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This study sought to determine the best method to quantify training based on heart rate data. It proposes a modification of Banister’s original performance model to improve the accuracy of predicted performance. The new formulation introduces a variable that accounts for changes in the subject’s initial performance as a result of the quantity of training. The two systems models were applied to a well-trained female monofin swimmer over a 24-week training period. Each model comprised a set of parameters unique to the individual and was estimated by fitting model-predicted performance to measured performance. We used the Alienor method associated to Optimization-Preserving Operators to identify these parameters. The quantification method based on training intensity zones gave a better estimation of predicted performance in both models. Using the new model in sports in which performance is generally predicted (running, swimming) will help us to define its real interest.

In this work, we discuss how to get instantaneous information (at fixed t = T ∈ [0, +∞[) on pollution terms in distributed systems of incomplete data in some ecology and/or meteorology problems, using the sentinel method of Lions [J.-L. Lions, Sentinelles Pour les Systemes Distribues a Donnees Incompletes, Recherches en mathematiques appliquees RMA No. 21, Masson, Paris, 1992] with control and observation having their supports in two different open sets as by Nakoulima [O. Nakoulima, Controlabilite a zero avec contraintes sur le controle, C. R. Acad. Sci. Paris, Ser. I 339 (2004), pp. 405–410]. Here, the ecological system is affected by pollution in the boundary of the domain (e.g. a border of a lake). We verify that if the initial data are completely unknown, the sentinel is null. If there is some information on the initial data, the instantaneous sentinel naturally exists if the control set is bigger than the one where the observation may be defined. Finally, we give the characterization of the instanta...

We consider a distributed system in which the state q is governed by a parabolic equation and a pair of controls v = (h,k) where h and k play two different roles: the control k is of controllability type while h expresses that the state q does not move too far from a given state. Therefore, it is natural to introduce the control point of view. In fact, there are several ways to state and solve optimal control problems with a pair of controls h and k , in particular the Least Squares method with only one criteria for the pair (h,k) or the Pareto Optimal Control for multicriteria problems. We propose here to use the notion of Hierarchic Control . This notion assumes that we have two controls h, k where h will be the leader while k will be the follower . The main tool used to solve the null-controllability problem with constraints on the follower is an observability inequality of Carleman type which is “adapted” to the constraints. The obtained results are applied to the sentinels theory of Lions [Masson (1992)].

Resume On examine un probleme de controlabilite a zero pour l'equation de la chaleur avec des contraintes lineaires (en nombre fini) sur le controle. L'outil essentiel pour resoudre les problemes d'existence et de convergence est une inegalite d'observabilite de type Carleman qui, ici, est adaptee aux contraintes . On applique ensuite les resultats obtenus a la theorie des sentinelles de Lions. Pour citer cet article : O. Nakoulima, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The Gardner or extended Korteweg-de Vries equation with the variable-sign cubic non-linear term is studied. This equation occurred in the theory of interfacial waves in the stratified shear flow at some certain conditions on the medium stratification. An asymptotic method is applied to study the solitary wave transformation when the coefficient of the cubic non-linear term is varied slowly. It is found the one branch of the solitary waves destroys in the vicinity of the bifurcation point, where the exponential soliton transforms into the breather through algebraic soliton. Other solitary waves change their amplitude accordingly to the cubic non-linear term variation. The structure of the tail formed behind the solitary wave is determined. Direct numerical simulation of the Gardner equation is performed to analyze the transformation and breaking of the solitary wave. It confirms most of the theoretical results. The calculated dynamics of the ''table-top'' soliton due to the large values of the mass and momentum integrals differs from the asymptotic prediction because the characteristic time of the variation of the cubic non-linear term has the same order as the duration of ''table-top'' soliton.

We study the control of distributed systems with incomplete data extending the work in [Nakoulima, et al., C. R. Acad. Sci. Paris Ser. I 330 (2000) 801–806] to the nonlinear case. In a first part, we discuss the regularity of the state solution to the problem in the general case, and we give an application. The second part deals with the optimal control problem. We prove the convergence of the low-regret control introduced in [Nakoulima, et al., C. R. Acad. Sci. Paris Ser. I 330 (2000) 801–806] to the no-regret control for which we obtain a singular optimality system. We also continue studying the application of the first part.

