Multiple Stationary Solutions and Global Stabilization of Reaction-diffusion Gilpin-Ayala Competition Model under Event-triggered Impulsive Control

In this paper, the author utilizes Saddle Theorem and variational methods to deduce existence of at least six stationary solutions for reaction-diffusion Gilpin-Ayala competition model (RDGACM). To obtain the global stabilization of the positive stationary solution of the RDGACM, the author designs a suitable impulsive event triggered mechanism (IETM) to derive the global exponential stability of the the positive stationary solution. It is worth mentioning that the new mechanism can exclude Zeno behavior and effectively reduce the cost of impulse control through event triggering mechanism. Besides, compared with existing literature, the restrictions on the parameters of the RDGACM are relaxed so that the methods used in existing literature can not be applied to the relaxed case of this paper, and so the author makes comprehensive use of Saddle Theorem, orthogonal decomposition of Sobolev space $H_0^1(\Omega)$ and variational methods to overcome the mathematical difficulty. Numerical examples show the effectiveness of the methods proposed in this paper.

Development of Damage Reconstruction Techniques From Impulsive Events Based on Measurements Made Remotely

Damage done to petroleum pipelines can be catastrophic if left undetected with time. Most of these damages results in financial losses, environmental pollution and frequent loss of life as a result of explosion due to prolong exposure of pipeline products. Therefore, damage detection and location techniques will play a key role in the overall integrity management of a pipeline system. This paper addresses the development and testing of mathematical techniques for locating an impulsive event on a pipeline and reconstructing the pressure pulse caused by it from measurements made remotely. When an impulsive event occurs along a pipeline, the pressure pulse propagates in both directions and can be detected and measured by sensors located at different positions along the pipeline. From these measurements the location of the event can be determined and its form reconstructed. Techniques for reconstructing the pulse at it source from the distorted pulses measured were developed using deconvolution theory and inverse methods. This theoretical work was validated by experiments using a simulated pipeline. The experimental work was carried out using a test rig comprising a flexible hose pipe 23 m long and 19 mm diameter with four pressure sensors distributed along the pipe and connected to a data acquisition system. The techniques were tested for static and flowing air in the pipe, and were found to give good results.

Modelling and Regularity of Nonlinear Impulsive Switching Dynamical System in Fed-Batch Culture

A hybrid system with state-based switchings is proposed to describe the fed-batch production of 1,3-propandiol from glycerol in our previous work. However, the on-off switching of alkali is too frequent, which greatly increases the computational cost of the numerical solution to the system so as to locate the state-based switchings in strict time order and implement the correct mode changes. To deal with this problem, we consider the switching of alkali pump as an impulsive event and present a nonlinear impulsive switching system to describe the fed-batch culture. It is proved that the impulsive switching system is non-Zeno. Some basic properties of solutions to the impulsive switching system are also explored. In order to overcome the discontinuities of the system, the Skorohod topology is induced and a specific form ofλis constructed to prove the main theorem. Additionally, a numerical simulation is carried out to show that the proposed system can describe the fed-batch culture properly and the essential difference with the previous work.

Computer Simulation of a Solar Microwave Burst

A simple impulsive event observed at microwave frequencies between 3.2 and 50 GHz is simulated with a homogeneous source model. Intensity and polarization of the 1984 April 26 event are shown in fig. 1a and 1b, respectively. The spectral maximum is close to the observing frequency of 11.8 GHz.The time profiles of the intensity I and the circular polarization ρc at this frequency as well as the spectrum at different characteristic phases (rise, maximum and decay) are calculated and compared with the observation in order to deduce parameters of the microwave source.