| Optimal control problems governed by PDEs cover various topics such as time optimal control, steady-state optimal control, feed-back control, analysis and control of fluids flow, optimal shape design which is the model of the application in material design, in crystal growth, chemical reaction and so on, which have been studied for a long time, for example, one can refer to [13,28,61,76]. The research in numerical methods for optimal control problems governed by PDEs is an active area in the last 30 years. To approximate the solutions of control problems, including flow control ones, the finite element approximation proves to be a flexible, powerful and most widely used method. Many people adopt it as an especially appropriate tool and use it as the main method in dealing with the numerical analysis for optimal control problems, the readers can find more details in [47,64,73,82] and references cited therein.Within the context of optimal control problems, the flow control, a quite lively field, is crucial for many engineering applications area. For example, one can find many useful model optimal control problems of flow motion with the purpose of achieving some desired objective in real-life applications such as in the fluids flow, aeronautical, chemical engineering, magnetic field, heat sources using radiation or the laser technol-ogy, see, for instance, [50] and [52]-[54]. In the last decade, flow control becomes a more and more active field and there develops much progress in its theoretical aspect, see, for example, [1,7,18,31,32,37,43,44,45,46,48,88] and references cited therein, which give the results in the existence of the optimal control, the optimality conditions, the existence and corresponding regularity of the adjoint state or the Lagrange multiplier, etc. It is obvious that efficient numerical methods and accurate convergence analysis are quite essential to many applications of flow control. The finite element method is undoubtedly the most appropriate tool to compute the flow control problems. Re-cently, there develop some results of the finite element approximation to flow control problems, see, for example, [18,43,45,46,56,65]. However, most of those works focus on control constrained problems, and there seem no works about state constrained ones and the super convergence analysis of nonlinear ones can be found.The state constrained control problem is also frequently met in many practical applications. It is a common knowledge that the state constrained optimal control problem is more difficult in numerical analysis than a control constrained one governed by even a linear state equation, for it is necessary to conquer the analysis of the Lagrange multiplier to the state constraint set, see, for example, [17,63,11,23,30,67, 75]. The multiplier related to the pointwise state constraint is proved to be a Radon measure when the state is a continuous function, and the active set contains some unknown boundary, which is studied by Casas in [22]. Although, there develop some progress in the finite element approximation, one can find these numerical results in [11,23,30,75]. In fact, one might pay more attention to the state constrained problem in the case of the average value or the energy-norm of the state is constrained. For example, one probably want to constrain the concentration, the temperature in the average sense in some domain, or the kinetic energy of the flow, and one can find many applications of this model problem in other area, see, for instance, in the noise removal of image processing in [69] and references cited therein etc. Those above problems, which are called the functional-type state constraint problems, are actually easier to be handled than the pointwise state constrained one because the related Lagrange multipliers are more regular than the one related to the pointwise type. Recently, there develop some theoretical results of those problems, the reader can find these results in [17,19,20] and references cited therein. For the numerical results of the functional-type state constrained optimal control problem, Casas give the analysis of the finite element approximation to the semilinear elliptic control problems in [17]. In [63,93], Liu, Yang and Yuan derive the optimal order a priori error estimates and expand the sub-gradient method, the Uzawa-type algorithm and the project gradient algorithm. However, there is no research on these problems in the flow control aspect.There has been so extensive research in theoretical analysis and numerical ap-proximation of the Stokes and the Navier-Stokes equations in the scientific literature that it is impossible to give even a very brief review here, yet we point out here that one can find the classical theoretical and numerical algorithm context in monographs [40,80,81]. Using those results, this dissertation will be mainly concerned to study the standard finite element approximation to the flow control problem. Adaptive finite element method, which is proposed by the pioneer work in [5], is becoming a central theme in scientific and engineer computations for its high efficiency among various kinds of finite element methods. It uses the a posteriori error indicator to guide the mesh