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Initial values for Riccati ODEs from variational PDEs 由变分偏微分方程得到Riccati偏微分方程的初值
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200005
V. Costanza, P. Rivadeneira
The recently discovered variational PDEs (partial differential equations) for finding missing boundary conditions in Hamilton equations of optimal control are applied to the extended-space transformation of time-variant linear-quadratic regulator (LQR) problems. These problems become autonomous but with nonlinear dynamics and costs. The numerical solutions to the PDEs are checked against the analytical solutions to the original LQR problem. This is the first validation of the PDEs in the literature for a nonlinear context. It is also found that the initial value of the Riccati matrix can be obtained from the spatial derivative of the Hamiltonian flow, which satisfies the variational equation. This last result has practical implications when implementing two-degrees-of freedom control strategies for nonlinear systems with generalized costs.
将最近发现的用于寻找Hamilton最优控制方程中缺失边界条件的变分偏微分方程应用于时变线性二次型调节器(LQR)问题的扩展空间变换。这些问题变得自治,但具有非线性动力学和成本。将微分方程的数值解与原LQR问题的解析解进行了对比。这是文献中首次对非线性环境下的偏微分方程进行验证。通过对哈密顿流的空间导数可以得到Riccati矩阵的初值,满足变分方程。最后的结果对于实现具有广义代价的非线性系统的二自由度控制策略具有实际意义。
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引用次数: 4
A universal constant for semistable limit cycles 半稳定极限环的一个通用常数
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200012
Joan C. Artés, J. Llibre, M. Teixeira
We consider one-parameter families of 2-dimensional vector fields Xµ having in a convenient region R a semistable limit cycle of multiplicity 2m when µ = 0, no limit cycles if µ 0. We show, analytically for some particular families and numerically for others, that associated to the semistable limit cycle and for positive integers n sufficiently large there is a power law in the parameter µ of the form µn ≈ Cnα< 0 with C, α ∈ R, such that the orbit of Xµn through a point of p ∈ R reaches the position of the semistable limit cycle of X0 after given n turns. The exponent α of this power law depends only on the multiplicity of the semistable limit cycle, and is independent of the initial point p ∈ R and of the family Xµ. In fact α = -2m/(2m - 1). Moreover the constant C is independent of the initial point p ∈ R, but it depends on the family Xµ and on the multiplicity 2m of the limit cycle Γ.
考虑二维向量场X μ的单参数族,当μ = 0时,在方便区域R中有一个多重2m的半稳定极限环,当μ = 0时没有极限环。我们用解析法和数值法证明了对于半稳定极限环和对于足够大的正整数n,在参数μ中存在一个幂律,其形式为μ n≈Cnα< 0,且C, α∈R,使得X μ n的轨道经过p∈R的一点,在给定n圈后到达X0的半稳定极限环的位置。幂律的指数α只依赖于半稳定极限环的多重性,与初始点p∈R和族Xµ无关。事实上,α = -2m/(2m - 1)。此外,常数C与初始点p∈R无关,但它取决于族Xµ和极限环的多重性2m Γ。
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引用次数: 0
Comparing stochastic optimization methods to solve the medium-term operation planning problem 比较随机优化方法解决中期运营计划问题
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200003
R. Gonçalves, E. Finardi, E. L. D. Silva, M. L. Santos
The Medium-Term Operation Planning (MTOP) of hydrothermal systems aims to define the generation for each power plant, minimizing the expected operating cost over the planning horizon. Mathematically, this task can be characterized as a linear, stochastic, large-scale problem which requires the application of suitable optimization tools. To solve this problem, this paper proposes to use the Nested Decomposition, frequently used to solve similar problems (as in Brazilian case), and Progressive Hedging, an alternative method, which has interesting features that make it promising to address this problem. To make a comparative analysis between these two methods with respect to the quality of the solution and the computational burden, a benchmark is established, which is obtained by solving a single Linear Programming problem (the Deterministic Equivalent Problem). An application considering a hydrothermal system is carried out.
