The exponential and asymptotic stability are studied for certain coupled systems
involving unbounded linear operators and linear infinitesimal semigroup generators.
Examples demonstrating the theory are also given from the field of partial differential equations.

Farid Ammar Khodja

Assia  Benabdallah

Djamel  Teniou

Stability of coupled systems

The embedding functions of an intermediate space <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> into a Banach couple
<mml:math alttext="$(A_0,A_1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> are defined as its embedding constants into the couples
<mml:math alttext="$(\frac{1}{\alpha}A_0,\frac{1}{\beta}A_1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&#945;</mml:mi></mml:mfrac><mml:msub><mml:mi>A</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>&#946;</mml:mi></mml:mfrac><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, <mml:math alttext="$\forall\alpha,\beta &#62; 0$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>&#8704;</mml:mo><mml:mi>&#945;</mml:mi><mml:mo>,</mml:mo><mml:mi>&#946;</mml:mi><mml:mo>&#62;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>. Using these functions,
we study properties and interrelations of different intermediate spaces, give a new description of all real
interpolation spaces, and generalize the concept of weak-type interpolation to
any Banach couple to obtain new interpolation theorems.

Evgeniy  Pustylnik

Embedding functions and their role  in interpolation theory

In this work we study a deformation of the minimal interface of two fluids in a vertical tube under the presence of gravitation.
We show that a symmetry of the base of tube let us to apply
a method developed earlier by the first author and based
on the Crandall-Rabinowitz bifurcation theorem.
Using the natural symmetry of the corresponding variational problem
defined by a symmetry of region and restricting the functional
to spaces of invariant functions we show the existence of bifurcation,
and describe its local picture,
for interfaces parametrized by the square and disc.

A. Y. Borisovich

W. Marzantowicz

Bifurcation of the equivariant minimal interfaces in a hydromechanics problem

Morse theory for isolated critical points at infinity is used for
the existence of multiple critical points for an asymptotically quadratic
functional. Applications are also given for the existence of multiple
nontrivial periodic solutions of asymptotically Hamiltonian systems.

Shujie Li

Jiabao Su

Existence of multiple critical points for an asymptotically
quadratic functional with applications

The existence, uniqueness, regularity and asymptotic
behavior of global solutions of semilinear heat equations in Hilbert spaces
are studied by developing new results in the theory of one-parameter strongly
continuous semigroups of bounded linear operators. Applications to special
semilinear heat equations in <mml:math alttext="$L^2(\mathbb{R}^n)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>L</mml:mi><mml:mrow><mml:mtext>&#8201;</mml:mtext><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>&#8477;</mml:mi><mml:mi>n</mml:mi></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> governed by pseudo-differential
operators are given.

G. Mihai Iancu

M. W. Wong

Global solutions of semilinear heat equations in Hilbert spaces

The invariance of a closed and convex set under a semigroup <mml:math alttext="$S(t)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> associated with a nonlinear form is investigated. Other properties of increase and domination 
of the semigroup <mml:math alttext="$S(t)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>S</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> are also derived. Examples are also given to demonstrate the power of the theoretical results.

Louise  Barth&#233;lemy

Invariance d'un convexe ferm&#233; par un semi-groupe associ&#233; &#224; une forme non-lin&#233;aire

This paper is concerned with the approximate and exact controllability      
properties of the wave equation with interior point controls entering       
via the concentrated force, the velocity of the displacement and the moment.
The emphasis is given to the moving point controls and their dual           
observations whose advantages and disadvantages, versus the static ones,    
are analyzed with respect to the space dimension, the duration of the       
control time interval and the function spaces involved.

A. Y. Khapalov

Interior point control and observation for the wave equation

We show here the uniform stabilization of a coupled system of hyperbolic and parabolic PDE's which describes a particular fluid/structure interaction system. This system has the wave equation, which is satisfied on the
interior of a bounded domain <mml:math alttext="$\Omega$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#937;</mml:mi></mml:math>, coupled to a &#8220;parabolic&#8211;like&#8221;
beam equation holding on <mml:math alttext="$\partial\Omega$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>&#8706;</mml:mo><mml:mi>&#937;</mml:mi></mml:mrow></mml:math>, and wherein the coupling is accomplished through velocity terms on the boundary. Our result is an analog of a recent result by Lasiecka and Triggiani which shows the exponential stability of the wave
equation via Neumann feedback control, and like that work, depends upon a
trace regularity estimate for solutions of hyperbolic equations.

George  Avalos

The exponential stability of a coupled hyperbolic/parabolic
system arising in structural acoustics

We study nonlinear semigroups of holomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we characterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog of the Hille exponential formula. We then apply our results to the null point theory of semi-plus complete vector fields. We study the structure of null point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of
the implicit function theorem and a discussion of some open problems are
also included.

Simeon Reich

David Shoikhet

Generation theory for semigroups of holomorphic mappings
in Banach spaces

About this journal

Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

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Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.

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Abstract and Applied Analysis - Table of contents 1996

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Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis