Klaus Krumbiegel

Error estimates for the Lavrentiev regularization of elliptic optimal control problems

We focus on optimal control problems governed by partial differential equations. The presence of constraints on the state provides numerical and analytical difficulties. We treat these obstacles introducing a Lavrentiev regularization. The key issue is addressed to the analytical investigation of the convergence order of solutions based on certain source representation arguments. A virtual control approach is investigated for problems with additional control constraints. Numerical experiments are presented.

A virtual control concept for state constrained optimal control problems

Abstract A linear elliptic control problem with pointwise state constraints is considered. These constraints are given in the domain. In contrast to this, the control acts only at the boundary. We propose a general concept using virtual control in this paper. The virtual control is introduced in objective, state equation, and constraints. Moreover, additional control constraints for the virtual control are investigated. An error estimate for the regularization error is derived as main result of the paper. The theory is illustrated by numerical tests.

Second Order Sufficient Optimality Conditions for Parabolic Optimal Control Problems with Pointwise State Constraints

In this paper we study optimal control problems governed by semilinear parabolic equations where the spatial dimension is two or three. Moreover, we consider pointwise constraints on the control and on the state. We formulate first order necessary and second order sufficient optimality conditions. We make use of recent results regarding elliptic regularity and apply the concept of maximal parabolic regularity to the occurring partial differential equations.

Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regularization parameter are derived.

A Priori Error Analysis for Linear Quadratic Elliptic Neumann Boundary Control Problems with Control and State Constraints

In this paper we consider a state-constrained optimal control problem with boundary control, where the state constraints are imposed only in an interior subdomain. Our goal is to derive a priori error estimates for a finite element discretization with and without additional regularization. We will show that the separation of the set where the control acts and the set where the state constraints are given improves the approximation rates significantly. The theoretical results are illustrated by numerical computations.

Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations

This paper is concerned with the discretization error analysis of semilinear Neumann boundary control problems in polygonal domains with pointwise inequality constraints on the control. The approximations of the control are piecewise constant functions. The state and adjoint state are discretized by piecewise linear finite elements. In a postprocessing step approximations of locally optimal controls of the continuous optimal control problem are constructed by the projection of the respective discrete adjoint state. Although the quality of the approximations is in general affected by corner singularities a convergence order of

SUFFICIENT OPTIMALITY CONDITIONS FOR THE MOREAU-YOSIDA-TYPE REGULARIZATION CONCEPT APPLIED TO SEMILINEAR ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH POINTWISE STATE CONSTRAINTS

h^2|\ln h|^{3/2}

Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints

h2|lnh|3/2 is proven for domains with interior angles smaller than