scholarly journals Profile of a touch-down solution to a nonlocal MEMS model

2019 ◽  
Vol 29 (07) ◽  
pp. 1279-1348 ◽  
Author(s):  
Giao Ky Duong ◽  
Hatem Zaag
Keyword(s):  
Finite Time ◽  
Index Theory ◽  
Mems Devices ◽  
Blowup Solution ◽  
Finite Dimensional ◽  
Topological Argument ◽  
Dimensional Parameters ◽  
The Stability ◽  
Final Profile ◽  
The Mathematical Model

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on [Formula: see text] : [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] and [Formula: see text]. In this work, we have succeeded to construct a solution which quenches in finite time [Formula: see text] only at one interior point [Formula: see text]. In particular, we give a description of the quenching behavior according to the following final profile [Formula: see text] The construction relies on some connections between the quenching phenomenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The method is inspired by the work of Merle and Zaag [Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] with a suitable modification. In addition to that, the proof relies on two main steps: A reduction to a finite-dimensional problem and a topological argument based on index theory. The main difficulty and novelty of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite-dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data.

scholarly journals Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

2019 ◽  
Vol 10 (4) ◽  
pp. 299-312
Author(s):  
Tej-Eddine Ghoul ◽  
Van Tien Nguyen ◽  
Hatem Zaag
Keyword(s):  
Blow Up ◽  
Index Theory ◽  
Parabolic Systems ◽  
Type I ◽  
Heat System ◽  
Finite Dimensional ◽  
Semilinear Parabolic Systems ◽  
Step Procedure ◽  
Topological Argument ◽  
Semilinear Parabolic

AbstractIn this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

A finite-time H-infinite adaptive fault-tolerant controller considering time delay for flutter suppression of airfoil flutter

2019 ◽  
Vol 234 (2) ◽  
pp. 293-307
Author(s):  
Gao Ming-Zhou ◽  
Chen Xin-Yi ◽  
Han Rong ◽  
Yao Jian-Yong
Keyword(s):  
Finite Time ◽  
Control Method ◽  
Controller Design ◽  
Fault Tolerant ◽  
Linear Matrix ◽  
Control Methods ◽  
Actuator Faults ◽  
Control Delay ◽  
Modeling Uncertainties ◽  
The Stability

To suppress airfoil flutter, a lot of control methods have been proposed, such as classical control methods and optimal control methods. However, these methods did not consider the influence of actuator faults and control delay. This paper proposes a new finite-time H∞ adaptive fault-tolerant flutter controller by radial basis function neural network technology and adaptive fault-tolerant control method, taking into account actuator faults, control delay, modeling uncertainties, and external disturbances. The theoretic section of this paper is about airfoil flutter dynamic modeling and adaptive fault-tolerant controller design. Lyapunov function and linear matrix inequality are employed to prove the stability of the proposed control method of this paper. The numeral simulation section further proves the effectiveness and robustness of the proposed control algorithm of this paper.

scholarly journals Insights into the dynamic trajectories of protein filament division revealed by numerical investigation into the mathematical model of pure fragmentation

2021 ◽  
Vol 17 (9) ◽  
pp. e1008964
Author(s):  
Magali Tournus ◽  
Miguel Escobedo ◽  
Wei-Feng Xue ◽  
Marie Doumic
Keyword(s):  
Mathematical Theory ◽  
Length Distribution ◽  
Numerical Approach ◽  
Distribution Profile ◽  
Fragmentation Dynamics ◽  
Mathematical Equations ◽  
Fragmentation Reactions ◽  
The Stability ◽  
Fragmentation Equation ◽  
The Mathematical Model

The dynamics by which polymeric protein filaments divide in the presence of negligible growth, for example due to the depletion of free monomeric precursors, can be described by the universal mathematical equations of ‘pure fragmentation’. The rates of fragmentation reactions reflect the stability of the protein filaments towards breakage, which is of importance in biology and biomedicine for instance in governing the creation of amyloid seeds and the propagation of prions. Here, we devised from mathematical theory inversion formulae to recover the division rates and division kernel information from time dependent experimental measurements of filament size distribution. The numerical approach to systematically analyze the behaviour of pure fragmentation trajectories was also developed. We illustrate how these formulae can be used, provide some insights on their robustness, and show how they inform the design of experiments to measure fibril fragmentation dynamics. These advances are made possible by our central theoretical result on how the length distribution profile of the solution to the pure fragmentation equation aligns with a steady distribution profile for large times.

