【作者】 袁磊；

【导师】 羊丹平； 刘文斌；

【作者基本信息】 山东大学 ， 计算数学， 2009， 博士

【摘要】 在近三十年来,分布参数最优控制问题的数值方法一直是一个非常活跃的研究领域.有限元方法已经被广泛的应用于数值求解不同类型的分布参数最优控制问题.并且很多学者都认为有限元方法特别适合处理这一类型的问题.关于这一主题可以参阅相关的专著.虽然,最优控制问题的有限元方法已经有了大量优秀的成果,但大部分的研究工作主要集中于控制受限的最优控制问题.近些年来,一些学者开始考虑状态受限的最优控制问题的有限元方法.这类问题在实际应用中经常出现,但却又非常难于处理.在这些学者中,大部分研究工作主要关注于一个比较特殊的问题一状态逐点受限问题.该问题具有约束形式:y≥φ,相关的工作参阅.在一些适当的条件下,对于状态逐点受限的最优控制问题,Casas在中证明了Lagrange乘子在测度意义上存在.一般情况下对于纯状态受限问题,乘子是一个Radon测度.同时接触集包含一些未知的自由边界,而且在自由边界附近解的正则性较低.因此,对于这个问题的有限元分析是非常困难的.然而,在近几年中,对于状态逐点受限的最优控制问题的有限元方法还是有了一些进展,见,例如.对于这个问题学者们还研究了一些其它的数值方法:拉格朗日函数方法,原始对偶(Primal-Dual)策略算法,水平集(Level Set)方法,Uzawa类型的算法,Lavrentiev正则化方法和变分不等式方法,等等.然而在实际的工程应用中,人们通常更为关心如何约束状态变量的平均值或者状态变量一些能量范数(本质上是积分类型的约束).例如,我们希望控制流体的浓度或者流体的动能.所以其实也存在很多其它类型的状态约束,如积分约束,L~2模约束,H~1模约束,等等.以前的有些学者研究了一些抽象形式的状态约束,参见.他们讨论了相应于问题的Lagrange乘子的存在性.但是对于这些问题的有限元逼近和误差分析,很少有系统的研究.近些年来,一些研究者开始关注这类问题的数值方法.在中,Tiba和Tr(o｜¨)ltzsch使用不精确的罚方法研究了一个状态积分形式受限,抛物方程作为状态方程的最优控制问题.由于他们使用了不精确的罚方法,因而讨论依赖于罚参数ε,并且对于观测状态正则性的一些假设在实际中也不太合适.另外一个相关工作是由Casas在中给出的.对于半线性椭圆方程作为状态方程,在有限个状态约束下的最优控制问题,Casas给出了有限元逼近的收敛性证明.随后Casas和Mateos在中扩展了他们的结论:降低了对于状态的正则性要求,并且也对半线性分布和边界控制问题的有限元逼近也给出了收敛性证明.在他们的讨论中,需要对解的局部性质做很多假设,并且没有给出有限元解的L~2和L~∞模的最优阶误差估计.在本篇论文中,我们将对几类整体型状态受限的最优控制问题及其有限元方法给出系统的研究.在分布参数最优控制问题的有限元方法研究中,另一个非常重要的方向是自适应方法的研究.最近的研究表明合适的自适应网格可以大量减少有限元离散解的误差,见.正如我们所知,在众多类型的有限元方法中,自适应有限元方法是极为重要的一类方法.关于这种方法的算法设计,理论分析和实际计算的相关研究也是近年来比较活跃的领域.为了得到精度可以接受的数值解,自适应有限元方法的本质是应用后验误差估计子去指导网格的加密生成过程.只有当后验误差估计子数值比较大的地方才会被加密,因而计算节点比较高密度的分布在精确解比较难于被逼近的地方.所以,对于具有奇性的解,可以使用最少的自由度得到较为精确的数值逼近解.自适应有限元方法目前已经被广泛的应用于各种科学计算.对于有效的处理偏微分方程的边值问题和初边值问题,自适应有限元方法的理论和应用已经到达了某种成熟的地步.相关的一些理论和技巧,可以参见.通常,最优控制问题中的最优控制具有一些奇性.例如在一个障碍类型的约束下,沿着接触集边界最优控制的梯度有间断.因此,数值计算的误差通常主要分布在这些解有奇性的地方,参见.显然,一个有效的离散格式应该有较多的计算节点分布在这些地方.相反地,如果计算网格不能适当的生成,那么在控制有奇性或状态有边界层的地方会产生较大的计算误差.所以大量的研究表明,自适应有限元方法应用于计算最优控制问题是非常有效的.已经有大量文献研究了控制受限最优控制问题的自适应方法.我们简要的回顾一些相关工作,基于残量方法的后验误差估计分别被:Liu和Yan,Hinterm(u｜¨)ller和Hinze,Gaevskaya、Hoppe和Repin研究过.将对偶含权残量方法应用于最优控制问题,可以参阅Becker和Rannacher的文献.近来的一些研究可以参阅.关于这一领域中的一些未解决的问题可以参阅.与控制受限的问题不同,自适应方法处理状态受限的最优控制问题也是最近才有了一些初步的进展.对于状态逐点受限问题,Hoppe和Kieweg在中给出一个基于残量的方法后验估计.G(u｜¨)ther和Hinze在中将对偶含权残量方法应用于状态受限的最优控制问题.Bendix和Vexler在中也给出了一个类似的方法.Wollner在中给出一个基于内部点方法的自适应方法,并且他还处理了状态梯度受限的问题.但是一般学者都认为,状态逐点受限问题的自适应有限元方法还是一个未解决的问题.另一方面,限于作者的知识,目前还没有关于积分或L~2模状态受限最优控制问题的自适应有限元方法的研究工作.此外,多套网格在计算最优控制问题中通常也是非常有用的,见Liu.在一个有约束的最优控制问题中,最优控制和状态通常具有不同的光滑性,因此它们奇性的分布位置也是不同的.这就意味着用一套网格的策略通常可能是效率很低的.多套自适应网格(即:根据不同的后验误差指示子,对不同变量分别给出不同的自适应网格)通常是必要的.由于通常最优控制问题是一个非线性问题,需要迭代求解.对控制和状态分别用不同的自适应网格,可以允许用较粗网格去求解状态方程和伴状态方程.因为计算最优控制主要的计算负载是在重复的求解状态方程和伴状态方程,所以大量的计算工作可以被节省,相关方面的研究,参见.在本篇论文中,结合使用多套网格,我们将对于状态受限积分约束和L~2模约束的最优控制问题给出相应的自适应有限元方法.求解最优控制需要将求解优化过程和求解状态方程统一结合起来.在现有的科学文献中,已经有大量关于最优控制问题的快速数值算法的研究.主要有两种方法:一种是着眼于最优性条件,直接求解最优性条件.这种方法通常需要求解一组偏微分方程.另外一种是直接离散原优化问题,使其转化成一个有限维的优化问题,然后可以用标准现成的优化软件求解.关于这一领域的最新进展可以参阅和.然而上述两种方法不能被视为完全无关.在本篇论文中,基于优化算法的思想,我们将介绍一个简单但却有效的梯度投影算法去求解离散后的有限元系统,并且我们给出了算法收敛性的证明.同时对于不易计算投影的问题,我们也给出了两个鞍点搜索算法,并且证明了算法的收敛性.本篇论文由一些关于状态受限最优控制问题的有限元方法的工作所构成.状态变量的约束在本质上是积分类型.同状态逐点受限问题不同,通常这类问题的解具有较高的正则性,所以可以预期能得到一些有限元方法的成果.然而,限于作者的知识,到目前为止还很少有对此类问题作系统有限元分析的工作.我们发展了一系列的技巧去研究这些不同类型的问题.显然,我们在研究过程中使用的技巧和前人的完全不同.下面我们逐章的介绍论文的创新点:在第二章中,讨论了积分状态受限的最优控制问题.首先,我们证明了Lagrange乘子是一个实数,这一点为我们的数值分析奠定了基础.其次,我们得到了有限元解的误差先验估计.再次,通过使用一个L~2投影,我们得到了一些超收敛性的结果.并且利用这些结果,得出了最优的L~2和L~∞模误差估计.最后,我们提出了一个简单但有效的梯度投影算法,并且证明了算法的收敛性.所有的结论都是基于使用多套网格,这种方式特别适合处理控制和状态具有不同奇性的问题.在第三章中,讨论了L~2模状态受限的最优控制问题.首先,我们证明了Lagrange乘子满足:λ=ty,其中t是一个实数,而y是状态.其次,我们得到了有限元解的误差先验估计.