11,672 research outputs found
On Finite-Time Stabilization of Evolution Equations: A Homogeneous Approach
International audienceGeneralized monotone dilation in a Banach space is introduced. Classical theorems on existence and uniqueness of solutions of nonlinear evolution equations are revised. A universal homogeneous feedback control for a finite-time stabilization of linear evolution equation in a Hilbert space is designed using homogeneity concept. The design scheme is demonstrated for distributed finite-time control of heat and wave equations
Singular stochastic integral operators
In this paper we introduce Calder\'on-Zygmund theory for singular stochastic
integrals with operator-valued kernel. In particular, we prove
-extrapolation results under a H\"ormander condition on the kernel. Sparse
domination and sharp weighted bounds are obtained under a Dini condition on the
kernel, leading to a stochastic version of the solution to the
-conjecture. The results are applied to obtain -independence and
weighted bounds for stochastic maximal -regularity both in the complex and
real interpolation scale. As a consequence we obtain several new regularity
results for the stochastic heat equation on and smooth and
angular domains.Comment: typos corrected. Accepted for publication in Analysis & PD
Spectral analysis of semigroups and growth-fragmentation equations
The aim of this paper is twofold: (1) On the one hand, the paper revisits the
spectral analysis of semigroups in a general Banach space setting. It presents
some new and more general versions, and provides comprehensible proofs, of
classical results such as the spectral mapping theorem, some (quantified)
Weyl's Theorems and the Krein-Rutman Theorem. Motivated by evolution PDE
applications, the results apply to a wide and natural class of generators which
split as a dissipative part plus a more regular part, without assuming any
symmetric structure on the operators nor Hilbert structure on the space, and
give some growth estimates and spectral gap estimates for the associated
semigroup. The approach relies on some factorization and summation arguments
reminiscent of the Dyson-Phillips series in the spirit of those used in
[87,82,48,81]. (2) On the other hand, we present the semigroup spectral
analysis for three important classes of "growth-fragmentation" equations,
namely the cell division equation, the self-similar fragmentation equation and
the McKendrick-Von Foerster age structured population equation. By showing that
these models lie in the class of equations for which our general semigroup
analysis theory applies, we prove the exponential rate of convergence of the
solutions to the associated remarkable profile for a very large and natural
class of fragmentation rates. Our results generalize similar estimates obtained
in \cite{MR2114128,MR2536450} for the cell division model with (almost)
constant total fragmentation rate and in \cite{MR2832638,MR2821681} for the
self-similar fragmentation equation and the cell division equation restricted
to smooth and positive fragmentation rate and total fragmentation rate which
does not increase more rapidly than quadratically. It also improves the
convergence results without rate obtained in \cite{MR2162224,MR2114413} which
have been established under similar assumptions to those made in the present
work
Almost automorphic delayed differential equations and Lasota-Wazewska model
Existence of almost automorphic solutions for abstract delayed differential
equations is established. Using ergodicity, exponential dichotomy and Bi-almost
automorphicity on the homogeneous part, sufficient conditions for the existence
and uniqueness of almost automorphic solutions are given.Comment: 16 page
Nonlinear L\'evy and nonlinear Feller processes: an analytic introduction
The program of studying general nonlinear Markov processes was put forward in
V. N. Kolokoltsov "Nonlinear Markov Semigroups and Interacting L\'evy Type
Processes" (Journ. Stat. Physics 126:3 (2007), 585-642), and was developed by
the author in monograph "Nonlinear Markov processes and kinetic equations".
Cambridge University Press, 2010, where, in particular, nonlinear L\'evy
processes were introduced.
The present paper is an invitation to the rapidly developing topic of
noninear Markov processes. We provide a quick (and at the same time more
abstract) introduction to the basic analytical aspects of the theory developed
in Part II of the above mentioned book.Comment: 20 page
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