A Posteriori Error Estimate Of Finite Element Method Of Optimal Control Problem With Integral Constraint

Optimal control problem of partial differential equations is a very lively and active mathematical field and also has been widely studied and applied in the last 30 years.This branch of mathematics covers various topics such as time optimal control, feedback control,analysis and control of flow equations,optimal shape design which are the models of applications in material design,in crystal growth,chemical reaction and others.The reader is referred to see examples of[56,73,74,80].It includes not only stationary and liner equations but also time-dependent and nonlinear ones.Especially, current research in the control of partial differential equations is driven by a multitude of applications in engineering and science that are modeled by coupled linear or nonlinear differential equations,for example,[4,36,37].Because of the complexity of problems in real applications,and what is also desirable and in fact necessary,to ensure that our mathematical theory can be widely and successfully applied to engineering and physical problems,it is a great challenge to form our optimal control models which are more reasonable and well-posed.On the other hand,with the development of computer,we need to compute and to implement these control problems to guide our real performance in engineering and manufacture.Now the becoming second task is that associated optimal control problems need efficient numerical methods to deal with the resulting problems.It needs a fast development of numerical methods and the associated analysis must keep track to justify them and to prepare the basis for further research.Among these numerical methods,as well known the finite element method is a powerful tool. The finite element approximation of optimal control problems has been extensively studied in the literature.Furthermore,there have been extensive studies in convergence of the standard finite element approximation of optimal control problems,see,some examples in[5,6,33],although it is impossible to give even a very brief review here. For optimal control problems governed by linear state equations,a prior error estimates of the finite element approximation were established long ago;see,for example[29].There has been so extensive research on developing fast numerical algorithms for optimal control in the scientific literature that it is simply impossible to give even a very brief review here.However there seems still some way to go before efficient solvers can be developed even for the constrained quadratic optimal control governed by an elliptic equation.The reason seems that there are so many computational bottlenecks in solving an optimal control problem numerically,and furthermore they are all closely related.There are at lease two key procedures in solving an optimal control problem: The optimal control problem has to be first discretised;then the discretised optimization problem(sometimes the KKT system) has to be solved,where it is normally needed to solve the state and co-state equations repeatedly.Among many kinds of finite element methods,adaptive finite element methods based on a posteriori error estimates have become a central theme in scientific and engineer computations for their high efficiency.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.In this sense efficiency and reliability of adaptive finite element approximation rely very much on the error indicator used.There exist several concepts including residual and hierarchical type estimators,error estimators that are based on local averaging,the so-called goal oriented dual weighted approach,and functional type error majorants(cf.[3,8,9]).The theory and application of adaptive finite element methods for the efficient numerical solution of boundary and initial-boundary value problems for partial differential equations have reached some state of maturity as documented by a series of monographs. On the other hand,as far as the development of adaptive finite element schemes for optimal control problems for PDEs is concerned,much less work has been done.The goal oriented dual weighted approach has been applied to unconstrained problems in [13].Residual-type a posteriori error estimators for control constrained problems have been derived and analyzed in[31,41,44,61,62].Also suitable adaptive meshes can quite efficiently reduce the control approximation error.It has been recently found that suitable adaptive meshes can greatly reduce discretisation errors,see[13,14,53,62].If the computational meshes are not properly generated,then there may be large error around the singularities of the control,which cannot be removed later on.It is pointed out only recently that the error indicators derived for the approximation of the state equation,which have been widely used in adaptive finite element schemes for optimal control,are not necessarily efficient for computing the optimal control problems,and purposed built error estimators for the control problem are essential in such schemes,see[13]and[53]for example.Further-more in a constrained control problem,the optimal control and the state usually have different regularity,and what is more,the locations of the singularities are very different. Usually the optimal control has only limited regularity.This indicates that the current all-in-one mesh strategy may be inefficient.Adaptive multi-mesh;that is, separate adaptive meshes which are adjusted according to different error indicators, are often necessary.Particularly it seems to be important to use multi-set adaptive meshes in applying adaptive finite element method to computing optimal control,see [46,52].However it is much more complicated to implement adaptive computational schemes for evolutional control problems,see[47,54].This dissertation is some research work on the residual type a posteriori error estimate for adaptive finite element method for the constraint optimal control problem. The dissertation is divided into six chapters.Firstly,we use the piecewise constant and piecewise linear discontinuous finite element to approximate the control and give sharp a posteriori error estimate for the optimal control problem which is only suitable for the same mesh.For this problem,there has been no other papers which have used the adaptive finite element to construct with it.Secondly,we give sharp a H~1 posteriori error estimation for the above optimal control problem which is suitable for the multi-meshes. Although[46]has given a posteriori error for the optimal control problem with obstacle constraint for control which is suitable for the multi-meshes,since the restrictions of the method and the difficulty of the problem,its a posterior error norm needs one term e which has been used to conform the equivalence of the estimation.The innovation of this chapter is that a H~1 norm posteriori error is equivalent,and we don't use other terms.Thirdly,we give sharp a L~2 posteriori error estimation for the above problem which is also suitable for the multi-meshes.The innovation of this chapter is that this is the sharp L~2 norm a posteriori error estimator for the optimal control problem.Although we only use the method to construct with the integral constraint optimal control problem,the method can also be used to construct the point-wise problem,and other optimal control problems.Fourthly,we used nonconforming finite element to construct the integral state constraint optimal control problem,and derive a priori error and a posteriori error estimation for the integral state constraint optimal control problem.The innovation of this chapter is that we use a lot of new methods for this problem,and obtain sharp a posteriori error estimation which is different from the others which have been derived for some other problems.Fifthly we transform the integral state-constraint problem to the integral control-state constraint problem and give a lot of analysis to this new problem,obtain the finite element a priori error estimation and a posteriori error estimator.Although this method has been used for another state constraint optimal control problem,the innovation of this chapter is that we are the one who give sharp a posteriori error estimator for this problem.Sixthly,we investigate indicators that are equivalent to the constrained optimal control problem governed by the parabolic equation.The innovation of this chapter is that we are the one who give equivalent a posteriori error estimation for the constrained optimal control problem governed by the parabolic equation and we can use the method to compute the example in different meshes with the time.Then,we show the summary of each chapter.Up to now it seems that all the existing a posteriori error estimates were derived for the case where the control constraints are of obstacle type:K={u∶u≥g},where g is an obstacle.It is well-known that for constrained optimal control problems,the formulas of a posteriori error estimators and the techniques used to derive them heavily depend on the structure of the control constraint set.K.In this work we investigate a posteriori error estimates for case where the control constraint is not given point-wisely, but K={u:∫_Ωu≥0}.It is interesting to derive sharp a posteriori error estimates for this type of optimal control problems.In the first chapter,we shall illustrate our main ideas through examining for a linear elliptic control problem.The purpose of this part is to investigate adaptive finite element in computing constrained optimal control problem.In the second chapter,we will illustrate our main ideas through examining for a linear elliptic control problem.The purpose of this part is to investigate adaptive multi-mesh finite element method for a constrained optimal control problem.We de- rived equivalent a posteriori error estimators for the control problem and then present numerical test results to confirm the effectiveness of the error estimators.The from the theory of adaptivity FE(see,[28]and[71]),then exponential convergence of adaptive FE can be guaranteed to be achieved by using the equivalent error estimators in the adaptive FE.This motivate us to derive and use our error estimators instead of those heuristically based.Although a posteriori error estimates equivalent to the H~1 norm of the approximation error(to be called H~1 norm equivalent a posteriori error estimates) have been derived for several elliptic optimal control problems,see[38,39,46],both for the control constraints of obstacle types and integral types,there seems no existing work on L~2 norm equivalent a posteriori error estimates,which are equivalent to the L~2 norm of the approximation error,although some upper bounds were derived using the L~2 norm for the control constraint of an obstacle type,see[46,62].It does not seem a trivial problem whether and how some lower bounds can be derived via the L~2 norm,although it seemed possible to adapt the existing duality techniques to derive upper bounds.In many engineering applications,one cares more about averaging values of the control and the states.In these cases,it seems to be more natural to use the L~2-norm of the approximation error as the stopping criteria in computations.Thus L~2 equivalent error indicators seem to be quite useful.Error indicators based on the L~2 norm error bounds tend to produce less over-reinterment