4,672 research outputs found
A survey on fuzzy fractional differential and optimal control nonlocal evolution equations
We survey some representative results on fuzzy fractional differential
equations, controllability, approximate controllability, optimal control, and
optimal feedback control for several different kinds of fractional evolution
equations. Optimality and relaxation of multiple control problems, described by
nonlinear fractional differential equations with nonlocal control conditions in
Banach spaces, are considered.Comment: This is a preprint of a paper whose final and definite form is with
'Journal of Computational and Applied Mathematics', ISSN: 0377-0427.
Submitted 17-July-2017; Revised 18-Sept-2017; Accepted for publication
20-Sept-2017. arXiv admin note: text overlap with arXiv:1504.0515
Discretely sampled signals and the rough Hoff process
We introduce a canonical method for transforming a discrete sequential data
set into an associated rough path made up of lead-lag increments. In
particular, by sampling a -dimensional continuous semimartingale at a set of times , we construct a piecewise
linear, axis-directed process comprised
of a past and future component. We call such an object the Hoff process
associated with the discrete data . The Hoff process can
be lifted to its natural rough path enhancement and we consider the question of
convergence as the sampling frequency increases. We prove that the It\^{o}
integral can be recovered from a sequence of random ODEs driven by the
components of . This is in contrast to the usual Stratonovich integral
limit suggested by the classical Wong-Zakai Theorem. Such random ODEs have a
natural interpretation in the context of mathematical finance
Multiple solutions of nonlinear equations involving the square root of the Laplacian
In this paper we examine the existence of multiple solutions of parametric
fractional equations involving the square root of the Laplacian in a
smooth bounded domain () and with
Dirichlet zero-boundary conditions, i.e. \begin{equation*} \left\{
\begin{array}{ll} A_{1/2}u=\lambda f(u) & \mbox{ in } \Omega\\ u=0 & \mbox{ on
} \partial\Omega. \end{array}\right. \end{equation*} The existence of at least
three -bounded weak solutions is established for certain values of
the parameter requiring that the nonlinear term is continuous and
with a suitable growth. Our approach is based on variational arguments and a
variant of Caffarelli-Silvestre's extension method
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