再次,通过使用一个L~2投影,我们得到了一些超收敛性的结果.并且利用这些结果,得出了最优的L~2和L~∞模误差估计.最后,我们给出了相应的梯度投影算法,并且证明了算法的收敛性.所有的结论也是基于使用多套网格.对于状态积分受限和L~2模受限的最优控制问题,第四章研究它们相应的自适应有限元方法.我们分别得到了这两类问题的等价的后验误差估计子.这些估计子特别适宜应用于多套自适应网格,去捕捉控制和状态的不同奇性分布.在第五章中,我们讨论了H~1模状态受限的最优控制问题.首先,证明了Lagrange乘子满足:λ=t(u+y),其中t是一个实数,u,y分别是控制和状态.其次,我们得到了有限元解的先验误差估计.最后,我们给出了相应的梯度投影算法,并且证明了算法的收敛性.所有的结论也是基于使用多套网格.基于第二章的一些结论,我们在第六章研究了一个积分形式控制和状态同时受限的最优控制问题.我们得到了有限元解的收敛性结论和误差先验估计.给出了两类鞍点搜索算法来处理同时受限的最优控制问题,并且证明了算法的收敛性.所有的结论也是基于使用多套网格.在第七章中,我们讨论了一个目标泛函不带罚项的L~2模控制受限的最优控制问题.首先,我们证明了伴状态满足:p=tu,其中t是一个实数,u是控制.其次,我们得到了有限元解的收敛性结论和误差先验估计.最后,我们给出相应的梯度投影算法,并且证明了算法的收敛性.在每一章中,我们都通过数值试验去验证理论分析的结果.更多还原

【Abstract】 The development of numerical methods for the solution of distributed optimal control problems is an active area of research in the last 30 years.Finite element method has been wildly used in computing numerical solutions of all kinds of distributed optimal control problems.And many researchers think that finite element methods in particular,are especially appropriate for these types of problems.Some monographs on this subject may be found in the[47,68,78,83].Although there were several excellent works on finite element approximation of optimal control problem,most of these works focused on control-constrained optimal control problems.In the recent past,some studies have been carried out to examine the finite element approximations of optimal control problems with state constraints, which are frequently met in engineering applications but much more difficult to handle. Most of these researches consider the specific case of point-wise constraints for the state of the form y≥φ,see,e.g.,[12,21,22,26,33].In[21],Casas have proved existence of a Lagrange multiplier in sense of measures for the optimal control problem with point-wise state constraint under some suitable conditions.In general,the multiplier is a Radon measure and the active set contains some unknown free boundary.Thus the finite element approximation is difficult to analyze.Nevertheless,in recent years,there has been some progress concerning the numerical approximation of optimal control problems with point-wise state constraints,see,e.g.,[25,32,33,79].For the problem there are the other numerical methods:Lagrangian multiplier method in[3,12],primaldual strategy in[11],level set approach in[45],Uzawa-type algorithms with or without block relaxation in[10],Lavrentiev-type regularization method in[27,74,75],variational inequality methods in[65],etc.In fact,in engineering applications,one often cares more about how to constrain the average value or some energy-norm of the state variable(essentially of integral type).For example,we want to control the concentration or kinetic energy of the flow. Hence there exist many other types of state constraints,such as integral constraint, L~2-norm constraint,H~1-norm constraint,etc.Generally speaking,these types of state constraints make the problem more easy for us to handle,since we can show the states are now more regular than in the point-wise state constraint case.Previous researchers discussed existence of Lagrange multipliers for some state-constrained optimal control problems with the abstract form,see,e.g.,[5,24,56].But there was a few systematical studies on its finite element approximation and error analysis.Tiba and Tr(o|¨)ltzsch studied a parabolic control problem with integral state constraint by using inexact