scholarly journals NECESSITY TO IMPROVE THE MATHEMATICAL MODEL OF FREIGHT CARS TO STUDY CASES OF ITS DERAILMENTS

2020 ◽  
pp. 442-451
Author(s):  
А.V. Batig ◽  
A. Ya. Kuzyshyn
Keyword(s):  
Mathematical Model ◽  
Mathematical Models ◽  
Computer Experiment ◽  
Rolling Stock ◽  
Software Systems ◽  
Freight Car ◽  
The Stability ◽  
The Many ◽  
The Impact ◽  
The Mathematical Model

One of the most important problems that pose a serious threat to the functioning of railways is the problem of freight cars derailment. However, according to statistics, the number of cases of the derailments of freight cars in trains annually grows. Тo prevent such cases, the necessary preventive measures are developed, and to study the causes of their occurrence, a significant number of mathematical models, programs and software systems created by leading domestic and foreign scientists. Studies of such mathematical models by the authors of this work have led to the conclusion that they are not sufficiently detailed to the extent that it is necessary for analyze the reasons of its derailment. At the same time, an analysis of the causes of the rolling stock derailments on the railways of Ukraine over the past five years showed that in about 20 % of cases they are obvious, and in 7 % of cases they are not obvious and implicitly expressed. The study of such cases of rolling stock derailment during an official investigation by the railway and during forensic railway transport expertises requires the use of an improved mathematical model of a freight car, which would allow a quantitative assessment of the impact of its parameters and rail track on the conditions of railway accidents. Therefore, taking into account the main reasons that caused the occurrence of such railroad accidents over the last five years on the railways of Ukraine, the article selected the main directions for improving the mathematical model of a freight car, allowing to cover all the many factors (explicit and hidden) and identify the most significant ones regarding the circumstances of the derailment rolling stock off the track, established on the basis of a computer experiment. It is proposed in the mathematical model of a freight car to take into account the guiding force, the value of which is one of the main indicators of the stability of the rolling stock. The authors of the article noted that not taking into account the influence of the guiding forces on the dynamics of the freight car can lead to an erroneous determination of the reasons for the rolling stock derailment or even to the impossibility of establishing them.

scholarly journals Выставка многопозиционной маячной системы на основе траекторных измерений

2020 ◽  
Author(s):  
A.S. Devyatisilnyi ◽  
V.M. Grinyak ◽  
A.V. Shurygin ◽  
Y.S. Ivanenko
Keyword(s):  
Measurement Errors ◽  
Dead Reckoning ◽  
Simulation Models ◽  
Point Of View ◽  
Observation System ◽  
Continuous Type ◽  
Mobile Objects ◽  
Finite Dimensional ◽  
State Measurement ◽  
The Mathematical Model

