| Since the great work from Lions[50], there have been active and systematical developments in optimal control problem governed by partial differential equations. Now as an important branch of applied mathematics, it has been widely used in many problems such as material design, crystal production, optimal time control, feedback control, rocket state control, optimal shape design (like wing shape of airplanes), oil production, and human population control. It is impossible to give even a very brief review here. Some applications can be found in, e.g.,[12,19,50,57,63,64,67,76].Nowadays, there is a conventional mathematical framework for optimal control problem governed by partial differential equations. and an extensive body of compu-tational software. The abstract mathematical model of optimal control mostly met in engineering reads: s.t. Here, J(u,y(u)) is objective functional, y is the state, u is the control variable, Uad is a control constraint set, and A(y, u)=0is a partial differential equation, including variational inequalities, or even state-constraints. In general, A(y, u)=0is referred to as the state equation.The state equations met in practical optimal control problems can be stationary or evolutional, linear or nonlinear, and more importantly combinations of those types, see for example,[2,12,19,21,35,36,50,57,63,64,67,76]. Furthermore optimal control problems can be classified into control constrained control and state constrained control, which both have been intensively studied in literature, see [17,31,42,47,57, 61,68,83] for control constrained control, and [11,14,15,16,25,52,62,65,80,85] for state constrained control.In many real applications, such as heat conduction in materials with memory, pop-ulation dynamics, and elastic-plastic mechanics, one often meets optimal control prob-lems governed by parabolic, quasi-parabolic and hyperbolic integro-differential partial differential equations. Although there has been existed systematical researches on these state equations, see [29,37,66], for classic theory and algorithms, and [33,58,59] for case of nonsmooth kernels, to our best knowledge there is lack of any research on op-timal control governed by them-either theoretical or computational. In particular, due to the complex nature of these equations, it is normally impossible to find their exact solutions. Therefore developing highly efficient algorithms now becomes one of the most important challenges in applying those control problems in practice.Although there are many useful computational methods such as the finite differ-ence method, spectral method, and finite volume method, which all have been actively used in computing numerical solutions of optimal control problems, there is no doubt that the finite element method has played a pivot role in today’s large scientific and engineering computations. There has been existed an extensive body of literature on the finite element approximation of optimal control problem governed by partial dif-ferential equations, including design of effective algorithms, convergence analysis and error estimates, see eg.,[3,4,5,32,34,57]. For instance, a priori error estimate has been given in [27] and [76] for the control governed by linear and nonlinear elliptic PDEs respectively. However even to simple control constrained cases, there are still many bottlenecks on developing effective algorithms for the finite element approximation of optimal control governed even by simple elliptic equations. What is more, these key factors are interlinked each other, and consequently it is hard to attack them separately. Among the key problems, the first is how to discrete the optimal control, and then is how to solve the discreted systems (sometimes, the KKT systems) very efficiently as they need to be solved repeatedly in computations.In the literature there already exists much work on the finite element approxima-tion for optimal control governed by parabolic and hyperbolic PDEs, see for example,[18] for error estimates of the mixed finite element approximation of optimal control problem governed by semi-linear parabolic equations;[56,74] for a posteriori error estimates of optimal control governed by parabolic equations;[60] for a posteriori er-ror estimates of the time-space finite element approximation of parabolic control, and [50] for hyperbolic control problems. Furthermore there exists systematical work on numerical solutions on integro-differential equations in the literature, see, for example,[23,29,33,37,58,59,66,81,82] and [8,13,20,26,45,49,71,75,84] for the finite element approximation of parabolic, quasi-parabolic and hyperbolic integro-differential equations. However to our best knowledge there is no reference on the finite element approximation of optimal control governed by these equations in the literature yet. Due to the extra integral terms presented in the state equations, it is more challenging to study the finite element approximation for optimal control governed by parabolic, quasi-parabolic and hyperbolic integro-differential equations.The purpose of this thesis is to study the finite element approximation of opti-mal control problems governed by parabolic, quasi-parabolic and hyperbolic integro-differential equations. For the first time we have established existence and regularity of the optimal controls, their finite element approximation schemes, and a priori error estimates of the optimal order in different norms.Among different finite element approaches in the literature, it proves that the adaptive finite element method can often greatly increase computational efficiency, and thus now has become a main tool in the finite element methods for large scientific and engineering computations. Adaptive finite element method includes the p-method where