penalty methods in[84].But their works depend on the penalization parameterεand the regularity hypothesis of observer state is not suitable in practice.Another work is also given by Casas in[23],which proved the convergence of finite element approximations to optimal control problems for semi-linear elliptic equations with finitely many state constraints.And Casas and Mateos extend these results in[25]to a less regular setting for the states and prove convergence of finite element approximations to semi-linear distributed and boundary control problems.In their discussion,some regular settings are needed and estimates of optimal order accuracy in L~2 and L~∞-norms for the states were not given.In this dissertation,we will give systematical studies on some optimal control problems with state constraint and their finite element approximations on multi-mesh.Another very important topic in fmite element approximation of distributed optimal control problem is its adaptive approach.It has been recently found that suitable adaptive meshes can greatly reduce discretization errors,see,e.g.,[7,8,58,59,68].As we know,among many kinds of finite element methods,adaptive finite element methods are among the most important classes of numerical methods.The study of this type of methods has been very active in recent years for algorithm design,theoretical analysis and applications to practical computations.In order to obtain a numerical solution of acceptable accuracy the adaptive finite element methods are essential in using a posteriori error indicator to guide the mesh refinement procedure.Only the area where the error indicator is larger will be refined so that a higher density of nodes is distributed over the area where the solution is difficult to approximate.Hence adaptive finite element methods are now widely used in the scientific computation to achieve better accuracy for a singular solution with minimum degree of freedom.The theory and application of adaptive finite element methods for the efficient numerical solution of boundary and initiai-boundary value problems for partial differential equations have reached some state of maturity.For some relevant theory and technique,one can see, e.g.,[2,31,34,43,77,82,86-88].In general,the optimal control of an optimal control problem has some singu- larities.For example,under the constraint of an obstacle type,typically the optimal control has gradient jumps around the free boundary of the active set.Thus the computational error is frequently concentrated around these singularises,as seen in[58]. Clearly an efficient discretization scheme should have more nodes in these areas.On the contrary,if the computational meshes are not properly generated,then there may be large error around the singularities of the optimal control or the boundary layer of the states,which can not be removed later on.Hence adaptive finite element approximation has been found very useful in computing the optimal control problems.There has been so extensive research on the topic.Let us also briefly mention some contributions to a posteriori adaptive concepts in PDEs constrained optimization.Residual based estimators for problems with control constraints are investigated by Liu and Yan in e.g.