| The optimal control problems governed by partial differential equations have devel-oped very fast in the last 30 years, and it has brought a promising and vital researching domain to the subject of mathematics. The optimal control problems governed by par-tial differential equations concern many applications in physics, chemistry, biology, etc., such as materials design, crystal growth, temperature control, petroleum exploitation, and so on. The relative details can be seen in [39,47,49,73], and so on. The partial differential equations involved in these problems include elliptic equations, parabolic equations and hyperbolic equations. In the meanings of constrained conditions, these optimal control problems can be divided into control constrained problems and state constrained problems. In each of the branches referred above, there are many excellent works and also many difficulties to be solved.The optimal control problems have had a rather complete theoretical framework in the last few decades' development, and the relative softwares have also improved rapidly. In project and mathematics, most of the optimal control problems which scientists care about can be represented by the following abstract mathematical model. where J is the object functional, and y is called the state variable, and u is the control variable. The notation Uad above is called the control set, and A(y;u) = 0 stands for partial differential equations, or variational inequalities, even contains some conditions such as state controls. Generally, A(y; u)= 0 is called the state equations.In recent years, in the area of optimal control problems governed by partial differ-ential equations, the following four aspects have got everyone's attention and in-depth exploration.(P1) The optimal control problems governed by more types of differential equa-tions, inequalities, and so on. How to build new mathematical model for the new projection problem, and how to analyze them theoretically and give effective algo-rithm.(P2) The further applications of self-adaptive finite element methods.(P3) How to deal with the problems with more complicated constrained conditions, no matter control constrained conditions or state constrained conditions.(P4) How to solve some more complicated partial differential equations fast and efficiently, such as 2-d or 3-d coupled nonlinear time-varied partial differential equation system.Obviously, these four aspects bring us a very broad and challenging area to re-search. Therefore, the discussions in these directions are very meaningful no matter in mathematical theory analysis, or in the practical engineering applications.For the first aspect (P1), only considering the elliptic control problem, there are still a lot of partial differential equations we can study. For the control prob-lems governed by second-order elliptic equations, there is a wide range of researches both in control constrained problems and in state constrained problems, such as in [2,3,6,14,16,17,18,19,31,61], and so on. Also, divided by control types, the optimal control problems have been studied broadly in the areas of distributed control, boundary control, parameter estimation, and so on, and relative results can be seen in [50,58,59,82,83], etc. However, for the optimal control problem governed by fourth-order elliptic equations, relatively less research is done. In this field, many related mathematical models have not been established, and the applications of a number of numerical algorithms, are also important topics for scientists to study. As we know, for solving the fourth-order elliptic equation, if we directly use conforming a finite element approximation, more freedom will be needed to determinate the piecewise polynomial basis function, and it will increase the calculation time greatly. Therefore, how to find more rapid and efficient numerical algorithms, and how to integrate these algorithms to the optimal control problems, are very significant work for us.There are many excellent works in the field of fourth order partial differential equa-tions("PDEs" for short), of course containing the bi-harmonic equations as well, such as in [7,8,22,25,26,28,30,40,53,63,68,71,81], and so on. The problems described in bi-harmonic equations arise from fluid mechanics and in solid mechanics, such as bending of elastic plates. In this paper, we just study the following first bi-harmonic equation, which is very representative fourth order partial differential equation.In many applications of the first bi-harmonic equations, such as in thin beams and plates, the term y in the above equation stands for the displacement, and the term Ay stands for the curvature, and the right hand side u in the equation is the external load or force. In some projects, engineers need to control the external force to change the displacement or curvature, or some other deformation properties. According to different goals, we can build a variety of optimal control models.For the fourth order partial differential equations, to decrease the freedom and solve more rapidly, the introduction of the mixed finite elements scheme is natu-ral. Many researches have been made about mixed finite element methods for the 2rd order