Editorial
Geometric and Polynomial Approaches of Complex Systems and
Control in Mathematics and Applied Sciences
Baltazar Aguirre-Hern´ andez ,
1
Jorge-Antonio L´ opez-Renter´ ıa ,
2
Alejandro Armando Hossian,
3
and Cutberto Romero-Mel´ endez
1
1
Universidad Aut´ onoma Metropolitana, Mexico City, Mexico
2
CONACYT, TecNM, Instituto Tecnol´ ogico de Tijuana, B.C., Tijuana, Mexico
3
Universidad Tecnol´ ogica Nacional, Neuqu´ en, Argentina
Correspondence should be addressed to Baltazar Aguirre-Hern´ andez; bahe@xanum.uam.mx
Received 27 February 2020; Accepted 27 February 2020; Published 14 May 2020
Copyright © 2020 Baltazar Aguirre-Hern´ andez et al. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
Complex dynamical systems are present in many theoretical
and practical domains of the science and engineering:
physical processes, man-made systems, deterministic and
stochastic control systems, distributed systems, and net-
works of leader-follower multiagent systems, between
others.
e matrix approach of state space has long been the way
to address many of the central problems of systems with
control. In recent decades, novel methods and approaches
for the study of systems, linear and nonlinear, with control
are based on a geometrical approach whose objective is to
reveal the properties of the geometric skeleton of the dy-
namic system. e geometric approach can convert a dif-
ficult nonlinear problem into a straight-forward linear one.
Geometric control theory and sub-Riemannian geom-
etry are areas that play a very important role in complex
dynamical systems, searching controllability, optimality, and
stability for linear and nonlinear control systems, applying
Lie theory techniques, Pontryagin maximum principle, and
other geometric and algebraic techniques for robotic con-
trol, motion planning problems, complexity on path plan-
ning, neurobiological visual processing models, and digital
image reconstruction.
On the contrary, the use of polynomial theory has been a
useful tool to explain the classical and complex behavior of
the solutions for a dynamical system, and it is largely
exploited in fundamental problems such as controllability,
stability, robustness, and other interesting applications in
uncertain systems, nonlinear systems, time-delay systems,
hybrid systems, and model predictive control.
In the paper “Poinacar´ e Map Approach to Global Dy-
namics of the Integrated Pest Management Prey-Predator
Model,” Zhenzhen Shi et al. studied, by means Poinacar´ e
maps, the existence of periodic solutions of an integrated
pest management prey-predator model with ratio-depen-
dent and impulsive feedback control. e existence and
stability of boundary order-one periodic solution is proved,
and the authors give conditions for the existence and global
stability of order r - 1 periodic solution and for the existence
of order k periodic solution.
e paper “Availability Equivalence Analysis of a Re-
pairable Bridge Network System,” by Jaafar Alghazo et al.,
discusses availability equivalence factors of a repairable
bridge network system where all components have constant
failure and repair rates. e authors derive the availability of
the original system and improved systems. Two types of
availability equivalence factors of the system are obtained in
order to compare different system designs. Numerical ex-
amples are included for illustrating the obtained results. e
proposed models and analysis method are very useful in
system engineering.
In the paper “On the Delay Interval in Which the
Control Delay Systems are Stabilizable,” Jiang Wei studies
the stability of single delay systems and uses two-variable
polynomials to derive some stability criteria. Besides, he uses
these results to compute an interval for the delay where a
Hindawi
Complexity
Volume 2020, Article ID 6281613, 2 pages
https://doi.org/10.1155/2020/6281613