Cauchy-Lipschitz theory for fractional multi-order dynamics -- State-transition matrices, Duhamel formulas and duality theorems

The aim of the present paper is to contribute to the development of the study of Cauchy problems involving Riemann-Liouville and Caputo fractional derivatives. Firstly existence-uniqueness results for solutions of non-linear Cauchy problems with vector fractional multi-order are addressed. A qualitative result about the behavior of local but non-global solutions is also provided. Finally the major aim of this paper is to introduce notions of fractional state-transition matrices and to derive fractional versions of the classical Duhamel formula. We also prove duality theorems relying left state-transition matrices with right state-transition matrices

Standard 1D solar atmosphere as initial condition for MHD simulations and switch-on effects

Many applications in Solar physics need a 1D atmospheric model as initial condition or as reference for inversions of observational data. The VAL atmospheric models are based on observations and are widely used since decades. Complementary to that, the FAL models implement radiative hydrodynamics and showed the shortcomings of the VAL models since almost equally long time. In this work, we present a new 1D layered atmosphere that spans not only from the photosphere to the transition region, but from the solar interior up to far in the corona. We also discuss typical mistakes that are done when switching on simulations based on such an initial condition and show how the initial condition can be equilibrated so that a simulation can start smoothly. The 1D atmosphere we present here served well as initial condition for HD and MHD simulations and should also be considered as reference data for solving inverse problems.Comment: 10 pages, 3 figures, published versio

Optimal sampled-data control, and generalizations on time scales

In this paper, we derive a version of the Pontryagin maximum principle for general finite-dimensional nonlinear optimal sampled-data control problems. Our framework is actually much more general, and we treat optimal control problems for which the state variable evolves on a given time scale (arbitrary non-empty closed subset of R), and the control variable evolves on a smaller time scale. Sampled-data systems are then a particular case. Our proof is based on the construction of appropriate needle-like variations and on the Ekeland variational principle.Comment: arXiv admin note: text overlap with arXiv:1302.351

Fractional fundamental lemma and fractional integration by parts formula -- Applications to critical points of Bolza functionals and to linear boundary value problems

In the first part of the paper, we prove a fractional fundamental (du Bois-Reymond) lemma and a fractional variant of the integration by parts formula. The proof of the second result is based on an integral representation of functions possessing Riemann-Liouville fractional derivatives, derived in this paper too. In the second part of the paper, we use the previous results to give necessary optimality conditions of Euler-Lagrange type (with boundary conditions) for fractional Bolza functionals and to prove an existence result for solutions of linear fractional boundary value problems. In the last case we use a Hilbert structure and the Stampacchia theorem.Comment: This is a preprint of a paper whose final and definite form is published in Advances in Differential Equation

Temperature structure of the intergalactic medium within seven nearby and bright clusters of galaxies observed with XMM-Newton

Aims. We map the temperature structure of the intra-cluster medium (ICM) within a nearly complete X-ray flux limited sample of galaxy clusters in the redshift range z=[0.045,0.096]. Our sample contains seven bright clusters of galaxies observed with XMM-Newton: Abell 399, Abell 401, Abell 478, Abell 1795, Abell 2029, Abell 2065, Abell 2256. Methods. We use a multi-scale spectral mapping algorithm especially designed to map spectroscopic observables from X-ray extended emission of the ICM. Derived from a former algorithm using Haar wavelets, our algorithm is now implemented with B-spline wavelets in order to perform a more regular analysis of the signal. Results. For the four clusters in our sample that are major mergers, we find a complex thermal structure with strong thermal variations consistent with their dynamics. For two of them, A2065 and A2256, we perform a 3-d analysis of cold front features evidenced from the gas temperature and brightness maps. Furthermore, we detect a significant non-radial thermal structure outside the cool core region of the other 3 more "regular" clusters, with relative amplitudes of about about 10%. We investigate possible implications of this structure on the mass estimates of the "regular" clusters A1795 and A2029, by extracting surface brightness and temperature profiles from sectors correspondings to the hottest and coldest regions in the maps. While compensating with surface brightness for A2029, leading to consistent mass profiles, the temperature structure leads to significant mass discrepancies in the innermost region of A1795.Comment: published in A&