[66],by Hinterm(u|¨)ller and Hinze in[44],and by Gaevskaya,Hoppe,and Repin in[37].For an excellent overview of the dual weighted residual method applied to optimal control problems we refer to the work[8]of Becker and Rannacher.Applications of these method in the presence of control constraints is provided in[50,90].Some open problem relate to this topic,one can refer to[67],where also a recent survey of the literature in the field is given.In contrast to control constraint problem,adaptive approaches to state constrained optimal control problems are only very recently reported.Hoppe and Kieweg present an residual based approach in[49].G(u|¨)ther and Hinze in[42]apply the dual weighted residual method to elliptic optimal control problems with state constraints.A related approach is presented by Bendix and Vexler in[9].Wollner in[91]presents an adaptive approach using interior point methods with applications to elliptic problems with state constraints,and he also considers problems with constraints on the gradient of the state.To the authors knowledge,there is no attempts have been made to develop adaptive finite element analysis for optimal control problems with integral or L~2-norm state constraint.Furthermore,it has been observed that multi-meshes are often useful in computing the optimal control,see,e.g.,[64].In a constrained problem,the optimal control and the states usually have different regularities,then the locations of the singularity are very different.This indicates that the all-in-one mesh strategy may be inefficient. Adaptive multi-meshes;that is,separate adaptive meshes which are adjusted according to different error indicators,are often necessary.In general,the optimal control problem is a nonlinear problem,iteration method is required.Using different adaptive meshes for the control and the states allows to use very coarse meshes in solving the state equation and the co-state equation.Thus much computational time can be saved since one of the major computational loads in computing optimal control is to solve the state and co-state equations repeatedly,see,e.g.,[50]and[57].In this dissertation,combining multi-meshes with the adaptive finite element approximations some optimal control problems with state constraint are discussed.Solving optimal control requires to combine optimization procedures with state equations solvers efficiently.There has been so extensive research on developing fast numerical algorithms for optimal control in the scientific literature.There are mainly two approaches.In the first one looks at the necessary optimality conditions and solves these.The approach often involves differential equations solver.The other approach consists in discretizing the optimal problem such that a fmite dimensional optimization problem appears which is solved by standard optimization software.Some of the recent progress in the area has been summarized in[46]and[85].However these two approaches outlined above should not be viewed totally separated.In this dissertation based on the ideas of optimization algorithms,a simple and yet efficient iterative gradient projection algorithm is proposed to solve finite element systems.And its convergence is proved.This dissertation is composed by a series of works on finite element approximation for the optimal control problems with state constraints.The constraint for state variable is essentially integral type.In contrast to the point-wise state constraint problem, the optimal solution of these problems has better regularity.As one might expect, some