in such cases.In the third chapter,we shall illustrate our main ideas through examining for a linear elliptic control problem. The purpose of this chapter is to investigate indicators that are equivalent to the L~2 norm of the approximation error for a constrained optimal control problem,where the control constraint is of an integral type.We derived L~2 norm equivalent a posteriori error estimators,which allow different meshes to be used for the states and the control. Then we performed some numerical tests to confirm the effectiveness of the error estimators.There are lots of papers about the state constrain optimal control problems,such as[15,17,19,27,59,67,72,76].Many authors have studied the specific case of point-wise constraints for the state,such as K={y:y≥φ} or K={y:y≤φ},e.g.,see[15, 17,19,27].Under some suitable conditions,Casas[17]proved existence of a multiplier in the sense of measures for the point-wise state-constrained optimal control problem. In general,the multiplier containsδmeasures on unknown free boundaries.Thus the finite element approximation is difficult to analyze.While convergence and a priori error analysis were studied in[17,18,27,59,75,77],a posteriori error estimates are only recently studied,see[34]and[45].An augmented Lagrangian method was proposed to solve the state and control constrained optimal control problems by Bergounioux and Kunisch in[11].They also proposed another method:a primal-dual strategy to solve the problem in[12].In applications of engineering,one often cares more about how to control the average value or L~2-norm of the state variable.So there exist many other type of state-constraints,such as integral constraint K={y:α≤∫_Ωy≤β}, L~2 norm constraint K={y:∫_Ωy~2≤β},and others.Barbu[7]and Lasiecka[51]only discussed existence of Lagrange multipliers for some state-constrained optimal control problems with the integral constraint such as∫_Ωg(y)≥0.Recently,W.B.Liu,D.P. Yang and L.Yuan,in[67,93],considered the elliptic optimal control problems with the integral constraint for state.They give the optimal condition of the optimal control problem and the a priori and a posteriori error estimation of the standard finite element approximation.Up to now,most parts of researches on finite element approximations for optimal control problems are focused upon the standard finite element approaches. In[59],W.B.Liu,W.Gong and N.N.Yan gave a new approach to the point-wise state-constrained control problem,which transformed the state-constrained optimal control problem into a fourth order variational inequality problem and then used the Morley nonconforming finite element to solve this problem.The a priori error estimates were proved.But it still is difficult to give the equivalent a posteriori error estimators for the nonconforming finite element approximation of the fourth order variational inequality problems.As we known,little work about equivalent a posteriori error estimators of nonconforming FEMs has been done.Recently,a posteriori estimate estimate for the fourth order equality problem was given in[94].In the fourth chapter,we use the nonconforming finite element to approach the elliptic optimal control problems with the integral constraint for state.We give a priori error estimation for this problem.And we derive and prove the equivalent a posteriori estimators of Morley's nonconforming finite element approximations for the fourth order variational inequality.In this process,we use a lot of new projections and some new bubble function,which has never been given in other papers.In the fifth chapter,we use another method to consider the integral state-constraint optimal control problem.We will transform the integral state-constraint problem to the integral control-state constraint problem.Unlike the above way which construct the control-state constraint problem using the optimal methods,we see it as a singular elliptic control problem which only has the control constrain.We also can prove this problem converging to the initial problem.Then we use the finite element to discrete this problem and obtain a priori error estimation and sharp a posteriori error estimator.Although a posteriori error estimates equivalent to the elliptic control problem,a posteriori error estimates for the parabolic control problem is still an open problem, which review in[63].There are also many works for the evolutional problems.In[54], they first give the time and space adaptive schemes for evolutional control problems. In[69],they give a posteriori error estimation for this problem.Although there are so much estimators for this problem,to our knowledge,there are no one which has given a proof lower bound for these estimators.In the sixth chapter is to investigate indicators that are equivalent to the constrained optimal control problem governed by the parabolic equation.The innovation of this paper is:1.We use adaptive finite element method to construct the optimal control problem with the integral constraint for control,and obtain a H~1 norm and L~2 norm posteriori error estimator which is suitable for the multi-meshes.2.We use two different methods to construct the integral state constraint optimal control problem.One method is that we transform the problem into a fourth order variational inequality problem,and use the Morley finite element to construct the new problem.We also obtain a priori error estimation and a posteriori error estimation. Another method is that we transform the problem into a mixed control-state constraint problem,and obtain a posteriori error estimation.3.We give equivalent a posteriori error estimation for the constrained optimal control problem governed by the parabolic equation.