CNES/1728/00-DCT094 see module \ref{filtrescnes}, \end{module} \begin{module}{contrats}{aspi-c}{Contract Alcatel Space (Cannes)} Contract \no 1 01 E 0726. \end{module} \begin{module}{international}{nat}{Scientific Committees} L. Baratchart is a member of the editorial board \end{module} \begin{module}{international}{nationale}{National Actions} Together with project-teams Caiman and Odyss�e \end{module} \begin{module}{international}{cee}{Actions Funded by the EC} The team is the recipient see \htmladdnormallink{\url{http://www.ladseb.pd.cnr.it/control/ercim/control.html}}{http://www.ladseb.pd.cnr.it/control/ercim/control.html}. \end{module} \begin{module}{international}{monde}{Extra-european International Actions} \textbf{NATO CLG} (Collaborative Linkage Grant), PST.CLG.979703, Constructive approximation and inverse diffusion problems'', with Vanderbilt Univ. (Nashville, USA) and LAMSIN-ENIT (Tunis, Tu.), 2003-2005. \textbf{EPSRC} grant (EP/C004418) Constrained approximation in function spaces, with applications'', with Leeds Univ. (UK) and Univ. Lyon I, 2005-2006. \textbf{STIC-INRIA} and \textbf{AireD�veloppement} grants with LAMSIN-ENIT (Tunis, Tu.), Probl�mes inverses du Laplacien et approximation constructive des fonctions'', \textbf{NSF EMS21} RTG students exchange program (with Vanderbilt University). \end{module} \begin{module}{international}{accueilx}{The Apics Seminar} The following scientists gave a talk at the seminar: \end{module} \begin{module}{diffusion}{dif-ens}{Teaching} \begin{description} \item [Courses] \ \begin{itemize} \item L. Baratchart, DEA G�om�trie et Analyse, LATP-CMI, Univ. de Provence \item M. Olivi, Math�matiques pour l'ing�nieur \end{itemize} \item [Trainees] \ \begin{itemize} \item Jonathan Chetboun (ENPC) \item Cristina Paduret \textit{R�solution de probl�mes inverses} \end{itemize} \item[Ph.D. Students] \ \begin{itemize} \item Alex Bombrun, �~Commande optimale de satellites~� (optimal etc) \item Imen Fellah, Data completion in inverse problems'', co-tutelle. \item Vincent Lunot, �~Probl�mes � la synth�se d'OMUX �, \item Moncef Mahjoub, Compl�tion de donn�es g�om�triques.'' co-tutelle. \item Erwin Mina Diaz, Asymptotic properties of urves.'' \item Maxim Yattselev, Meromorphic orthogonality.'' \end{itemize} \item[defended Ph.D. thesis] \ \begin{itemize} \item David Avanessoff, �~Lin�arisation dynamique des solutions~� (dynamic trajectories). June 8, 2005.\cite{th-david} \end{itemize} \item[Jurys] \ % \item L.~Baratchart sat on \item J.~Leblond has been sitting \item F.~Seyfert has been sitting \item J.-B. Pomet has been sitting \end{description} \end{module} \begin{module}{diffusion}{dif-anim}{Community service} L. Baratchart was a member of the bureau'' of the CP (Comit� des Projets) of INRIA-Sophia Antipolis untill July. He is a member of the commission de sp�cialistes'' (section 25) of the Universit� de Provence. J.~Leblond and J. Grimm are co-editors of the proceedings (to appear in 2006) of the CNRS-INRIA summer school Harmonic analysis and rational approximation: their r\^oles in signals, control and dynamical systems theory'' (Porquerolles, 2003) \htmladdnormallink{\url{http://www-sop.inria.fr/apics/anap03/index.en.html}} {http://www-sop.inria.fr/apics/anap03/index.en.html} \footcite{anap}. \end{module} \begin{module}{diffusion}{dif-conf}{Conferences and workshops} A. Bombrun, B. Atfeh and L. Baratchart have presented a communication at CMFT2005 (Computational Methods and Function Theory), Joensuu, Finland (June). \refercite{th-david} \end{module} \loadbiblio \end{document}