A hybrid approach consisting of two neural networks is used to
model the oscillatory dynamical behavior of the
Kuramoto-Sivashinsky (KS) equation at a bifurcation parameter
<mml:math alttext="$\alpha=84.25$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>84.25</mml:mn></mml:math>. This oscillatory behavior results from a fixed
point that occurs at <mml:math alttext="$\alpha=72$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>72</mml:mn></mml:math>
 having a shape of two-humped
curve that becomes unstable and undergoes a Hopf bifurcation at
<mml:math alttext="$\alpha=83.75$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>83.75</mml:mn></mml:math>. First, Karhunen-Loève (KL) decomposition was
used to extract five coherent structures of the oscillatory
behavior capturing almost 100% of the energy. Based on the five
coherent structures, a system offive ordinary differential
equations (ODEs) whose dynamics is similar to the original
dynamics of the KS equation was derived via KL Galerkin
projection. Then, an autoassociative neural network was utilized
on the amplitudes of the ODEs system with the task of reducing
the dimension of the dynamical behavior to its intrinsic
dimension, and a feedforward neural network was usedto model
the dynamics at a future time. We show that by combining KL
decomposition and neural networks, a reduced dynamical model of
the KS equation is obtained.

Nejib Smaoui

A hybrid neural network model for the dynamics of the Kuramoto-Sivashinsky equation

We present a dynamic modelfor access control mechanism used in
computer communication network applied to MPEG video transmission
over Internet. This modelis different fromthosedeveloped
inthe previous works related to this topic. In our
model, token buckets supported by data buffersare used to shape
incoming traffic and one multiplexor, serving all the token
pools, multiplexes all theconforming
traffic. The model is governed by a system of discrete nonlinear
difference equations. Weuse neural network as the feedback
controller which receives at its input (measurable) available
information and provides at its output the optimal control. The
simulated annealing algorithm isusedto optimize the system
performance by adjusting the weights. For illustration, we
presentnumerical results which show that the system performance
of MPEG video server can be improved by using neural network and
simulated annealing approach.

N. U. Ahmed

Hong  Yan

Access control for MPEG video applications using neural network and simulated annealing

This paper gives localization and nonexistence conditions of
periodic orbits in some subsets of the state space. Mainly, our
approach is based on high-order extremum conditions, on
high-order tangency conditions of a nonsingular solution of a
polynomial system with an algebraic surface, and on some ideas
related to algebraically-dependent polynomials. Examples of the
localization analysis of periodic orbits are presented including
the Blasius equations, the generalized mass action (GMA) system,
and the mathematical model of the chemical reaction with
autocatalytic step.

Konstantin E. Starkov

Localization of periodic orbits of autonomous systems based on high-order extremum conditions

We consider optimum feedback control strategy for computer
communication network, in particular, the access control
 mechanism. The dynamic model
representing the source and the access control system is
described by a system of stochastic differential equations
developed in our previous works. Simulated annealing (SA) was used
to optimize the parameters of the control law based on neural
network. This technique was found to be computationally
intensive. In this paper, we have proposed to use a more powerful
algorithm known as recursive random search (RRS). By using this
technique, we have been able to reduce the computation time by a
factor of five without compromising the optimality. This is very
important for optimization of high-dimensional systems serving a
large number of aggregate users. The results show that the
proposed control law can improve the network performance by
improving throughput, reducing multiplexor and TB losses, and
relaxing, not avoiding, congestion.

Cheng Li

Optimum feedback strategy for access control mechanism modelled as stochastic differential equation in computer network

Modern complex large-scale impulsive systems involve multiple
modes of operation placing stringent demands on controller
analysis of increasing complexity. In analyzing these large-scale
systems, it is often desirable to treat the overall impulsive
system as a collection of interconnected impulsive subsystems.
Solution properties of the large-scale impulsive system are then
deduced from the solution properties of the individual impulsive
subsystems and the nature of the impulsive system
interconnections. In this paper, we develop vector dissipativity
theory for large-scale impulsive dynamical systems. Specifically,
using vector storage functions and vector hybrid supply rates,
dissipativity properties of the composite large-scale impulsive
systems are shown to be determined from the dissipativity
properties of the impulsive subsystems and their
interconnections. Furthermore, extended Kalman-Yakubovich-Popov
conditions, in terms of the impulsive subsystem dynamics and
interconnection constraints, characterizing vector
dissipativeness via vector system storage functions, are derived.
Finally, these results are used to develop feedback
interconnection stability results for large-scale impulsive
dynamical systems using vector Lyapunov functions.

Wassim M. Haddad

VijaySekhar Chellaboina

Qing Hui

Sergey Nersesov

Vector dissipativity theory for large-scale impulsive dynamical systems

In this study, the buffer allocation problem (BAP) in
homogeneous, asymptotically reliable serial production lines is
considered. A known aggregation method, given by Lim, Meerkov, and
Top (1990), for the performance evaluation (i.e., estimation of
throughput) of this type of production lines when the buffer
allocation is known, is used as an evaluative method in
conjunction with a newly developed dynamic programming (DP)
algorithm for the BAP. The proposed algorithm is applied to
production lines where the number of machines is varying
from four
up to a hundred machines. The proposed algorithm is fast because
it reduces the volume of computations by rejecting allocations
that do not lead to maximization of the line's throughput.
Numerical results are also given for large production lines.

