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- ItemFinite element error analysis for state-constrained optimal control of the Stokes equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Reyes, Juan Carlos de los; Meyer, Christian; Vexler, BorisAn optimal control problem for 2d and 3d Stokes equations is investigated with pointwise inequality constraints on the state and the control. The paper is concerened with the full discretization of the control problem allowing for different types of discretization of both the control and the state. For instance, piecewise linear and continuous approximations of the control are included in the present theory. Under certain assumptions on the $L^infty$-error of the finite element discretization of the state, error estimates for the control are derived which can be seen to be optimal since their order of convergence coincides with the one of the interpolation error. The assumptions of the $L^infty$-finite-element-error can be verified for different numerical settings. The theoretical results are confirmed by numerical examples.
- ItemOptimal control for the thermistor problem(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Hömberg, Dietmar; Meyer, Christian; Rehberg, Joachim; Ring, WolfgangThis paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem.
- ItemA priori error analysis for state constrained boundary control problems : Part II: Full discretization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krumbiegel, Klaus; Meyer, Christian; Rösch, ArndThis is the second of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [26] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems. The theoretical results are illustrated by numerical computations.
- ItemNumerical analysis of Lavrentiev-regularized state constrained elliptic control problems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Hinze, Michael; Meyer, ChristianIn the present work, we apply semi-discretization proposed by the first author in [13] to Lavrentiev-regularized state constrained elliptic control problems. We extend the results of [17] and prove weak convergence of the adjoint states and multipliers of the regularized problems to their counterparts of the original problem. Further, we prove error estimates for finite element discretizations of the regularized problem and investigate the overall error imposed by the finite element discretization of the regularized problem compared to the continuous solution of the original problem. Finally we present numerical results which confirm our analytical findings.
- ItemOptimal control of static plasticity with linear kinematic hardening(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Griesse, Roland; Meyer, ChristianAn optimal control problem for the static problem of infinitesimal elastoplasticity with linear kinematic hardening is considered. The variational inequality arising on the lower-level is regularized using a Yosida-type approach, and an optimal control problem for the so-called viscoplastic model is obtained. Existence of a global optimizer is proved for both the regularized and original problems, and strong convergence of the solutions is established
- ItemState-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Meyer, Christian; Yousept, IrwinWe consider a control- and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The nonlocal radiation interface condition and the pointwise state-constraints represent the particular features of this problem. To deal with the state-constraints, continuity of the state is shown which allows to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an $L^2$-neighborhood.
- ItemA priori error analysis for state constrained boundary control problems : Part I: Control discretization(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krumbiegel, Klaus; Meyer, Christian; Rösch, ArndThis is the first of two papers concerned with a state-constrained optimal control problems with boundary control, where the state constraints are only imposed in an interior subdomain. We apply the virtual control concept introduced in [20] to regularize the problem. The arising regularized optimal control problem is discretized by finite elements and linear and continuous ansatz functions for the boundary control. In the first part of the work, we investigate the errors induced by the regularization and the discretization of the boundary control. The second part deals with the error arising from discretization of the PDE. Since the state constraints only appear in an inner subdomain, the obtained order of convergence exceeds the known results in the field of a priori analysis for state-constrained problems
- ItemRegularization for optimal control problems associated to nonlinear evolution equations(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2019) Meinlschmidt, Hannes; Meyer, Christian; Rehberg, JoachimIt is well-known that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard ``calculus of variations'' proof for the existence of optimal controls. For time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space and a suitable regularization- or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacle-type in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand.
- ItemRegularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Meyer, Christian; Yousept, IrwinA state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.
- ItemStability of infinite dimensional control problems with pointwise state constraints(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Hinze, Michael; Meyer, ChristianA general class of nonlinear infinite dimensional optimization problems is considered that covers semi-linear elliptic control problems with distributed control as well as boundary control. Moreover, pointwise inequality constraints on the control and the state are incorporated. The general optimization problem is perturbed by a certain class of perturbations, and we establish convergence of local solutions of the perturbed problems to a local solution of the unperturbed optimal control problem. These class of perturbations include finite element discretization as well as data perturbation such that the theory implies convergence of finite element approximation and stability w.r.t. noisy data.