The realization problem for positive single-input single-output
 discrete-time systems with one time
delay is formulated and solved. Necessary and sufficient
conditions for the solvability of the realization problem are
established. A procedure for computation of a minimal positive
realization of a proper rational function is presented and
illustrated by an example.

Tadeusz  Kaczorek

Realization problem for positive linear systems with time delay

Functionally graded materials have gained considerable attention
in the high-temperature applications. A study of parametric
vibrations of functionally graded plates subjected to in-plane
time-dependent forces is presented. Moderately large deflection
equations taking into account a coupling of in-plane and
transverse motions are used. Material properties are graded in the
thickness direction of the plate according to volume fraction
power law distribution. An oscillating temperature causes
generation of in-plane time-dependent forces destabilizing the
plane state of the plate equilibrium. The asymptotic stability and
almost-sure asymptotic stability criteria involving a damping
coefficient and loading parameters are derived using Liapunov's
direct method. Effects of power law exponent on the stability
domains are studied.

Andrzej  Tylikowski

Dynamic stability of functionally graded plate under
in-plane compression

We consider the mathematical model of interaction of a vibrating surface with
the load placed on it. With the purpose of accounting for
influence on behavior of not only system interaction of a blade
with a load  but also internal interaction of particles of a
material, the load is submitted as a finite number of strips with
zero thickness. The carrying blade is represented as a vibrating
membrane. It is supposed that the weight of the material is
comparable to or considerably surpasses the weight of the blade.
Therefore, the model takes into account the inertia of the
material. In the model with joint movement of the blade and the
load, the separation opportunity of the load from the blade is
provided. Therefore, there is a phase of separate movement of the
blade and the load, with their subsequent connection accompanied
with impact. The process of system movement is represented as
alternating sequences of joint and separate movements of the load
and the blade. The modeling of the process of the interaction of
the load and the blade is represented as an initial-boundary value
problem. The method of solution is developed and the exact
solution of the set problem is obtained in a class of generalized
functions.

V. A. Ostapenko

Problems of interaction of vibrating surfaces with
processable materials

A novel iteration procedure for dynamical problems, where in each
time step, a contacting plates' zone is improved, is proposed.
Therefore, a zone and magnitude of a contact load are
also improved. Investigations of boundary conditions' influence on
externally driven vibrations of uncoupled two-layer plates, where
for each of the layers, the Kirchhoff hypothesis holds, are
carried out.

J. Awrejcewicz

V. A. Krysko

O. Ovsiannikova

Novel procedure to compute a contact zone magnitude of
vibrations of two-layered uncoupled plates

Jan Awrejcewicz

Guest editorial

The minimization control problem of quadratic functionals for the
class of affine nonlinear systems with the hypothesis of nilpotent
associated Lie algebra is analyzed. The optimal control
corresponding to the first-, second-, and third-order nilpotent
operators is determined. In this paper, we have considered the
minimum fuel problem for the multi-input nilpotent control and for
a scalar input bilinear system for such systems. For the
multi-input system, usually an analytic closed-form solution for
the optimal control <mml:math alttext="$u^{\star}_{i}(t)$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>u</mml:mi><mml:mi>i</mml:mi><mml:mo>∗</mml:mo></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math> is not possible and it is
necessary to use numerical integration for the set of <mml:math alttext="$m$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math>
nonlinear coupled second-order differential equations. The optimal
control of bilinear systems is obtained by considering the Lie
algebra generated by the system matrices. It should be noted that
we have obtained an open-loop control depending on the initial
value of the state <mml:math alttext="$x_{0}$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>x</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math>.

M. Popescu

A. Dumitrache

On the optimal control of affine  nonlinear systems

Mechanical models are governed either by partial differential equations with boundary conditions and initial conditions (e.g., in the frame of continuum mechanics) or by ordinary differential equations (e.g., after discretization via Galerkin procedure or directly from the model description) with the initial conditions. In order to study dynamical behavior of mechanical systems with a finite number of degrees of freedom including nonsmooth terms (e.g., friction), we consider here problems governed by differential inclusions. To describe effects of particular constitutive laws, we add a delay term. In contrast to previous papers, we introduce delay via a Volterra kernel. We provide existence and uniqueness results by using an Euler implicit numerical scheme; then convergence with its order is established. A few numerical examples are given.

Claude-Henri Lamarque

Jérôme Bastien

Matthieu Holland

Maximal monotone model with delay term of convolution

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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

Dynamical Engineering Systems

Dynamical Engineering Systems

Articles

Realization problem for positive linear systems with time delay

Dynamic stability of functionally graded plate under in-plane compression

Problems of interaction of vibrating surfaces with processable materials

Novel procedure to compute a contact zone magnitude of vibrations of two-layered uncoupled plates

Guest editorial

On the optimal control of affine nonlinear systems

Maximal monotone model with delay term of convolution