Resume On traite du controle de systemes distribues a donnees manquantes d'apres la notion de controle sans regrets utilisee par J.-L. Lions [2]. Au controle sans regrets, on associe comme dans J.-L. Lions [4], une suite de controles a moindres regrets definis par une perturbation quadratique. On montre ici que le systeme perturbe qui correspond a une suite de problemes de controle standard, converge vers le controle sans regrets pour lequel on obtient un systeme d'optimalite singulier. On donne aussi quelques applications.

In this article, we study the null-controllability problem with two constraints on a pair of controls.We apply these results to a discriminating sentinel with given sensitivity to detect some parameters in a pollution problem, governed by a semi-linear parabolic equation with Dirichlet boundary condition.

International audience; Modelling environmental problems leads to mathematical systems with missing data. For instance, weather problems have generally missing initial conditions. The paper is concerned with identifying pollution terms arising in the state equation of some dissipative system with incomplete initial condition. To this aim the so-called sentinel method is used. Here, the problem of determining a sentinel is equivalent to a null-controllability problem for which Carleman inequalities are revisited.

Resume Dans cette Note, on traite de la controlabilite a zero pour le probleme de chaleur a donnee initiale manquante, avec un terme source v∈L2. On cherche ensuite le meilleur des controles v dans le sous ensemble convexe ferme des controles admissibles U ad de L2 par le moyen de la notion de controle a moindres regrets introduite par Lions. On obtient un systeme d'optimalite singulier sans hypothese supplementaire sur U ad . Pour citer cet article : R. Dorville et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

International audience; We discuss the control of distributed systems with incomplete data following the notion of no-regret control (or, equivalently, Pareto control) used by Lions in [C. R. Acad. Sci. Paris Ser. I Math., 302 (1986), pp. 223-227] and [C. R. Acad. Sci. Paris Ser. I Math., 302 (1992), pp. 1253-1257]. We associate with the no-regret control a sequence of low-regret controls defined by a quadratic perturbation previously used by Nakoulima, Omrane, and Velin in [C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), pp. 801-806]. In the first part, we prove that the perturbed system corresponds to a sequence of standard control problems and converges to the no-regret (or Pareto) control for which we obtain a singular optimality system. We give also some applications. In the second part, we show how the method can be extended to the evolution case. Equations of parabolic type, Petrowsky type, or hyperbolic type are considered.

The problem of long-wave scattering by piecewise-constant periodic topography is studied both for a linear solitary-like wave pulse, and for a weakly nonlinear solitary wave [Korteweg-de Vries (KdV) soliton]. If the characteristic length of the topographic irregularities is larger than the pulse length, the solution of the scattering problem is obtained analytically for a leading wave in the framework of linear shallow-water theory. The wave decrement in the case of the small height of the topographic irregularities is proportional to delta2, where delta is the relative height of the topographic obstacles. An analytical approximate solution is also obtained for the weakly nonlinear problem when the length of the irregularities is larger than the characteristic nonlinear length scale. In this case, the Korteweg-de Vries equation is solved for each piece of constant depth by using the inverse scattering technique; the solutions are matched at each step by using linear shallow-water theory. The weakly nonlinear solitary wave decays more significantly than the linear solitary pulse. Solitary wave dynamics above a random seabed is also discussed, and the results obtained for random topography (including experimental data) are in reasonable agreement with the calculations for piecewise topography.

We consider a nonlinear model for age-dependent population dynamics subject to a density dependent factor which regulates the selection of newborn at age zero. The initial-boundary value problem is studied using a vanishing viscosity method (in the age direction) together with the fixed point theory. Existence and uniqueness are obtained, and also the positivity of the solution to the problem.

International audience; We use the low-regret notion of Lions for the control of a class of singular distributed systems: the ill-posed problems. A regularization approach is applied to the backwards heat equation, and we obtain a problem of incomplete data, for which the method of Nakoulima et al. is developed. Passing to the limit, a singular optimality system is obtained for the low-regret control of the original problem without any Slater hypothesis.

Cette thèse est consacrée à l’étude du contrôle de trois problèmes singuliers associés à l’équation de la chaleur. Pour le contrôle de chacun de ces trois problèmes, on propose la notion de contrôle sans regret. Pour y parvenir, nous utilisons la méthode de régularisation et à la théorie des systèmes distribués à données manquantes. Nous présentons nos résultatsafin qu’ils soient applicables aux problèmes linéaires de type parabolique., This thesis is devoted to the study of the control of three singular problems associated with the heat equation. To control each of these three problems, we propose the notion of no-regret control.To this end, we use the regularisation method and the theory of distributed systems with incomplete data. We present our results in order to apply them to parabolic-type linear problems.