Finite Element Approximations Of Optimal Control Problems With State Constraints

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.