4,141 research outputs found
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
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