refinement procedure, which leads to boost the accuracy and efficiency of the finite element discretization. More precisely, the decision of whether the further refinement of the mesh is necessary is based on the error indicator derived from the discrct solutions, only the area where the error indicator is larger will be refined if the further refinement is to be performed so as to a higher density of nodes distribute over the area where the solution is difficult to approximate. To achieve a higher accuracy and save the computational work as much as possible, the adaptive finite element method is undoubtedly the particularly appropriate finite element tool in solving PDEs as yet. The theory and application of adaptive finite element methods for the numerical solutions of boundary and initial-boundary value problems for PDEs have reached some state of maturity. The literature in this area is huge, some results which are directly relevant to our works can be found in [3,6,16,64,65,74,79,85,86,87,89]. There develops many kinds of a posteriori estimates including hierarchical type, gaol-oriented dual weighted approach and functional type, besides the so-called residual type and recovery type, which both are most widely used methods. In [65,68], Liu and Yan give the residual type a posteriori error estimates for the control constrained Stokes flow problems, we will systematically study the residual type estimates for the state constrained ones of Stokes and steady-state Navier-Stokes flow in this dissertation.In general, the optimal control and the state of an optimal control problem may have some singularities. A typical situation is that with an obstacle constraint the variable probably has gradient jump around the free boundary of the active set, thus the computational error is frequently gathered in those singularities, as seen in [59]. It should be pointed out that the optimal control and state usually have different reg-ularities in a control problem, then the singularities probably distribute to different positions. That leads to the all-in-one mesh strategy may be inefficient. Adaptive multi-mesh method, which is to use many separate adaptive meshes adjusted by differ-ent error indicators, is much more efficient to deal with that case. It allows us to use very coarse mesh to solve the state and co-state equations via using adaptive multi-mesh method when we only care the approximation accuracy of the control. Thus much computational work can be saved because the major computational works in solving optimal control problems is to solve the state equations and co-state ones repeatedly, see, for instance, [51] and [58]. Accordingly, most of the finite element approximations used in this dissertation are constructed on a set of multi-mesh, although the relation between the adjoint state and control may be an equation when the control is uncon-strained, we prefer adopting a set of multi-mesh to approximate the variables to using one mesh, for this type of approach includes actually the case of adopting one mesh strategy.Research of the a posteriori estimate and further the corresponding superconver-gence of the control constrained Stokes flow problems can be found in [65] and [62]. This dissertation analyze the numerical approximation to the velocity L2-norm con-strained optimal control problem, in which the technique we use to derive the analysis of the Lagrange multiplier is obviously different from the method used in [93]. The error estimates we obtain are more precise than the results in [17], which discuss a general problem with many constraints on state and control.Recently, there develops some research of the state constrained control problems governed by the steady-state Navicr-Stokes equations, De Los Reyes and Griesse study the first order necessary and the second sufficient optimality conditions, the regularity of the adjoint state and the state constraint multiplier, and the corresponding Lipschitz stability with respect to the perturbations for the pointwise state constrained problem in [31,32], but there is no results of numerical analysis for the state constrained prob-lem as yet. In this dissertation we will analyze the numerical approximation of the functional-type state constrained optimal control problem governed by steady-state Navier-Stokes equations.In [1], Abergel and Temam study the first order optimality conditions and give a gradient algorithm. Wang obtain some theoretical results of state constrained con-trol problems governed by the 3-d instantaneous Navier-Stokes equations in [88]. And Gunzburger, Hou and Svobodny firstly study the finite element approximation of opti-mal control problems governed by the steady-state Navier-Stokes equations in [44]-[46], however, both the control and the state in the problems studied there arc without any constraints, and the error estimates obtained there arc based on the assumption that the whole optimality system defines a diffeomorphism. In [18], Casas, Matcos and Raymond discuss the finite element approximation to an optimal control problem gov-erned by the steady-state Navier-Stokes equations with control constraints, they use the sufficient second optimality conditions to obtain the a