量测带噪声的多个体二阶系统聚集控制

考虑在含有量测噪声情况下的二阶多个体系统聚集控制问题.目的是使得系统中每个个体根据邻居信息构造控制,在只有部分个体能够观测到目标的情况下到达目标.和以前许多多个体同步及聚集问题的研究模型中所考虑的一阶系统不同, 系统的每个个体都只能量测到其邻居个体的部分状态信息,如位置, 并且这些量测还带有噪声.根据这些信息设计了基于局部规则的分散控制律,并且证明当量测噪声和状态量测本身相关时,只要系统在任意给定的时间区域段之内能够保持联合连通,就能够实现系统对目标的跟踪和达到目标.     

Symplectic Euler Method for Nonlinear High Order Schrödinger Equation with a Trapped Term

In this paper, we establish a family of symplectic integrators for a class of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization. Then we apply the symplectic Euler method to the Hamiltonian system. It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system, but also does not require to resolve coupled nonlinear algebraic equations which is different from the general implicit symplectic schemes. The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated. It shows that the semi-explicit scheme is conditionally stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones, such as backward Euler integrators.     

A symplectic, symmetric algorithm for spatial evolution of particles in a time-dependent field

A symplectic, symmetric, second-order scheme is constructed for particle evolution in a time-dependent field with a fixed spatial step. The scheme is implemented in one space dimension and tested, showing excellent adequacy to experiment analysis. Higher order schemes are deduced by composition.     

VARIATIONAL INTEGRATORS FOR HIGHER ORDER DIFFERENTIAL EQUATIONS

We analyze three one parameter families of approximations and show that they are sympectic in Largrangian sence and can be related to symplectic schemes in Hamiltonian sense by different symplectic mapping.We also give a direct generalization of Veselov variational principlc for construction of scheme of higher order differential equations.At last,we present numerical experiments.     

On error estimates of an exponential wave integrator sine pseudospectral method for the Klein–Gordon–Zakharov system

In this article, we propose an exponential wave integrator sine pseudospectral (EWI‐SP) method for solving the Klein–Gordon–Zakharov (KGZ) system. The numerical method is based on a Deuflhard‐type exponential wave integrator for temporal integrations and the sine pseudospectral method for spatial discretizations. The scheme is fully explicit, time reversible and very efficient due to the fast algorithm. Rigorous finite time error estimates are established for the EWI‐SP method in energy space with no CFL‐type conditions which show that the method has second order accuracy in time and spectral accuracy in space. Extensive numerical experiments and comparisons are done to confirm the theoretical studies. Numerical results suggest the EWI‐SP allows large time steps and mesh size in practical computing. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 266–291, 2016     

Multi-symplectic variational integrators for the Gross–Pitaevskii equations in BEC

In this paper, a variational integrator is constructed for Gross–Pitaevskii equations in Bose–Einstein condensate. The discrete multi-symplectic geometric structure is derived. The discrete mass and energy conservation laws are proved. The numerical tests show the effectiveness of the variational integrator, and the performance of the proved discrete conservation law.     

Stability analysis of a predictor/multi-corrector method for staggered-grid Lagrangian shock hydrodynamics

This article presents the complete von Neumann stability analysis of a predictor/multi-corrector scheme derived from an implicit mid-point time integrator often used in shock hydrodynamics computations in combination with staggered spatial discretizations. It is shown that only even iterates of the method yield stable computations, while the odd iterates are, in the most general case, unconditionally unstable. These findings are confirmed by, and illustrated with, a number of numerical computations. Dispersion error analysis is also presented.     

更一般高阶非完整系统的指数调节

研究了一类更具有一般性的强漂移高阶非完整系统的指数调节问题.首先通过指数形式的input-state-scaling不连续变换将原x子系统的指数调节问题转化为变换后z子系统的镇定问题,并有效克服x子系统在控制输入u₀=0时不能控的问题.然后基于"加幂积分器"反推方法设计鲁棒控制器,在新的切换策略下,使闭环系统状态指数收敛到零点.仿真例子验证了所提方法的有效性.