Рассматривается проблема построения маячной дальномерной системы наблюдения. В статье обсуждается постановка и подходы к решению двухкоординатной задачи выставки (местной координатной привязки) многопозиционной маячной системы, предназначенной для наблюдения подвижных объектов различного целевого назначения (подводных, надводных, наземных, воздушных и др.). Такого рода системы актуальны как для традиционных сфер решения навигационных задач, так и для задач наблюдения нового типа, например, навигация мобильных устройств в статье моделируется гидроакустическая маячная система, предназначенная для позиционирования подводных аппаратов. Сформулирована математическая модель задачи выставки, основанная на уравнениях типа состояние-измерение и конечномерных представлениях метода наименьших квадратов. В силу исходной нелинейности задачи предлагается её линеаризация около некоторого опорного решения, характеризующего априорные представления о состоянии системы наблюдения. Особое внимание в статье уделено вопросу разрешимости задачи в трёх аспектах: принципиальной разрешимости (наблюдаемости), разрешимости в условиях инструментальных погрешностей измерений, разрешимости в условиях конечной точности вычислений. Первый аспект разрешимости интерпретируется полнотой ранга соответствующей системы линейных алгебраических уравнений, второй и третий обусловленностью задачи и сходимостью итерационной процедуры оценивания. Приведены результаты численного моделирования для типичных ситуаций. Показано, что могут быть достигнуты точности выставки, достаточные для качественного решения широкого круга навигационных задач.Current paper is about problem observation system based on range measurer. The paper discusses the formulation and approaches to the solution of the two-coordinate task of the adjustment (local coordinate binding) of a multi-position system intended for monitoring mobile objects for various special purposes (underwater, surface, air, etc.). The problem of refining the configuration and spatial orientation for a multiposition observing system during a correction of the solution of a navigation problem by the dead reckoning method is considered. The mathematical model of the exhibition problem based on equations of the state-measurement type of continuous type and finite-dimensional representations of the method of least squares is formulated. Attention is paid to the problem of resolvability of the problem from the point of view fundamental resolvability (observability) and resolvability under instrumental measurement errors and finite accuracy of machine computations. The question of solvability is discussed. The results of a numerical experiment with the use of simulation models are presented. These results adequately illustrate the possibility of an efficient solution of the problem.

scholarly journals Vision-Based Target Detection and Tracking for a Miniature Pan-Tilt Inertially Stabilized Platform

Electronics ◽  
2021 ◽  
Vol 10 (18) ◽  
pp. 2243
Author(s):  
Jianchuan Guo ◽  
Chenhu Yuan ◽  
Xu Zhang ◽  
Fan Chen
Keyword(s):  
Target Detection ◽  
Feedback Linearization ◽  
Visual Servoing ◽  
Control Method ◽  
Sliding Mode ◽  
Optical Axis ◽  
Detection And Tracking ◽  
The Stability ◽  
Stabilized Platform ◽  
The Mathematical Model

This paper presents a novel visual servoing sheme for a miniature pan-tilt intertially stabilized platform (ISP). A fully customized ISP can be mounted on a miniature quadcopter to achieve stationary or moving target detection and tracking. The airborne pan-tilt ISP can effectively isolate a disturbing rotational motion of the carrier, ensuring the stabilization of the optical axis of the camera in order to obtain a clear video image. Meanwhile, the ISP guarantees that the target is always on the optical axis of the camera, so as to achieve the target detection and tracking. The vision-based tracking control design adopts a cascaded control structure based on the mathematical model, which can accurately reflect the dynamic characteristics of the ISP. The inner loop of the proposed controller employs a proportional lag compensator to improve the stability of the optical axis, and the outer loop adopts the feedback linearization-based sliding mode control method to achieve the target tracking. Numerical simulations and laboratory experiments demonstrate that the proposed controller can achieve satisfactory tracking performance.

Walking Stability Analysis of Multi-Legged Crablike Robot over Slope

2013 ◽  
Vol 572 ◽  
pp. 636-639
Author(s):  
Xi Chen ◽  
Gang Wang
Keyword(s):  
Mathematical Model ◽  
Stability Analysis ◽  
Stability Criterion ◽  
Stability Margin ◽  
Static Stability ◽  
Matlab Simulation ◽  
And Performance ◽  
The Stability ◽  
The Mathematical Model ◽  
The Relationship

This paper deals with the walking stability analysis of a multi-legged crablike robot over slope using normalized energy stability margin (NESM) method in order to develop a common stabilization description method and achieve robust locomotion for the robot over rough terrains. The robot is simplified with its static stability being described by NESM. The mathematical model of static stability margin is built so as to carry out the simulation of walking stability over slope for the crablike robot that walks in double tetrapod gait. As a consequence, the relationship between stability margin and the height of the robots centroid, as well as its inclination relative to the ground is calculated by the stability criterion. The success and performance of the stability criterion proposed is verified through MATLAB simulation and real-world experiments using multi-legged crablike robot.

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