In this paper, a Lyapunov function candidate is introduced for multivariable systems with inner delays, without assuming a priori stability for the nondelayed subsystem. By using this Lyapunov function, a controller is deduced. Such a controller utilizes an input&#x2013;output description of the original system, a circumstance that facilitates practical applications of the proposed approach.

Carlos F. Alastruey

Manuel de la Sen

Stability of time-delay systems via Lyapunov functions

In this paper, we revise the classical formulation of the problem depriving it of the concepts that are superfluous from the mathematical point of view. We observe that a number of power stations can be substituted by a single one that behaves equivalently to the entire set. Proceeding in this way, we obtain a variational formulation in its purest sense (without restrictions). This formulation allows us to employ the theory of calculus of variations to the highest degree. We then calculate the equivalent minimizer in the case where the cost functions are second-order polynomials. We prove that the equivalent minimizer is a second-order polynomial with piece-wise 
constant coefficients. Moreover, it belongs to the class <mml:math alttext="$C^1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:math>. Finally, we present various examples prompted by real systems and perform the proposed algorithms using Mathematica.

L.  Bay&#243;n

J. M. Grau

P.  Su&#225;rez

A new formulation of the equivalent thermal in optimization of
hydrothermal systems

A new treatment of kinematical constraints and potential energies arising in the dynamics of systems of rigid bodies and particles is presented which is suited to Newtonian and Lagrangian formulations. Its novel feature is the imposing of invariance requirements on the constraint functions and potential energy functions. These requirements are extensively used in continuum mechanics and, in the present context, one finds certain generalizations of Newton's third law of motion and an elucidation of the nature of constraint forces and moments. One motivation for such a treatment can be found by considering approaches where invariance 
requirements are ignored. In contrast to the treatment presented in this paper, it is shown that this may lead to a difficulty in formulating the equations governing the motion of the system.

Oliver M. O'reilly

Arun R. Srinivasa

On potential energies and constraints in the dynamics of rigid
bodies and particles

A numerical procedure for an inverse problem of identification of an unknown source in a heat equation is presented. Approach of proposed method is to approximate unknown function by polygons linear pieces which are determined consecutively from the solution of minimization problem based on the overspecified data. Numerical examples are presented.

Afet Golayo&#x11F;lu Fatullayev

Numerical method of identification of an unknown source term in a
heat equation

Dispersion relations are obtained for the propagation of symmetric and antisymmetric modes in a free transversely isotropic plate. Dispersion curves are plotted for the first four symmetric modes for a magnesium plate immersed in water. The first mode is highly damped and switches over to the second mode when the normalized frequency exceeds 12.

F. Ahmad

N. Kiyani

F. Yousaf

M. Shams

Guided waves in a fluid-loaded transversely isotropic plate

We present existence results for some boundary value problems defined on infinite intervals. In particular our discussion includes a problem which arises in the theory of colloids.

Ravi P. Agarwal

Donal  O&#39;Regan

Boundary value problems on the half line in the theory of colloids

This paper presents an upper and lower solution theory for boundary value problems modelled from the Thomas&#8211;Fermi equation subject to a boundary condition corresponding to the neutral atom with Bohr radius.

An upper and lower solution approach for a generalized
Thomas&#8211;Fermi theory of neutral atoms

Inequalities, relations and stochastic orderings, as well as useful ageing notions for weighted distributions are established. Also presented are preservation and stability results and comparisons for weighted and length-biased distributions. Relations for length-biased and equilibrium distributions as examples of weighted distributions are also presented.

Broderick O. Oluyede

E. Olusegun George

On stochastic inequalities and comparisons of reliability
measures for weighted distributions

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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 2002

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Mathematical Problems in Engineering