Pub Date : 2016-03-02DOI: 10.4236/IJMNTA.2016.51008
Ruijin Lou, Penghui Lv, Guoguang Lin
In this paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori estimation, we first prove the existence and uniqueness of solutions to the initial boundary value conditions, and then we study the global attractors of the equation.
本文考虑了一类广义非线性kirchhoff - sin - gordon方程。通过先验估计，首先证明了初始边值条件解的存在唯一性，然后研究了方程的全局吸引子。
{"title":"Global Attractors for a Class of Generalized Nonlinear Kirchhoff-Sine-Gordon Equation","authors":"Ruijin Lou, Penghui Lv, Guoguang Lin","doi":"10.4236/IJMNTA.2016.51008","DOIUrl":"https://doi.org/10.4236/IJMNTA.2016.51008","url":null,"abstract":"In this \u0000paper, we consider a class of generalized nonlinear Kirchhoff-Sine-Gordon equation . By a priori \u0000estimation, we first prove the existence and uniqueness of solutions to the \u0000initial boundary value conditions, and then we study the global attractors of \u0000the equation.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"5 1","pages":"73-81"},"PeriodicalIF":0.0,"publicationDate":"2016-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-11-12DOI: 10.4236/IJMNTA.2015.44018
Yoshiki Abe, G. Nishida, N. Sakamoto, Y. Yamamoto
This paper proposes a system representation for unifying control design and numerical calculation in nonlinear optimal control problems with inequality constraints in terms of the symplectic structure. The symplectic structure is derived from Hamiltonian systems that are equivalent to Hamilton-Jacobi equations. In the representation, the constraints can be described as an input-state transformation of the system. Therefore, it can be seamlessly applied to the stable manifold method that is a precise numerical solver of the Hamilton-Jacobi equations. In conventional methods, e.g., the penalty method or the barrier method, it is difficult to systematically assign the weights of penalty functions that are used for realizing the constraints. In the proposed method, we can separate the adjustment of weights with respect to objective functions from that of penalty functions. Furthermore, the proposed method can extend the region of computable solutions in a state space. The validity of the method is shown by a numerical example of the optimal control of a vehicle model with steering limitations.
{"title":"Symplectic Numerical Approach for Nonlinear Optimal Control of Systems with Inequality Constraints","authors":"Yoshiki Abe, G. Nishida, N. Sakamoto, Y. Yamamoto","doi":"10.4236/IJMNTA.2015.44018","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.44018","url":null,"abstract":"This paper proposes a system representation for unifying control design and numerical calculation in nonlinear optimal control problems with inequality constraints in terms of the symplectic structure. The symplectic structure is derived from Hamiltonian systems that are equivalent to Hamilton-Jacobi equations. In the representation, the constraints can be described as an input-state transformation of the system. Therefore, it can be seamlessly applied to the stable manifold method that is a precise numerical solver of the Hamilton-Jacobi equations. In conventional methods, e.g., the penalty method or the barrier method, it is difficult to systematically assign the weights of penalty functions that are used for realizing the constraints. In the proposed method, we can separate the adjustment of weights with respect to objective functions from that of penalty functions. Furthermore, the proposed method can extend the region of computable solutions in a state space. The validity of the method is shown by a numerical example of the optimal control of a vehicle model with steering limitations.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"234-248"},"PeriodicalIF":0.0,"publicationDate":"2015-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-11-12DOI: 10.4236/IJMNTA.2015.44019
Kun Yao
In a physical system, phase point trajectory is impossible to be space-filling curve, of which the dimension is not greater than one. Equipotential map concept is proposed. When phase point trajectory dimension is 0, calculus tool is no longer applicable. System state can be changed instantly. When phase point trajectory dimension is 1, differential equation can be used to handle this case.
{"title":"Dimension of Phase Point Trajectory","authors":"Kun Yao","doi":"10.4236/IJMNTA.2015.44019","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.44019","url":null,"abstract":"In a physical system, phase point trajectory is impossible to be space-filling curve, of which the dimension is not greater than one. Equipotential map concept is proposed. When phase point trajectory dimension is 0, calculus tool is no longer applicable. System state can be changed instantly. When phase point trajectory dimension is 1, differential equation can be used to handle this case.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"22 1","pages":"249-253"},"PeriodicalIF":0.0,"publicationDate":"2015-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706264","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-11-12DOI: 10.4236/IJMNTA.2015.44016
Gendai Gu, Hongxiao Peng
Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.
