最优控制问题有限体积元法的收敛性分析
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摘要
最优控制问题是微分方程约束下的一个约束优化问题,如同微分方程一样,最优控制问题应用广泛,比如大气污染的控制,癌症化疗,金融投资,流体控制等.有限体积元法是一种具有守恒性(质量,能量和动量守恒)的微分方程数值近似的方法,特别适合用于具有守恒性的问题的数值近似,比如流体类问题.本文研究几类基本最优控制问题有限体积元法的收敛性分析,虽是研究三类基本最优控制问题的有限体积元法近似,但是它为进一步研究最优控制问题的有限体积元法打下了基础.
本文基于先优化后离散的方式,采用变分离散技巧(variational discretization)研究了求解线性二次椭圆、抛物、二阶双曲最优控制问题的有限体积元法,给出了相应离散系统解的存在唯一性和收敛性分析,对于发展型方程最优控制问题,还给出了其全离散格式及相应的先验误差估计,对所有问题都给出了数值算例.本文的主要工作如下：
首先,基于先优化后离散的方式,对线性二阶椭圆最优控制问题,采用变分离散技巧得到了线性有限体积元近似的离散最优性系统,证明了此离散最优系统解的存在唯一性,给出了由此离散系统得出的状态,对偶状态和控制的收敛性分析,并用数值算例检验理论结果的正确性.
其次,对线性抛物最优控制问题采用先优化后离散的方式得到了线性有限体积元近似的半离散最优性系统,证明了此半离散最优系统解的存在唯一性；对由此得出的半离散解给出了的先验误差估计；在半离散格式的基础上,给出了向后Euler有限体积元格式和Crank-Nicolson有限体积元格式,对此两种全离散近似解作出了收敛性分析,得出了理想的收敛阶；还提供了数值算例来检验理论结果的正确性.
最后,对二阶双曲最优控制问题得到了线性有限体积元近似的半离散最优性系统,证明了此离散最优系统解的存在唯一性；对由此离散系统得出的半离散解给出了先验误差估计；以半离散最优性系统为基础,给出了带参数的全离散有限体积元格式和Crank-Nicolson有限体积元格式,对Crank-Nicolson有限体积元全离散近似解作出了收敛性分析,得出了最优的收敛阶；用数值算例检验理论结果的正确性.
The optimal control problem is a optimization problem under the constraints of diferentialequations. As the diferential equations do, the optimal control problem is widely used in oursociety, such as air pollution control, cancer chemotherapy, financial investment, fluid control etc.Finite volume element method is a method with conservation (mass, energy and momentum). It isparticularly suitable for the numerical approximation with conservation issues, such as fluid flowproblems. In this paper we study the convergent analysis of finite volume element method foroptimal control problems. Though we study the finite volume element approximation of severalbasic classes optimal control problems, it is the foundation for further studying the finite volumeelement approximation of some other optimal control problems.
In this paper, we apply the optimize-then-discretize technique and the variational discretiza-tion approach to study the finite volume element approximation of the elliptic optimal control,parabolic optimal control and second order hyperbolic optimal control problems. We obtain thecorresponding discrete optimal system, verify the existence and uniqueness of these systems, andderive some convergent results (i.e. some a prior error estimates) for the numerical solution. Forthe evolutional equations optimal problems, we also develop some fully-discrete schemes and getthe corresponding a priori error estimates. Numerical examples are also presented to test thetheoretical analysis. The main results are as followsμ
Firstly, we use the optimize-then-discreize technique to discretize the second order ellipticoptimal problems and obtain a finite dimensional optimal system. The existence and uniquenessof the system are proved. A priori error estimates are derived. A numerical example is listed toillustrate the theoretical results.
Secondly, the semi-discrete finite volume element optimal system of the parabolic optimalproblem is got. We verify the existence and uniqueness of the discrete system and get some apriori error estimates. Back Euler finite volume element scheme and Crank-Nicolson finite volumeelement scheme are used to approximate the semi-discrete system. Convergent analysis of the fully-discrete solution is carried out and some convergent orders are achieved. Numerical experimentsare provided to test all the theoretical results at the end of these contents.