A. C. Diamantidis

C. T. Papadopoulos

A dynamic programming algorithm for the buffer
allocation problem in homogeneous asymptotically reliable serial production lines

We study systems that are subject to sudden structural changes
due to either changes in the operational mode of the system or
failure. We consider linear dynamicalsystems that depend on a
modal variable which is either modeled as a finite-state Markov
chain or generated by an automaton that is subject to an external
disturbance. In the Markov chain case, the objective of the
control is to minimize a risk-sensitive cost functional. The
risk-sensitive cost functional measures the risk sensitivity of
the system to transitions caused by the random modal variable. In
the case when a disturbed automaton describes the modal variable,
the objective of the control is to make the system as robust to
changes in the external disturbance as possible. Optimality
conditions for both problems are derived and it is shown that the
disturbance rejection problem is closely related to a certain
risk-sensitive control problem for the hybrid system.

Thordur Runolfsson

Robust control of discrete-time hybrid systems with uncertain modal dynamics

This paper presents an integral solution of the
generalized one-dimensional equation of energy transport
with the convective term.The solution of the problem has been
achieved by the use of a novel technique that involves
generalized derivatives (in particular, derivatives of noninteger
orders). Confluent hypergeometric functions, known as Whittaker's
functions, appear in the course of the solution procedure upon
applying the Laplace transform to the original transport
equation.The analytical solution of the problem is written in
the integral form and provides a relationship between the
local values of the transported property (e.g.,
temperature, mass, momentum, etc.) and its flux.The solution is
valid everywhere within the domain, including the domain boundary.

Vladimir V. Kulish

An analytical solution of the generalized equation of energy transport in one-dimensional semi-infinite domains

In the area of stress-strength models there has been a large amount of work as regards estimation of the reliability <mml:math alttext="$R = \PPr (X_2 \lt X_1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mo>Pr</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#60;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> when <mml:math alttext="$X_1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> and <mml:math alttext="$X_2$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math> are independent random variables belonging to the same univariate family of distributions. The algebraic form for <mml:math alttext="$R = \PPr (X_2 \lt X_1)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi><mml:mo>=</mml:mo><mml:mo>Pr</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>X</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>&#60;</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math> has been worked out for the majority of the well-known distributions in the standard forms. However, there are still many other distributions (including generalizations of the well-known distributions) for which the form of <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math> has not been derived. In this paper, we consider several Laplace distributions and derive the corresponding forms for the reliability <mml:math alttext="$R$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>R</mml:mi></mml:math>. The calculations involve the use of special functions.

Saralees  Nadarajah

Reliability for Laplace distributions

A Melnikov analysis of single-degree-of-freedom (DOF) oscillators is performed by taking into account the first (classical) and higher-order Melnikov functions, by considering Poincar&#233; sections nonorthogonal to the flux, and by explicitly determining both the distance between perturbed and unperturbed manifolds (&#147;one-half&#148; Melnikov functions) and the distance between perturbed stable and unstable manifolds (&#147;full&#148; Melnikov function). The analysis is developed in an abstract framework, and a recursive formula for computing the Melnikov functions is obtained. These results are then applied to various mechanical systems. Softening versus hardening stiffness and homoclinic versus heteroclinic bifurcations are considered, and the influence of higher-order terms is investigated in depth. It is shown that the classical (first-order) Melnikov analysis is practically inaccurate at least for small and large excitation frequencies, in correspondence to degenerate homo/heteroclinic bifurcations, and in the case of generic periodic excitations.

Stefano  Lenci

Giuseppe  Rega

Higher-order Melnikov functions for single-DOF mechanical oscillators: theoretical treatment and applications

Stability of a switched system that consists of a set of linear
time invariant subsystems and a periodic switching rule is
investigated. Based on the Floquet theory, necessary and
sufficient conditions are given for exponential stability. It is
shown that there exists a slow switching rule that achieves
exponential stability if at least one of these subsystems is
asymptotically stable. It is also shown that there exists a fast
switching rule that achieves exponential stability if the average
of these subsystems is asymptotically stable. The results are
illustrated by examples.

Cevat  G&#246;k&#231;ek

Stability analysis of periodically switched linear systems using Floquet theory

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Mathematical Problems in Engineering is a broad-based journal publishing&nbsp;results of rigorous engineering research across all disciplines, carried out using mathematical tools.

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Chief Editor, Professor Guangming Xie, is currently a full professor of dynamics and control with the College of Engineering, Peking University. His research interests include complex system dynamics and control and intelligent and biomimetic robots.

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Mathematical Problems in Engineering - Table of contents 2004

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Mathematical Problems in Engineering