状态受限最优控制问题的谱方法
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摘要
偏微分方程最优控制问题的理论分析和数值方法一直是一个非常活跃的研究领域.虽然关于采用有限元方法分析控制变量受限的最优控制问题已经有了大量很好的成果,但是,目前关于控制变量受限最优控制问题采用谱方法进行分析的研究工作还很少,文献[32]就控制变量积分受限型最优控制问题的谱方法分析进行了讨论.最近,一些学者开始考虑状态变量受限的最优控制问题.这类最优控制问题在实际问题中会经常遇到,但是非常难处理.目前,大多文献都是采用有限元方法分析状态变量受限的最优控制问题.就作者所知,目前关于状态变量受限最优控制问题谱方法分析的研究工作还很少.
本文研究了状态变量积分受限最优控制问题的谱方法分析.为了能在形式上便于说明我们的方法和技巧,我们仅选择了以Poisson方程和双调和方程为状态方程的两大类,当然,本文的方法和结论可以推广到一般的模型问题.文章讨论了先验误差估计和后验误差估计,给出了有效的梯度投影算法,并证明了其收敛性.
文中通过利用Legendre多项式的正交性就一维Poisson方程模型给出了较[38]改进型的后验误差估计子,我们推导出该后验误差估计子的显式表达式,这样为工程问题中的实际应用提供了极大的便利.特别地,我们给出的后验误差估计子中不包含数值解的信息,只依赖于模型问题中方程的右端项按照Legendre多项式展开系数中的两项,并构造了p-有限元方法的离散格式和相应的后验误差估计子.
采用相似的技巧和方法,根据二维离散空间基函数的特点,我们同样得到了二维空间中Poisson方程的显式后验误差估计子,在该后验误差估计子的显式表达式中,仅包含模型问题状态方程的右端项按照Legendre多项式展开系数中的四项(事实上,因为对称性只是三项),并就二维空间中Poisson方程的p-有限元方法给出了离散格式及显式后验误差估计子.
由于最优控制问题日益得到重视,其中大多数的文献都是采用有限元方法进行分析.如果模型问题的解具有任意光滑性,通过选择适当的谱方法就可以得到”谱精度”,因此,我们研究了最优控制问题的谱方法分析.我们给出了一维空间中状态变量积分受限最优控制问题的谱方法分析,得到了状态变量积分受限最优控制问题的最优性条件,讨论了相应的先验误差估计和后验误差估计,构造了近似等价的后验误差估计子,并得到其显式的表达式.
在工程应用中,有很多的模型方程可以用双调和方程来描述,因此,对于以双调和方程为状态方程的状态变量积分受限最优控制问题如何采用谱方法来离散是我们必须考虑的问题,利用KKT条件,我们证明了该模型问题的最优性条件.此外,在讨论了相应的先验误差估计的同时,我们构造了有效的梯度投影算法,并证明了该算法的收敛性.
同样地,在很多的控制问题模型中,我们经常会遇到包含状态变量的二阶导数项的目标泛函,因此,如何保证得到该项的高精度逼近在状态方程的数值模拟中显得尤为重要.混合有限元方法是通过引入中间变量来提高梯度项的逼近精度,类似地,我们在进行谱方法分析的过程中也引进辅助变量来构造混合元谱方法.从而,我们采用混合元谱方法分析以双调和方程为状态方程的状态变量积分受限最优控制问题,我们得到了相应的最优性条件及先验误差估计,并构造有效的梯度投影算法,讨论了该算法的收敛速度.
我们通过具体的数值算例验证了上述理论的正确性.
There is an active and attractive area of the research about theoretical analysis and numerical methods for the optimal control problems governed by partial differential equa-tions. Although there are lots of research about the control constrained optimal control problems with finite element methods, not with the spectral methods. In [32], the authors investigated the control integral constrained optimal control problems. Recently, many re-searchers concern about the state constrained optimal control problems, which are met in applications, by the finite element methods. However, to the author's knowledge, it seems that there are few works have been made to systematically discuss spectral methods for these problems.
