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two-point boundary value problems for First Order Causal Difference Equations

INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS 2020年第4期51卷 1399-1416页

作者： Wang, Wen-Li Tian, Jing-Feng Cheung, Wing-Sum China Univ Geosci Fac Fundamental Courses Great Wall Coll Baoding 071000 Hebei Peoples R China North China Elect Power Univ Dept Math & Phys Baoding 071003 Hebei Peoples R China Univ Hong Kong Dept Math Pokfulam Rd Hong Kong Peoples R China

This paper focuses on two-point boundary value problem for first order causal difference equations. We will start with two new comparison theorems. Then, by utilizing these theorems and fixed point theorems, we obtain the existence of solutions for the corresponding linear problem. By applying monotone iterative technique, sufficient conditions for the existence of extremal solutions are also established. The results of this paper extend some existing results in the literature. Finally, two examples to show the usefulness of our results are exhibited.

关键词： two-point boundary value problem causal difference equations monotone iterative technique extremal solutions

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two-point boundary value problems for finite fractional difference equations

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JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS 2011年第4期17卷 445-456页

作者： Atici, Ferhan M. Eloe, Paul W. Univ Dayton Dept Math Dayton OH 45469 USA Western Kentucky Univ Dept Math & Comp Sci Bowling Green KY 42101 USA

In this paper, we introduce a two-point boundary value problem for a finite fractional difference equation. We invert the problem and construct and analyse the corresponding Green's function. We then provide an application and obtain sufficient conditions for the existence of positive solutions for a two-point boundary value problem for a nonlinear finite fractional difference equation.

关键词： discrete fractional calculus Green's function existence of solutions two-point boundary value problem

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Linear Quadratic Mean Field Games with Quantile-Dependent Cost Coefficients

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Journal of Systems Science & Complexity 2025年第1期38卷 495-510页

作者： GAO Shuang MALHAME Roland P. Department of Electrical Engineering Polytechnique MontrealMontreal H3C 3A7Canada Group for Research in Decision Analysis(GERAD) MontrealQuebec H3T 1J4Canada

This paper studies a class of linear quadratic mean field games where the coefficients of quadratic cost functions depend on both the mean and the variance of the population’s state distribution through its quantile *** a formulation allows for modelling agents that are sensitive to not only the population average but also the population *** potential mean field game equilibria are *** calculation involves solving two nonlinearly coupled differential equations:one is a Riccati equation and the other the variance evolution *** conditions for the existence and uniqueness of a mean field equilibrium are ***,numerical results are presented to illustrate the behavior of two coupled differential equations and the performance of the mean field game solution.

关键词： Mean field games quantile Riccati equation two-point boundary value problem

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Fast point-To-point Transportation of the Portal Crane via Double-Input Inversion-Based Feedforward and Feedback Control

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OPTIMAL CONTROL APPLICATIONS & METHODS 2025年第0期

作者： Zhang, Yongli Li, Mengjie Tianjin Univ Technol & Educ Sch Automat & Elect Engn Tianjin Peoples R China Tianjin Key Lab Informat Sensing & Intelligent Con Tianjin Peoples R China

This paper describes a double-input double-output inversion-based feedforward control (DIDOIBFC) scheme for fast point-to-point (PTP) transportation of the portal crane. Luffing and rotation of the portal crane are two mutually coupled motions in the process of PTP transportation, which are actuated by two control input, respectively. In this paper, the Bessel curve is used to obtain the PTP space transportation trajectory, and the relationship between the luffing length and the rotation angle is fitted into a polynomial function. The control strategy based on inversion system is a two-point boundary value problem (TPBVP) about the input-output coordinates of the portal crane. Then, the time-varying control nominal trajectory (CNT) of the system is obtained by solving a TPBVP for the internal dynamics of the portal crane, and the feedforward controller is designed. In order to realize the anti-swinging transportation, the gain scheduling feedback controller (GSFC) is designed to stabilize the motion along the feedforward trajectories in real time. When the crane is staying at the start/end point of the transportation or at any other position, the linear quadratic regulator (LQR) controller optimized by the particle swarm optimization (PSO) algorithm is used to keep the system stabilizing. Moreover, in order to improve the robustness of the system, the feedback controller is improved by an H infinity feedback control, and the numerical experiments are carried out. Finally, the numerical experiments are carried out. The experimental results verify the effectiveness of the control scheme proposed in this paper, which can realize fast PTP transportation with high efficiency and good stability.

