Éditeur(s) : HAL CCSD Elsevier
Résumé : 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.
ISSN: 0893-9659

hal-00763938
https://hal.univ-antilles.fr/hal-00763938
DOI : 10.1016/j.aml.2003.04.006

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : 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.
ISSN: 1270-900X

hal-00720818
https://hal.univ-antilles.fr/hal-00720818
https://hal.univ-antilles.fr/hal-00720818/document
https://hal.univ-antilles.fr/hal-00720818/file/esaimpr2.pdf

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : 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.
ISSN: 0363-0129

hal-00764259
https://hal.univ-antilles.fr/hal-00764259
https://hal.univ-antilles.fr/hal-00764259/document
https://hal.univ-antilles.fr/hal-00764259/file/38018.pdf
DOI : 10.1137/S0363012900380188

Éditeur(s) : HAL CCSD Nature Publishing Group
Résumé : International audience
High 3He/4He ratios in some basalts have generally been interpreted as originating in an incompletely degassed lower-mantle source1, 2, 3, 4, 5, 6, 7, 8, 9. This helium source may have been isolated at the core-mantle boundary region since Earth's accretion4, 5, 6. Alternatively, it may have taken part in whole-mantle convection and crust production over the age of the Earth7, 8, 9; if so, it is now either a primitive refugium at the core-mantle boundary8 or is distributed throughout the lower mantle7, 9. Here we constrain the problem using lavas from Baffin Island, West Greenland, the Ontong Java Plateau, Isla Gorgona and Fernandina (Galapagos). Olivine phenocryst compositions show that these lavas originated from a peridotite source that was about 20 per cent higher in nickel content than in the modern mid-ocean-ridge basalt source. Where data are available, these lavas also have high 3He/4He. We propose that a less-degassed nickel-rich source formed by core-mantle interaction during the crystallization of a melt-rich layer or basal magma ocean5, 6, and that this source continues to be sampled by mantle plumes. The spatial distribution of this source may be constrained by nickel partitioning experiments at the pressures of the core-mantle boundary.
ISSN: 0028-0836

hal-00941025
https://hal.archives-ouvertes.fr/hal-00941025
DOI : 10.1038/nature11771