This paper investigates the state constrained optimal control problems with spec-tral methods. For simplicity, we choose the Poisson equation and the first bi-harmonic equation as the state equations. Obviously, the same techniques and conclusions can be expanded to general model problems. We focus on the a priori error estimates and a posteriori error estimates, specially, a gradient projection algorithm and its convergent property are investigated.
With the orthogonal property of the Legendre polynomials, we get an improved a posteriori error estimator to [38], which is given with explicit formulations. Then, the ex-plicit error estimator is convenient for engineering applications. Especially, the improved a posteriori error estimator only depends on the right hand side item in the state equation. In fact, there are the two coefficients of the Legendre expansion. Similarly, we discuss the p-version finite element methods and its a posteriori error estimator.
According to the property of the basis for two dimension and the orthogonal property of Legendre polynomials, we study the Poisson equation in two-dimension and deduce the explicit a posteriori error estimator, which depends on four items of the coefficients for the Legendre expansion(with the symmetric property, there are only three items). Easily, we get the discrete formulation for p-version finite element methods and the explicit formulation for the a posteriori error estimator in two-dimensional domain.
There are lots of works applying finite element methods to approximate the fashion-able optimal control problems. Obviously, if the initial data are sufficient smooth, we can choose suitable spectral methods to obtain "spectral accuracy". So, we investigate the optimal control problems with spectral methods. Firstly, we systematically analyze state integral constrained optimal control problems in one-dimension. We deduce the optimal conditions, the a priori error estimates and the a posteriori error estimator with explicit formulations.
In engineering applications, the state equations can be described by the first bi-harmonic equations. So, it is the point for us to investigate state integral constrained optimal control problems subjecting to the first bi-harmonic equations. With the help of KKT conditions, we declare the optimal conditions easily. Moreover, we investigate the a priori error estimates and derive an efficient projection gradient algorithm, specially, we discuss the convergence of the algorithm.
In many optimal control problems, the objective functions include two-order deriva-tive of the state variables. Then, how to ensure a high accuracy of this item is the key point in numerical approximation for the state equations. As we all known, the mixed finite element methods introduce an intervening variable to enhance the accuracy of ap-proximation for gradient item. Following the same ideals, we introduce an auxiliary vari-able to construct mixed spectral methods to investigate state integral constrained optimal control problems with the first bi-harmonic equation. We derive the optimal conditions, a priori error estimates and an efficient gradient projection algorithm, especially, we obtain the convergent rate of the algorithm.
We perform numerical examples to confirm our theoretical results.
本文研究了状态变量积分受限最优控制问题的谱方法分析.为了能在形式上便于说明我们的方法和技巧,我们仅选择了以Poisson方程和双调和方程为状态方程的两大类,当然,本文的方法和结论可以推广到一般的模型问题.文章讨论了先验误差估计和后验误差估计,给出了有效的梯度投影算法,并证明了其收敛性.
文中通过利用Legendre多项式的正交性就一维Poisson方程模型给出了较[38]改进型的后验误差估计子,我们推导出该后验误差估计子的显式表达式,这样为工程问题中的实际应用提供了极大的便利.特别地,我们给出的后验误差估计子中不包含数值解的信息,只依赖于模型问题中方程的右端项按照Legendre多项式展开系数中的两项,并构造了p-有限元方法的离散格式和相应的后验误差估计子.
采用相似的技巧和方法,根据二维离散空间基函数的特点,我们同样得到了二维空间中Poisson方程的显式后验误差估计子,在该后验误差估计子的显式表达式中,仅包含模型问题状态方程的右端项按照Legendre多项式展开系数中的四项(事实上,因为对称性只是三项),并就二维空间中Poisson方程的p-有限元方法给出了离散格式及显式后验误差估计子.
由于最优控制问题日益得到重视,其中大多数的文献都是采用有限元方法进行分析.如果模型问题的解具有任意光滑性,通过选择适当的谱方法就可以得到”谱精度”,因此,我们研究了最优控制问题的谱方法分析.我们给出了一维空间中状态变量积分受限最优控制问题的谱方法分析,得到了状态变量积分受限最优控制问题的最优性条件,讨论了相应的先验误差估计和后验误差估计,构造了近似等价的后验误差估计子,并得到其显式的表达式.