PDEs(for example, Raviart-Thomas, Brezzi-Douglas-Marini, Brezzi-Douglas-Fortin-Marini elements based mixed methods) and the 4th order PDEs(for example, Ciarlet-Raviart, Herrmann-Miyoshi, Hellan-Herrmann-Johnson mixed methods). More details can be found in [11,25,26,28,51,63,68,71] and the references there. In this paper, we make use of the mixed method when discretizing our optimal control prob-lem. Among the mixed finite element methods, Ciarlet-Raviart mixed finite element method of the piecewise linear elements is the special case, for which the weaker con-vergent rate was proved by Scholz in [71]. Optimal control problems governed by the fourth order PDEs also are encountered in many engineering applications. In [52], Li and Liu introduced a mixed finite element method for the optimal boundary control problem governed by the bi-harmonic equation. In the analysis of the a priori error estimates, we improve Scholz's results in [71] for the piecewise linear C-R mixed ele-ments and other C-R mixed elements of polynomial of higher degree in [7,28,37,68]. In the analysis of the a posteriori error estimates, we give equivalent a posteriori er-ror estimators for the piecewise linear C-R mixed finite element approximation of the optimal control problem in a modified norm, and this result is also an improvement.As we know, the self-adaptive mesh refinement techniques based on a posteriori error estimators have been widely used in numerical methods of PDEs over the last few years(see [3], [50], for example). Also, the adaptive mixed finite element methods for the fourth-order PDEs (see [22], for example) and optimal control problems governed by many kinds of PDEs(see [23], [24], et..) have been introduced. However, the research about the mixed finite element method for the optimal control problem governed by the first bi-harmonic equation hasn't been payed many attentions. In [52], Li and Liu introduced a mixed finite element method for optimal boundary control problem governed by the bi-harmonic equation. In our paper, we give the C-R mixed method for optimal distributed control problem governed by the bi-harmonic equations. In [22], Charbonneau, Dossou and Pierre obtained equivalent a posteriori error estimators for the C-R mixed finite element approximation errors of the bi-harmonic equation using piecewise quadric element approximation, however, in their form of approximation errors, it seems impossible to have the equivalence property using piecewise linear approximation. In our this paper, by piecewise linear approximation, we obtained equivalent a posteriori error estimators for a new form of the approximation errors.The optimal control problems governed by PDEs have been widely and deeply studied by scholars all over the world, especially the control constrained problems, such as in [6,16,17,43,50,59], and so on. Recent years, people begin to pay more attention to the numerical methods of the state constrained optimal control problems, and there have been many excellent work shown. Casas, Deckelnick, and Hinze studied the pointwise state constrained optimal control problem governed by elliptic PDEs in [14,16,17,31] and the references cited there. In [60,85,86], Yang, Liu and Yuan studied the integral state constrain, L2 norm state constraint and some other state constrains problems. In [84], Yuan and Yang studied some kinds of control-state double constraints optimal control problems. Meyer, Prufert, Troltzsch and Weiser in [61] and [67] gave the numerical analysis of control-state mixed constrained elliptic optimal control problems.Most of the work for optimal control problem are governed by the second order PDEs, however, optimal control problems governed by fourth order PDEs have not been widely researched. Some researchers begin to study these problems in recent a few years. Li and Liu in [52] studied the optimal boundary control problem gov-erned by the first bi-harmonic equation, Yang and Cao studied the distributed control constrained optimal control problem governed by the first bi-harmonic equations in [13]. As the authors know, there hasn't any work in the area of control-state dou-ble constrained optimal control problem governed by the first bi-harmonic equation. This paper gives Ciarlet-Raviart mixed finite element approximation to the pointwise control and integral state constraints optimal control problem governed by the first bi-harmonic equation.The field of petroleum engineering is concerned with the search for ways to extract more oil and gas from the earth's subsurface. In a world in which an increase in production of tenths of a percentage may result into a growth in profit of millions of dollars, no stone is left unturned.In the field of petroleum engineering, experts in reservoir simulation have adopted conservation of mass and momentum conservation equations to describe the transport of the underground chemical in porous media, such as oil, water, gas, and polymers, etc. This type of equations is often related to a large number of coupled nonlinear elliptic and parabolic equations, and there are a lot of difficulties in numerical sim-ulation. Currently, most of the oil field is still mainly uses water injection flooding of the