热液系统的中期运行规划(MTOP)旨在确定每个电厂的发电量,使规划期内的预期运行成本最小化。在数学上,这项任务可以被描述为一个线性的、随机的、大规模的问题,需要应用合适的优化工具。为了解决这个问题,本文建议使用嵌套分解,经常用于解决类似的问题(如巴西的情况),以及渐进式套期保值,一种替代方法,它具有有趣的特征,使其有望解决这个问题。为了对这两种方法在求解质量和计算量方面进行比较分析,建立了一个基准,并通过求解单个线性规划问题(确定性等价问题)得到该基准。以热液系统为例进行了应用。
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引用次数: 11
A study of axi-symmetric waves through an isotropic thermoelastic diffusive medium 通过各向同性热弹性扩散介质的轴对称波的研究
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200001
Rajesh Kumar, M. Panchal
The paper deals with the propagation of axial symmetric cylindrical surface waves in a cylindrical bore through a homogeneous isotropic thermoelastic diffusive medium of infinite extent. The three theories of thermoelasticity namely, Coupled Thermoelasticity (CT), Lord and Shulman (L-S) and Green and Lindsay (G-L) are used to study the problem. The frequency equations, connecting the phase velocity with wave number, radius of bore and other material parameters, for empty and liquid filled bore are derived. The numerical results obtained have been illustrated graphically to understand the behaviour of phase velocity and attenuation coefficient versus wave number of a wave. A particular case of interest has also been deduced from the present investigation.
研究了轴对称柱面波在圆柱孔中通过无限宽均匀各向同性热弹性扩散介质的传播。本文采用了三种热弹性理论,即耦合热弹性理论(CT)、Lord and Shulman理论(L-S)和Green and Lindsay理论(G-L)来研究该问题。导出了空孔和充液孔相速度与波数、孔半径及其它材料参数的频率方程。为了理解相速度和衰减系数随波数的变化规律,本文用图形说明了所得到的数值结果。从目前的调查中还推断出一个特别的案件。
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引用次数: 3
A semi-analytical computation of the Kelvin kernel for potential flows with a free surface 自由表面势流开尔文核的半解析计算
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200002
J. D'Elía, L. Battaglia, Mario Alberto Storti
A semi-analytical computation of the three dimensional Green function for seakeeping flow problems is proposed. A potential flow model is assumed with an harmonic dependence on time and a linearized free surface boundary condition. The multiplicative Green function is expressed as the product of a time part and a spatial one. The spatial part is known as the Kelvin kernel, which is the sum of two Rankine sources and a wave-like kernel, being the last one written using the Haskind-Havelock representation. Numerical efficiency is improved by an analytical integration of the two Rankine kernels and the use of a singularity subtractive technique for the Haskind-Havelock integral, where a globally adaptive quadrature is performed for the regular part and an analytic integration is used for the singular one. The proposed computation is employed in a low order panel method with flat triangular elements. As a numerical example, an oscillating floating unit hemisphere in heave and surge modes is considered, where analytical and semi-analytical solutions are taken as a reference.
提出了一种三维格林函数的半解析计算方法。假设了一个随时间谐波依赖的势流模型和一个线性化的自由表面边界条件。乘法格林函数表示为时间部分和空间部分的乘积。空间部分被称为开尔文核,它是两个朗肯源和一个波状核的和,是最后一个使用Haskind-Havelock表示的。通过对两个Rankine核的解析积分和对Haskind-Havelock积分使用奇异减法技术来提高数值效率,其中正则部分使用全局自适应求积分,奇异部分使用解析积分。所提出的计算方法采用了一种具有平面三角形单元的低阶面板法。作为数值算例,考虑了浮单元半球在升沉和激波模式下的振动,并以解析解和半解析解为参考。
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引用次数: 17
A family of uniformly accurate order Lobatto-Runge-Kutta collocation methods 一类均匀精确阶Lobatto-Runge-Kutta配置方法
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200004
D. G. Yakubu, N. Manjak, S. Buba, A. I. Maksha
We consider the construction of an interpolant for use with Lobatto-Runge-Kutta collocation methods. The main aim is to derive single symmetric continuous solution(interpolant) for uniform accuracy at the step points as well as at the off-step points whose uniform order six everywhere in the interval of consideration. We evaluate the continuous scheme at different off-step points to obtain multi-hybrid schemes which if desired can be solved simultaneously for dense approximations. The multi-hybrid schemes obtained were converted to Lobatto-Runge-Kutta collocation methods for accurate solution of initial value problems. The unique feature of the paper is the idea of using all the set of off-step collocation points as additional interpolation points while symmetry is retained naturally by integration identities as equal areas under the various segments of the solution graph over the interval of consideration. We show two possible ways of implementing the interpolant to achieve the aim and compare them on some numerical examples.