{"title":"Numerical Simulation of Reaction-Diffusion Systems of Turing Pattern Formation","authors":"Gendai Gu, Hongxiao Peng","doi":"10.4236/IJMNTA.2015.44016","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.44016","url":null,"abstract":"Differential method and homotopy analysis method are used for solving the two-dimensional reaction-diffusion model. And the structure of the solutions is analyzed. Finally, the homotopy series solutions are simulated with the mathematical software Matlab, so the Turing patterns will be produced. Overall analysis and experimental simulation of the model show that the different parameters lead to different Turing pattern structures. As time goes on, the structure of Turing patterns changes, and the final solutions tend to stationary state.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"215-225"},"PeriodicalIF":0.0,"publicationDate":"2015-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-11-12DOI: 10.4236/IJMNTA.2015.44017
W. Aziz, N. Merentes
We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.
{"title":"Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz","authors":"W. Aziz, N. Merentes","doi":"10.4236/IJMNTA.2015.44017","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.44017","url":null,"abstract":"We show that the lateral regularizations of the generator of any uniformly bounded set-valued composition Nemytskij operator acting in the spaces of functions of bounded variation in the sense of Riesz, with nonempty bounded closed and convex values, are an affine function.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"226-233"},"PeriodicalIF":0.0,"publicationDate":"2015-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-08-07DOI: 10.4236/IJMNTA.2015.43015
Zhaojuan Wang, Shengfan Zhou
In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with multiplicative noise under homogeneous Dirichlet boundary condition. We investigate the existence of a compact random attractor for the random dynamical system associated with the equation.
{"title":"Asymptotic Behavior of Stochastic Strongly Damped Wave Equation with Multiplicative Noise","authors":"Zhaojuan Wang, Shengfan Zhou","doi":"10.4236/IJMNTA.2015.43015","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.43015","url":null,"abstract":"In this paper we study the asymptotic dynamics of the stochastic strongly damped wave equation with multiplicative noise under homogeneous Dirichlet boundary condition. We investigate the existence of a compact random attractor for the random dynamical system associated with the equation.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"204-214"},"PeriodicalIF":0.0,"publicationDate":"2015-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706407","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-08-07DOI: 10.4236/IJMNTA.2015.43013
T. Kai
This paper considers an obstacle avoidance control problem for the compass-type biped robot, especially circular obstacles are dealt with. First, a sufficient condition such that the swing leg does not collide the circular obstacle is derived. Next, an optimal control problem for the discrete compass-type robot is formulated and a solving method of the problem by the sequential quadratic programming is presented in order to calculate a discrete control input. Then, a transformation method that converts a discrete control input into a continuous zero-order hold input via discrete Lagrange-d’ Alembert principle is explained. From the results of numerical simulations, it turns out that obstacle avoidance control for the continuous compass-type robot can be achieved by the proposed method.
{"title":"Circular Obstacle Avoidance Control of the Compass-Type Biped Robot Based on a Blending Method of Discrete Mechanics and Nonlinear Optimization","authors":"T. Kai","doi":"10.4236/IJMNTA.2015.43013","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.43013","url":null,"abstract":"This paper considers an obstacle avoidance control problem for the compass-type biped robot, especially circular obstacles are dealt with. First, a sufficient condition such that the swing leg does not collide the circular obstacle is derived. Next, an optimal control problem for the discrete compass-type robot is formulated and a solving method of the problem by the sequential quadratic programming is presented in order to calculate a discrete control input. Then, a transformation method that converts a discrete control input into a continuous zero-order hold input via discrete Lagrange-d’ Alembert principle is explained. From the results of numerical simulations, it turns out that obstacle avoidance control for the continuous compass-type robot can be achieved by the proposed method.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"179-189"},"PeriodicalIF":0.0,"publicationDate":"2015-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-08-07DOI: 10.4236/IJMNTA.2015.43014
Zhaoqin Yuan, Liang Guo, Guoguang Lin
In this paper, we prove the existence of inertial manifolds for 2D generalized MHD system under the spectral gap condition.