results on finite element approximation may be obtained.However,to the authors knowledge,up to now it seems that there is few work have been made to develop a systematic finite element analysis for these problem.We develop a series of techniques to study these types of problem.It is clear that the techniques used in our studies are quite different from those used in the previous studies.Let’s show the novelty by chapter:In Chapter 2,an optimal control problem with integral state constraint is discussed. First,we proved Lagrange multiplierλis a number.Second,the a priori error estimates for all variable are obtained.Third,by using an L~2-projection we derived some super-convergence results.And applying these results we obtained the optimal order accuracy in L~2-norm for the states and almost optimal order accuracy in L~∞-norm for all variables.At final,a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence is shown.All results are obtained under the multi-meshes,which are suitable to treat different regularities of the control and the states.In Chapter 3,an optimal control problem with L~2-norm state constraint is dis- cussed.First,we proved that Lagrange multiplier such thatλ=ty,where t is a number.Second,a priori error estimates for all variable are obtained.Third,by using an L~2-projection we derived some super-convergence results.And applying these results we obtained the optimal order accuracy in L~2-norm for the states and almost optimal order accuracy in L~∞-norm for all variables.At final,a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence is shown. All results are obtained under the multi-meshes,which are suitable to treat different regularities of the control and the states.In Chapter 4,we study adaptive approaches for our problems in Chapter 2 and Chapter 3,respectively.We derive the equivalent a posteriori error estimators for the finite element approximations,which particularly suit adaptive multi-meshes to capture different singularities of the control and the states.In Chapter 5,an optimal control problem with H~1-norm state constraint is studied. First,we proved that Lagrange multiplier such thatλ= t(u+y),where t is a number.Second,we discuss the convergence and a priori error estimates for finite element approximation.At final,a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence is shown.All results are obtained under the multi-meshes,which are suitable to treat different regularities of the control and the states.Based on the results in Chapter 2,an optimal control problem with integral control and state constraint is studied in Chapter 6.We discussed the convergence and a priori error estimates for finite element approximation.Two kinds of saddle-point search algorithm are given for the problem and their convergence are analyzed.All results are obtained under the multi-meshes,which are suitable to treat different regularities of the control and the states.In Chapter 7,an L~2-norm control constrained optimal control problem without penalty term is studied.First,we proved that the co-state such that p=tu,where t is a number.Second,we discussed the convergence and a priori error estimates for finite element approximation.At final,a simple and yet efficient iterative gradient projection algorithm is proposed and its convergence is shown.In each chapter,we perform numerical experiments to confirm the theoretical results.更多还原