secondary oil recovery methods. In particular, the main domestic oil fields has entered to the later stage of secondary phase of oil production, and even begun to enter the tertiary oil recovery phase. In this stage, the way for secondary oil recovery, often needs a large amount of water injected, but only a small amount of oil can be produced. As we know, the water used for injection needs high quality for reducing the corrosion of pipes, and it needs to go through several purification and processing procedures before injection. Then the production cost is worth our consideration. How to use the least water to exploit the largest oil so as to enhance oil recovery, is a great challenge ahead of reservoir engineers. In mathematics, this issue is to determine an optimal control strategy, in detail, to develop an optimal control model governed by the partial differential equations satisfied by oil, gas. water, and so on. For example, we can introduce an optimal control problem governed by the two-phase miscible (or immiscible)equations. There have been many works on this problem, such as Brouwer and Jansen's results in [12], etc. In our this paper, we use the way of deducing the co-states equation and getting the exact optimal conditions, and give an theoretical analysis and some numerical results.Under the guidance of Professor Danping Yang and Professor Wenbin Liu, the author did some research on optimal control problems governed by the first bi-harmonic equation and the two-phase immiscible displacement equations. For the former, i.e. the optimal control problem governed by the first bi-harmonic equation, we analyzed the distributed control constrained case and distribute control constrained, integral state constrained double-constrained case. For both of the two cases, we gives the a priori error estimates, respectively. In the distributed control constrained case, our innovations is to have obtained a improved convergence order, and for the piecewise linear Ciarlet-Raviart element approximation of this problem, we obtained sub-optimal the a posteriori error estimates. Some of the results in this part have been published on the Journal of Computational and Applied Mathematics. For the double-constrained situation, we combined a number of literature's methods, and gave the convergence and the a priori error estimates of the control problem for the first time. We also gave some numerical examples. For the latter, we proposed the mathematical model of optimal control problem, and had a comprehensive analysis about the existence of solutions, optimality conditions, finite element approximation, the a priori error estimates, and also did some numerical experiments. Some of these results have been submitted to the SIAM. The full-text is divided into three chapters, and the followings are descriptions of the main contents for each chapter.In Chapter 1, our purpose is to research the C-R mixed finite element method for the optimal distributed control problem governed by the bi-harmonic equations. We investigate the a priori error estimate and the a posteriori error estimates of the mixed finite element approximation. In the analysis of the a priori error estimates, we improve Scholz's results in [71] for the piecewise linear C-R mixed elements and other C-R mixed elements of polynomial of higher degree in [28,68]. In the analysis of the a posteriori error estimates, we obtained equivalent a posteriori error estimators with a modified norm. At the end of each error estimates, we gave some numerical experiments to confirm the error estimates.In Chapter 2, we studied the Ciarlet-Raviart mixed finite element approximation of the control-state double constrained optimal control problem governed by the first bi-harmonic equation. We learn from the analysis of [17,61,84], etc., and deduce the convergence results and the a priori error estimates. Some numerical experiments are performed to confirm the theoretical analysis for the a priori error estimate.In Chapter 3, with the aim to maximize production of oil from petroleum reservoirs in the field of oil recovery, we construct the optimal control problem governed by the modeling system describing the two-phase incompressible flow in porous media in this paper. Then we give the proof of the existence of solutions of our control problem. The optimality conditions are obtained. After that we considered the finite element approximation and prove the existence of solutions of co-state equations. We also deduced the a priori error estimates and carried out some numerical tests at the end of the Chapter. These results are obtained through cooperation of our research group, which is led by Professor Danping Yang and Professor Wenbin Liu, and involves Professor Tongjun Sun and Professor Ning Du, and also Dr. Yanzhen Chang and me. Some of these results have been shown in the doctoral dissertation of Yanzhen Chang, and we just list the conclusion in Chapter 3 but do not give the proof in detail there. This Chapter update Chang's contents in the aspect of the proof for the existence of co-state equations' solutions, the a priori error estimates, and also the numerical results.
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