the orders of base polynomials is adjusted; the moving-mesh method where the position of the meshes is adjusted, and the h-method where the meshes of the elements are refined or coarsen. Unless specially indicated, the adaptive finite element method used in this thesis is the h-method. This method uses a posteriori error indicator, whose value is positively correlated to singularity of the solutions to be approximated, to guide local mesh refinement in computation in such a way that denser meshes are only used where the value of the indicator is larger so solution is more singular, and consequently the degree of freedoms may be greatly reduced on these only locally refined meshes. Therefore quality of the posteriori error indicators is vital for effectiveness of adaptive finite element method. There are different kinds of a posteriori error indicators such as local average type, gradient type, and residual type, see e.g.,[1.6.7] for the details. Although adaptive finite element has been extensively studied and widely applied for finite element approximation of boundary value problems of PDEs, it has only been properly studied and used in finite element approximation of optimal control governed by PDEs quite recently. Until90s, some a posteriori error estimators for approximation error of the state equations are used in guiding local mesh refinement in the adaptive finite element method. In work [9,54,55], it was pointed that the error indicators for the finite element approximation of the state equations are general not suitable for that of optimal control problems. In their work, the objective orientated duality method [9] mainly suites the unconstrained optimal control problems, while the residual method in [54,55] is applicable for control-constrained optimal control. In this thesis we will derive a posteriori error indicators for the finite element approximation of optimal control problem governed by parabolic, quasi-parabolic and hyperbolic integro-differential PDEs. For these control problems, we carried out very detailed analysis and obtained sharp a posteriori error estimates which are equivalent to the approximation error of the finite element discretization of the optimal control problems on multi-meshes.References on effectiveness of the adaptive finite element in reducing computa-tional work for optimal control problem governed by PDEs can be found in e.g.,[9,10,47,55].It is important to point out that in general constrained optimal control has very different regularity from that of the state and the co-state. Therefore it would be very inefficient to use the same meshes in approximating the control, the states. Generally speaking the states are more regular than the optimal control that often has some singularity. For example, for the common obstacle control constraints, the optimal control often has gradient jumps around the boundary of the contact sets, see [47]. Thus if we use the same meshes to discrete the states and the control, we may have to use dense meshes to approximate the smooth states due to singularity of the optimal control, and thus greatly reduce its computational efficiency. The adaptive multi-mesh approach can show the unique advantages in such cases-it uses different indicators and meshes for the states and the control, and thus the meshes can be refined separately. If we are only interested in obtaining accurate optimal control, we only need to use relative coarse meshes for computing the states to save a great deal of computation work as the state equations often need to be solve repeatedly in seeking the numerical solution of the optimal control for a given error tolerance. Thus for the optimal control constrained problems we often need to use multi-mesh approach, as in this thesis for the optimal control of integro-differential equations. The ideas can also be used for other control problems, see e.g.,[41,46,48].Although equivalent a posteriori error estimates have been given in [18,56,60] for the finite element approximation of optimal control problem governed by elliptic and parabolic equations, it is much more complicated to obtain such estimates for the opti-mal control governed by parabolic, quasi-parabolic and hyperbolic integro-differential equations due to the greater complexities of these state equations, and consequently it is still an open problem to obtain sharp a posteriori error estimates for the finite element approximation of the optimal control governed by such state equations. In this thesis we derive such a posteriori error estimates in L2(0,T; H1(Ω))-norm for optimal control governed by those state equations, and present some numerical tests to confirm their equivalence.The brief outlines of the subsequent chapters are as follows:In Chapter One, we study the finite element approximation for optimal control problem governed by parabolic integro-differential equations under two kinds of control constraints:Obstacle type:Uad={v∈X; v≥0}; Integral type:Uaa={v∈X;∫n u≥0}. We firstly present the weak formula for our optimal control problem. We then establish existence and regularity of the optimal control, by utilizing a priori estimation and Gronwall inequalities. By following the ways of deriving the optimality conditions for optimal control governed by elliptic equations, we derive the optimality conditions for our control problems. We then consider a semi-discrete finite element approximation for our control problem, establish its well-postness and then derive the a priori error estimates for the states and the control in L2(0, T; H1(Ω))-norm and L2(0, T; L2(Ω))-norm. To this end we firstly introduce intermeddle variables and equations, and then derive the a priori error estimates by extending the techniques used in [57]. In order to derive L2(0,T; L2(Ω))-norm