本文证明了二维广义MHD系统在谱隙条件下惯性流形的存在性。
{"title":"Inertial Manifolds for 2D Generalized MHD System","authors":"Zhaoqin Yuan, Liang Guo, Guoguang Lin","doi":"10.4236/IJMNTA.2015.43014","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.43014","url":null,"abstract":"In this paper, we prove the existence of inertial manifolds for 2D generalized MHD system under the spectral gap condition.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"4 1","pages":"190-203"},"PeriodicalIF":0.0,"publicationDate":"2015-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70706248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-05-15DOI: 10.4236/IJMNTA.2015.42007
Baishun Liu
This paper proposes an output feedback nonlinear general integral controller for a class of uncertain nonlinear system. By solving Lyapunov equation, we demonstrate a new proposition on Equal ratio gain technique. By using Equal ratio gain technique, Singular perturbation technique and Lyapunov method, theorem to ensure regionally as well as semi-globally exponential stability is established in terms of some bounded information. Moreover, a real time method to evaluate the ratio coefficients of controller and observer are proposed such that their values can be chosen moderately. Theoretical analysis and simulation results show that not only output feedback nonlinear general integral control has the striking robustness but also the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions.
{"title":"Output Feedback Nonlinear General Integral Control","authors":"Baishun Liu","doi":"10.4236/IJMNTA.2015.42007","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.42007","url":null,"abstract":"This paper proposes an output feedback nonlinear general integral controller for a class of uncertain nonlinear system. By solving Lyapunov equation, we demonstrate a new proposition on Equal ratio gain technique. By using Equal ratio gain technique, Singular perturbation technique and Lyapunov method, theorem to ensure regionally as well as semi-globally exponential stability is established in terms of some bounded information. Moreover, a real time method to evaluate the ratio coefficients of controller and observer are proposed such that their values can be chosen moderately. Theoretical analysis and simulation results show that not only output feedback nonlinear general integral control has the striking robustness but also the organic combination of Equal ratio gain technique and Singular perturbation technique constitutes a powerful tool to solve the output feedback control design problem of dynamics with the nonlinear and uncertain actions.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"101-116"},"PeriodicalIF":0.0,"publicationDate":"2015-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70705764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2015-05-15DOI: 10.4236/IJMNTA.2015.42009
Jiangyong Hou, Wenjing Yan, Jie Chen
The upwind scheme is very important in the numerical approximation of some problems such as the convection dominated problem, the two-phase flow problem, and so on. For the fractional flow formulation of the two-phase flow problem, the Penalty Discontinuous Galerkin (PDG) methods combined with the upwind scheme are usually used to solve the phase pressure equation. In this case, unless the upwind scheme is taken into consideration in the velocity reconstruction, the local mass balance cannot hold exactly. In this paper, we present a scheme of velocity reconstruction in some H(div) spaces with considering the upwind scheme totally. Furthermore, the different ways to calculate the nonlinear coefficients may have distinct and significant effects, which have been investigated by some authors. We propose a new algorithm to obtain a more effective and stable approximation of the coefficients under the consideration of the upwind scheme.
{"title":"Velocity Projection with Upwind Scheme Based on the Discontinuous Galerkin Methods for the Two Phase Flow Problem","authors":"Jiangyong Hou, Wenjing Yan, Jie Chen","doi":"10.4236/IJMNTA.2015.42009","DOIUrl":"https://doi.org/10.4236/IJMNTA.2015.42009","url":null,"abstract":"The upwind scheme is very important in the numerical approximation of some problems such as the convection dominated problem, the two-phase flow problem, and so on. For the fractional flow formulation of the two-phase flow problem, the Penalty Discontinuous Galerkin (PDG) methods combined with the upwind scheme are usually used to solve the phase pressure equation. In this case, unless the upwind scheme is taken into consideration in the velocity reconstruction, the local mass balance cannot hold exactly. In this paper, we present a scheme of velocity reconstruction in some H(div) spaces with considering the upwind scheme totally. Furthermore, the different ways to calculate the nonlinear coefficients may have distinct and significant effects, which have been investigated by some authors. We propose a new algorithm to obtain a more effective and stable approximation of the coefficients under the consideration of the upwind scheme.","PeriodicalId":69680,"journal":{"name":"现代非线性理论与应用(英文)","volume":"04 1","pages":"127-141"},"PeriodicalIF":0.0,"publicationDate":"2015-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70705840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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