在工程应用中,有很多的模型方程可以用双调和方程来描述,因此,对于以双调和方程为状态方程的状态变量积分受限最优控制问题如何采用谱方法来离散是我们必须考虑的问题,利用KKT条件,我们证明了该模型问题的最优性条件.此外,在讨论了相应的先验误差估计的同时,我们构造了有效的梯度投影算法,并证明了该算法的收敛性.
同样地,在很多的控制问题模型中,我们经常会遇到包含状态变量的二阶导数项的目标泛函,因此,如何保证得到该项的高精度逼近在状态方程的数值模拟中显得尤为重要.混合有限元方法是通过引入中间变量来提高梯度项的逼近精度,类似地,我们在进行谱方法分析的过程中也引进辅助变量来构造混合元谱方法.从而,我们采用混合元谱方法分析以双调和方程为状态方程的状态变量积分受限最优控制问题,我们得到了相应的最优性条件及先验误差估计,并构造有效的梯度投影算法,讨论了该算法的收敛速度.
我们通过具体的数值算例验证了上述理论的正确性.
There is an active and attractive area of the research about theoretical analysis and numerical methods for the optimal control problems governed by partial differential equa-tions. Although there are lots of research about the control constrained optimal control problems with finite element methods, not with the spectral methods. In [32], the authors investigated the control integral constrained optimal control problems. Recently, many re-searchers concern about the state constrained optimal control problems, which are met in applications, by the finite element methods. However, to the author's knowledge, it seems that there are few works have been made to systematically discuss spectral methods for these problems.
This paper investigates the state constrained optimal control problems with spec-tral methods. For simplicity, we choose the Poisson equation and the first bi-harmonic equation as the state equations. Obviously, the same techniques and conclusions can be expanded to general model problems. We focus on the a priori error estimates and a posteriori error estimates, specially, a gradient projection algorithm and its convergent property are investigated.
With the orthogonal property of the Legendre polynomials, we get an improved a posteriori error estimator to [38], which is given with explicit formulations. Then, the ex-plicit error estimator is convenient for engineering applications. Especially, the improved a posteriori error estimator only depends on the right hand side item in the state equation. In fact, there are the two coefficients of the Legendre expansion. Similarly, we discuss the p-version finite element methods and its a posteriori error estimator.
According to the property of the basis for two dimension and the orthogonal property of Legendre polynomials, we study the Poisson equation in two-dimension and deduce the explicit a posteriori error estimator, which depends on four items of the coefficients for the Legendre expansion(with the symmetric property, there are only three items). Easily, we get the discrete formulation for p-version finite element methods and the explicit formulation for the a posteriori error estimator in two-dimensional domain.
There are lots of works applying finite element methods to approximate the fashion-able optimal control problems. Obviously, if the initial data are sufficient smooth, we can choose suitable spectral methods to obtain "spectral accuracy". So, we investigate the optimal control problems with spectral methods. Firstly, we systematically analyze state integral constrained optimal control problems in one-dimension. We deduce the optimal conditions, the a priori error estimates and the a posteriori error estimator with explicit formulations.
In engineering applications, the state equations can be described by the first bi-harmonic equations. So, it is the point for us to investigate state integral constrained optimal control problems subjecting to the first bi-harmonic equations. With the help of KKT conditions, we declare the optimal conditions easily. Moreover, we investigate the a priori error estimates and derive an efficient projection gradient algorithm, specially, we discuss the convergence of the algorithm.
In many optimal control problems, the objective functions include two-order deriva-tive of the state variables. Then, how to ensure a high accuracy of this item is the key point in numerical approximation for the state equations. As we all known, the mixed finite element methods introduce an intervening variable to enhance the accuracy of ap-proximation for gradient item. Following the same ideals, we introduce an auxiliary vari-able to construct mixed spectral methods to investigate state integral constrained optimal control problems with the first bi-harmonic equation. We derive the optimal conditions, a priori error estimates and an efficient gradient projection algorithm, especially, we obtain the convergent rate of the algorithm.
We perform numerical examples to confirm our theoretical results.
引文
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