我们考虑了一个用于Lobatto-Runge-Kutta搭配方法的插值构造。主要目的是推导出单对称连续解(插值)在步进点和离步点的均匀精度,在考虑的区间内处处均匀阶数为6。我们在不同的离步点处评估连续格式,得到多混合格式,如果需要,可以同时求解密集近似。将得到的多重混合格式转换为Lobatto-Runge-Kutta配置方法,以精确求解初值问题。本文的独特之处在于将所有的离步配点集用作额外的插值点,同时通过积分恒等式在考虑的区间内的解图的各个部分下作为相等的面积自然地保持对称性。给出了实现插值的两种可能方法,并通过数值算例进行了比较。
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引用次数: 2
Solving of time varying quadratic optimal control problems by using Bézier control points 利用bsamizier控制点求解时变二次型最优控制问题
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-07-18 DOI: 10.1590/S1807-03022011000200007
M. Gachpazan
In this paper, linear quadratic optial control probles are solved by applying least square method based on Bezier control points. We divide the interval which includes t, into k subintervals and approximate the trajectory and control functions by Bezier curves. We have chosen the Bezier curves as piacewise polynomials of degree three, and determined Bezier curves on any subinterval by four control points. By using least square ethod, e introduce an optimization problem and compute the control points by solving this optimization problem. Numerical experiments are presented to illustrate the proposed method.
本文采用基于Bezier控制点的最小二乘法求解线性二次型光控制问题。我们将包含t的区间分成k个子区间,用Bezier曲线近似轨迹和控制函数。我们选择Bezier曲线作为三次的逐次多项式,并通过四个控制点确定任意子区间上的Bezier曲线。利用最小二乘法引入一个优化问题,并通过求解该优化问题计算控制点。数值实验说明了所提出的方法。
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引用次数: 20
An inexact subgradient algorithm for Equilibrium Problems 平衡问题的非精确次梯度算法
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-01-17 DOI: 10.1590/S1807-03022011000100005
P. J. S. Santos, S. Scheimberg
We present an inexact subgradient projection type method for solving a nonsmooth Equilibrium Problem in a finite-dimensional space. The proposed algorithm has a low computational cost per iteration. Some numerical results are reported.
提出了一种求解有限维空间非光滑平衡问题的非精确亚梯度投影法。该算法具有较低的每次迭代计算量。报道了一些数值结果。
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引用次数: 150
Derivative-free methods for nonlinear programming with general lower-level constraints 具有一般低级约束的非线性规划的无导数方法
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-01-17 DOI: 10.1590/S1807-03022011000100003
M. A. Diniz-Ehrhardt, J. Martínez, L. G. Pedroso
Augmented Lagrangian methods for derivative-free continuous optimization with constraints are introduced in this paper. The algorithms inherit the convergence results obtained by Andreani, Birgin, Martinez and Schuverdt for the case in which analytic derivatives exist and are available. In particular, feasible limit points satisfy KKT conditions under the Constant Positive Linear Dependence (CPLD) constraint qualification. The form of our main algorithm allows us to employ well established derivative-free subalgorithms for solving lower-level constrained subproblems. Numerical experiments are presented.
本文介绍了带约束的无导数连续优化问题的增广拉格朗日方法。该算法继承了Andreani, Birgin, Martinez和Schuverdt对存在和可用解析导数情况下的收敛性结果。特别是可行极限点在恒定正线性相关(CPLD)约束条件下满足KKT条件。我们的主要算法的形式允许我们使用完善的无导数子算法来解决较低层次的约束子问题。给出了数值实验结果。
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引用次数: 43
Two derivative-free methods for solving underdetermined nonlinear systems of equations 求解欠定非线性方程组的两种无导数方法
IF 2.6 3区 数学 Q1 MATHEMATICS, APPLIED Pub Date : 2011-01-01 DOI: 10.1590/S1807-03022011000100011
N. Echebest, M. L. Schuverdt, R. Vignau
In this paper, two different approaches to solve underdetermined nonlinear system of equations are proposed. In one of them, the derivative-free method defined by La Cruz, Martinez and Raydan for solving square nonlinear systems is modified and extended to cope with the underdetermined case. The other approach is a Quasi-Newton method that uses the Broyden update formula and the globalized line search that combines the strategy of Grippo, Lampariello and Lucidi with the Li and Fukushima one. Global convergence results for both methods are proved and numerical experiments are presented.
本文提出了求解欠定非线性方程组的两种不同方法。在其中一种方法中,对La Cruz, Martinez和Raydan定义的求解平方非线性系统的无导数方法进行了修正和扩展,以处理欠定情况。另一种方法是准牛顿方法,它使用Broyden更新公式和全球化线搜索,将Grippo, Lampariello和Lucidi的策略与Li和Fukushima的策略结合起来。证明了两种方法的全局收敛性,并给出了数值实验。
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引用次数: 5
期刊
Computational & Applied Mathematics
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