关键词： Bessel curve H infinity feedback control inversion-based feedforward control portal crane two-point boundary value problem

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Fast Gradient Evaluation and Mapping Method for Bi-Impulsive Transfers Within Periodic Orbits

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JOURNAL OF GUIDANCE CONTROL AND DYNAMICS 2025年第1期48卷 168-176页

作者： Caleb, Thomas Armellin, Roberto Lizy-Destrez, Stephanie ISAE SUPAERO Dept Design & Control Space & Aeronaut Vehicles F-31055 Toulouse France Univ Auckland Te Punaha Atea Space Inst Auckland 1142 New Zealand ISAE SUPAERO DCAS SaCLaB F-31055 Toulouse France

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Exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2001年第2期46卷 181-197页

作者： Liu, ZL Shandong Univ Dept Math Jinan 250100 Peoples R China

The exact number of solutions of a class of two-point boundary value problems involving concave and convex nonlinearities was discussed. Theorems that proves that proposed solutions can solve boundary value problems i... 详细信息

关键词： a problem raised by Ambrosetti, Brezis and Cerami two-point boundary value problem concave and convex nonlinearity exact structure of all solutions properties of solutions

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Enclosing all solutions of two-point boundary value problems for ODES

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COMPUTERS & CHEMICAL ENGINEERING 2008年第8期32卷 1714-1725页

作者： Lin, Youdong Enszer, Joshua A. Stadtherr, Mark A. Univ Notre Dame Dept Chem & Biomol Engn Notre Dame IN 46556 USA

The two-point boundary value problem (TPBVP) occurs in a wide variety of problems in engineering and science, including the modeling of chemical reactions, heat transfer, and diffusion, and the solution of optimal control problems. A TPBVP may have no solution, a single solution, or multiple solutions. A new strategy is presented for reliably locating all solutions of a TPBVP. The method determines narrow enclosures of all solutions that occur within a specified search interval. Key features of the method are the use of a new solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of nonlinear dynamic systems with interval-valued parameters and initial states, and the use of a constraint propagation strategy on the Taylor models used to represent the solutions of the dynamic system. Numerical experiments demonstrate the use and computational efficiency of the method. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： two-point boundary value problem ordinary differential equations initial value problem interval analysis Taylor model

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Sinc collocation method with boundary treatment for two-point boundary value problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2006年第1期196卷 229-240页

作者： Wu, Xionghua Kong, Wenbin Li, Chen Tongji Univ Dept Appl Math Shanghai 200092 Peoples R China

Sinc collocation method is proven to provide an exponential convergence rate in solving linear differential equations, even in the presence of singularities. But in order to treat the derivatives on boundaries, people often relied on the finite difference method, which would be expected to limit the accuracy. The present paper develops a Sinc collocation method with boundary treatment for two-point boundary value problems. Numerical results show that the method can directly and efficiently handle the boundary derivatives. (c) 2005 Elsevier B.V. All rights reserved.

关键词： Sinc collocation method boundary derivative two-point boundary value problem

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Positive solutions of two-point boundary value problems for 2n-order differential equations dependent on all derivatives

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APPLIED MATHEMATICS LETTERS 2005年第2期18卷 209-218页

作者： Liu, YJ Ge, WG Beijing Inst Technol Dept Math Appl Beijing 100081 Peoples R China Hunan Inst Technol Dept Math Yueyang 414000 Hunan Peoples R China

In this work, we consider the higher-order differential equation (-1)(n)x((2n))(t) = f(t, x(t), x(1)(t), ...,x((2n-1))(t)), 0 = 0 for all t is an element of [0, 1] and (x(0), x(1), ..., x(2n-1)) is an element of R-2n.... 详细信息

In this work, we consider the higher-order differential equation (-1)(n)x((2n))(t) = f(t, x(t), x(1)(t), ...,x((2n-1))(t)), 0 < t < 1, subject to one of the following multi-point boundary value conditions: x((2i))(0) = x((2i))(1) = 0 for i = 0, 1, ..., n-1, or x((2i+1)) (0) = x((2i))(1) = 0 for i = 0,1, ..., n-1, where f (t, x(0), x(1), ..., x(2n-1)) is continuous with f (t, x(0), x(1), ..., x(2n-1)) >= 0 for all t is an element of [0, 1] and (x(0), x(1), ..., x(2n-1)) is an element of R-2n. Sufficient conditions for the existence of at least one positive solution of the BVP (1) and (2) and BVP (1) and (3) are established, respectively. The emphasis in this work is on f depending on all higher-order derivatives. Examples are given to illustrate the main results. (c) 2004 Elsevier Ltd. All rights reserved.

关键词： solvability two-point boundary value problem higher-order differential equation positive solution

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Cubic spline for a class of singular two-point boundary value problems

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期170卷 733-740页

作者： Kanth, ASVR Reddy, YN Eritrean Inst Technol Dept Math Asmera Eritrea Natl Inst Technol Dept Math Warangal 506004 Andhra Pradesh India

In this paper we have presented a method based on cubic splines for solving a class of singular two-point boundary value problems. The original differential equation is modified at the singular point then the boundary value problem is treated by using Cubic spline approximation. The tridiagonal system resulting from the spline approximation is efficiently solved by Thomas algorithm. Some model problems are solved, and the numerical results are compared with exact solution. (c) 2005 Published by Elsevier Inc.

关键词： ordinary differential equations singular point two-point boundary value problem cubic spline

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