Thirdly, we obtain the semi-discrete optimal system of the second order hyperbolic optimalproblems. The existence and uniqueness of the discrete system is proved. Some a priori errorestimates are got for the solution of the semi-discrete optimal system. Two fully-discrete schemesare proposed to approximate the semi-discrete system. One is a finite volume element scheme witha parameter. The other one is Crank-Nicolson finite volume element scheme. Convergent analysisof the Crank-Nicolson finite volume element approximation is carried out and optimal convergent order is obtained. Some numerical experiments are presented to test the theoretical results.
本文基于先优化后离散的方式,采用变分离散技巧(variational discretization)研究了求解线性二次椭圆、抛物、二阶双曲最优控制问题的有限体积元法,给出了相应离散系统解的存在唯一性和收敛性分析,对于发展型方程最优控制问题,还给出了其全离散格式及相应的先验误差估计,对所有问题都给出了数值算例.本文的主要工作如下：
首先,基于先优化后离散的方式,对线性二阶椭圆最优控制问题,采用变分离散技巧得到了线性有限体积元近似的离散最优性系统,证明了此离散最优系统解的存在唯一性,给出了由此离散系统得出的状态,对偶状态和控制的收敛性分析,并用数值算例检验理论结果的正确性.
其次,对线性抛物最优控制问题采用先优化后离散的方式得到了线性有限体积元近似的半离散最优性系统,证明了此半离散最优系统解的存在唯一性；对由此得出的半离散解给出了的先验误差估计；在半离散格式的基础上,给出了向后Euler有限体积元格式和Crank-Nicolson有限体积元格式,对此两种全离散近似解作出了收敛性分析,得出了理想的收敛阶；还提供了数值算例来检验理论结果的正确性.
最后,对二阶双曲最优控制问题得到了线性有限体积元近似的半离散最优性系统,证明了此离散最优系统解的存在唯一性；对由此离散系统得出的半离散解给出了先验误差估计；以半离散最优性系统为基础,给出了带参数的全离散有限体积元格式和Crank-Nicolson有限体积元格式,对Crank-Nicolson有限体积元全离散近似解作出了收敛性分析,得出了最优的收敛阶；用数值算例检验理论结果的正确性.
The optimal control problem is a optimization problem under the constraints of diferentialequations. As the diferential equations do, the optimal control problem is widely used in oursociety, such as air pollution control, cancer chemotherapy, financial investment, fluid control etc.Finite volume element method is a method with conservation (mass, energy and momentum). It isparticularly suitable for the numerical approximation with conservation issues, such as fluid flowproblems. In this paper we study the convergent analysis of finite volume element method foroptimal control problems. Though we study the finite volume element approximation of severalbasic classes optimal control problems, it is the foundation for further studying the finite volumeelement approximation of some other optimal control problems.
In this paper, we apply the optimize-then-discretize technique and the variational discretiza-tion approach to study the finite volume element approximation of the elliptic optimal control,parabolic optimal control and second order hyperbolic optimal control problems. We obtain thecorresponding discrete optimal system, verify the existence and uniqueness of these systems, andderive some convergent results (i.e. some a prior error estimates) for the numerical solution. Forthe evolutional equations optimal problems, we also develop some fully-discrete schemes and getthe corresponding a priori error estimates. Numerical examples are also presented to test thetheoretical analysis. The main results are as followsμ
Firstly, we use the optimize-then-discreize technique to discretize the second order ellipticoptimal problems and obtain a finite dimensional optimal system. The existence and uniquenessof the system are proved. A priori error estimates are derived. A numerical example is listed toillustrate the theoretical results.
Secondly, the semi-discrete finite volume element optimal system of the parabolic optimalproblem is got. We verify the existence and uniqueness of the discrete system and get some apriori error estimates. Back Euler finite volume element scheme and Crank-Nicolson finite volumeelement scheme are used to approximate the semi-discrete system. Convergent analysis of the fully-discrete solution is carried out and some convergent orders are achieved. Numerical experimentsare provided to test all the theoretical results at the end of these contents.
Thirdly, we obtain the semi-discrete optimal system of the second order hyperbolic optimalproblems. The existence and uniqueness of the discrete system is proved. Some a priori errorestimates are got for the solution of the semi-discrete optimal system. Two fully-discrete schemesare proposed to approximate the semi-discrete system. One is a finite volume element scheme witha parameter. The other one is Crank-Nicolson finite volume element scheme. Convergent analysisof the Crank-Nicolson finite volume element approximation is carried out and optimal convergent order is obtained. Some numerical experiments are presented to test the theoretical results.
引文
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