priori error estimates. In general, it depends on discussing the corresponding second order optimality conditions to make the numerical analysis of the optimal nonlinear control problem, one can find these theories in the monograph [13] of Bonnans and Shapiro, there are also many other references on this aspect which are impossible to give even a very brief review here. As well known, the first order optimality conditions of the linear problem are both necessary and sufficient, as the results in [63,70,77,93]. But, to investigate an optimal control problem governed by the nonlinear PDEs, one may refer to the results of Casas as in [17]-[23] and references cited therein. And it should be pointed out that the discrete second conditions must be used when the super convergence analysis of the nonlinear control problems are studied, however, we can hardly find these works in [63,70,77,93] and other references. This dissertation attempt to discuss the discrete second order optimality conditions of the state constrained steady-state Navier-Stokes flow control problems, give the superconvergence analysis and derive the optimal order a priori error estimates.Usually, the optimal control is the project of the dual state to the control con-straints set, and the dual state in the regular domain has a better regularity (for ex-ample, the dual state can be imbedded into the continuous function space for the second elliptic control problem), therefore, the optimal control is often a continuous function. Accordingly, the superconvergence analysis of the elliptic control problem is cared about by many people, the results about the control constrained linear problems are given in [70,77]. Further, the control constrained bilinear type problem is studied and the optimal order a priori error estimates for the finite element approximation are derived in [90]. A more general semilinear problem, in which both the control and the state are constrained, is discussed in [17], this dissertation will make the superconver-gence analysis about that problem, in the meanwhile, more precise error estimates for the same problem than the results in [17] are obtained.This dissertation is composed of a series of research in the finite clement approx-imation of optimal control problems governed by the Stokes equations or the steady-state Navier-Stokes with state constrained, and the superconvergence analysis of the semilinear control problems with control and state constraints. We pay attention to the error estimate of the standard finite element approximation to those control prob-lems, most of which are put into the multi-mesh. This dissertation attempt to derive the optimal or more precise order error estimates than before results. Meanwhile, the a posteriori error estimates for the flow control problems and the equivalent error indicators are given. Let us show the summary of each chapter as follows.In Chapter 1, finite element approximation to a semilinear elliptic control control problem with control and state constraints is studied, some better a priori error esti-mates of the same problem than [17] are obtained. The super convergence analysis is discussed using the similar assumption to [70], which holds in R2 and R3.In Chapter 2, we study an optimal control problem governed by the Stokes equa-tions with L2-norm velocity-constrained. The main results are the optimal order a priori L2-norm error estimates for both control and statesIn Chapter 3, the a posteriori estimates for an optimal control problem governed by the Stokes equations with L2-norm velocity-constrained is studied, the equivalent indicators arc given. It allows us to deduce the the equivalent posteriori estimates that we can derive an explicit expression of the Lagrange multiplier.In Chapter 4, an optimal control problem governed the by steady-state Navier-Stokes equations with many finite functional type velocity constraints is studied. As the flow obeys a nonlinear PDEs, it seems more troublesome to get the optimal-order error estimates. We adopt and develop some well known techniques, where we consider the case of the optimal state is a so-called nonsingular solution to the optimal control. As a sequel, the results fit for a general Reynolds number.In Chapter 5, we study the adaptive multi-mesh approach to a similar control problems as in Chapter 3. The equivalent a posteriori estimates arc obtained, which especially suit be used to carry out the adaptive multi-mesh computational work to approximate the control and states with different singularities. The results hold for a general Reynolds number, where the optimal state is a nonsingular solution to the optimal control.In Chapter 6, we investigate an elliptic optimal control problem with a single functional type control constraint, by changing a constrained control problem into a unconstrained nonlinear control problem, we present the Newton-algorithm based on the explicit expression of the Lagrange multiplier and the uniqueness in the half-space, numerical experiments show its fast convergence compared to the classical Uzawa algorithm.On the other hand, we perform some numerical experiments at the end of each chapter.
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