estimates, we sub-divide the contact set in order to obtain sharper error estimates. Finally we carried out numerical tests to verify our theoretical estimates.In Chapter Two, we still deal with optimal control problem governed by parabolic integro-differential equations, but now we focus on dealing with the a posteriori error estimates used in adaptive finite element method. It is well-known that for control constraint problems, the formulation of the estimators and the techniques needed in deriving them very much depend on the control constraint sets K. In this chapter we consider the case:Integral case:{K=∫Ωu(t)≥0,(?) t∈[0,T]}.For these cases we study the adaptive finite element approximation on multi-meshes, and derive equivalent a posteriori error estimates in L2(0,T;H(Ω))-norm. Based on the semi-discrete finite element approximation, we further apply the Backwards Euler scheme to obtain the full discretization, and then derive the well-postness and the optimality conditions for the scheme. By introducing intermeddle variables and generating the method in [57], we obtain the sharp upper a posteriori a error estimates. Among the main difficulties, the term containing the integral needs delicate estimates in order to obtain sharp results. For lower a posteriori error estimates, we apply the conventional bubble function method in [1,78] to establish the equivalence. Then we construct the a posteriori error estimators and design the adaptive finite element scheme using multi-meshes. We then carry out numerical tests to verify our theoretical analysis. The empirical results show that the adaptive meshes can indeed to increase computational efficiency greatly.In Chapter Three, we apply similar ideas to the finite element approximation of optimal control problem governed by quasi-parabolic integro-differential equations with the obstacle and integral type of control constraints. Since now the the state equation has gradient of yt, we need to treat the mathematical framework in different state spaces. We then establish regularity up to the second order derivatives with respect to time. Then we derive optimality conditions similarly, and examine well-postness and the finite element semi-discretization of the optimal control problems. We then derive the a priori error estimates for the approximation error in H1(0, T; H1(Ω))-norm and L2(0,T; L2(Ω))-norm for the state and the control. Finally, we carried out numerical tests to verify our theoretical estimates.In Chapter Four, we continue the investigation on optimal control governed by quasi-parabolic integro-differential equations, but now focus on adaptive finite element method. We obtain the equivalent a posteriori error estimates in L2(0, T; H1(Ω))-norm for the control constraint of the integral type. We again apply the Backwards Euler scheme to obtain the full discretization, and then derive the well-postness and the opti-mality conditions for the scheme. By introducing intermeddle variables and generating the method in [57], we also obtain the sharp upper a posteriori error estimates. We still apply the conventional bubble function method in [1,78] to establish the lower a posteriori error estimates for the equivalence. It is of course much harder now in deriving these estimates. Then we again construct the a posteriori error estimators and design the adaptive finite element scheme using multi-meshes. We then carry out numerical tests to verify our theoretical analysis. The empirical results show that the adaptive meshes can indeed to increase computational efficiency again. In Chapter Five, we examine the finite element approximation for optimal con-trol problem governed by hyperbolic integro-differential equations under the control constraints of Obstacle type and Integral type as before. We firstly give the weak for-mula for the control problem. We then derive existence and regularity of the optimal control. The methods used here are quite different from those for the parabolic integro-differential equations, and the regularity results are stronger. By following the classic ways of deriving the optimality conditions for optimal control, we derive the optimality conditions for the control problems. We then consider a semi-discrete finite element approximation for the control problem, establish its well-postness and then derive the a priori error estimates for the control and the states in L∞(0, T; H1(Ω))-norm. To this end we firstly introduce intermeddle variables and equations, and then derive the a priori error estimates by extending the techniques used in [57]. In order to derive L2(0,T; L2(Ω))-norm estimates, we also sub-divide the contact set in order to obtain sharper error estimates.Also we present numerical results at the end of Chapter1-Chapter4.The main innovations of this thesis are:1、Systematically studied the finite element approximation of the optimal con-trol problems governed by parabolic, quasi-parabolic and hyperbolic integro-differential PDEs, with existence, optimality conditions, well-postness of finite element schemes, a priori error estimates of the optimal orders, and consistent numerical test results.2、Systematically studied the adaptive finite element approximation of the optimal control problems governed by parabolic, quasi-parabolic integro-differential equations, with the equivalent a posteriori error estimates, error estimators, and well-designed adaptive finite element schemes with consistent numerical results. These computational results confirms that for a given control approximation error, the adaptive meshes can greatly reduce work load than the uniform meshes, and our posteriori error estimators are reliable and efficient in the adaptive finite element implementations. |
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