Axel Kröner

Semismooth Newton Methods for Optimal Control of the Wave Equation with Control Constraints

In this paper optimal control problems governed by the wave equation with control constraints are analyzed. Three types of control action are considered: distributed control, Neumann boundary control, and Dirichlet control, and proper functional analytic settings for them are discussed. For treating inequality constraints, semismooth Newton methods are discussed and their convergence properties are investigated. In the case of distributed and Neumann control, superlinear convergence is shown. For Dirichlet boundary control, superlinear convergence is proved for a strongly damped wave equation. For numerical realization, a space-time finite element discretization is discussed. Numerical examples illustrate the results.

Adaptive Finite Element Methods For Optimal Control Of Second Order Hyperbolic Equations

Abstract In this paper we consider a posteriori error estimates for space-time finite element discretizations for optimal control of hyperbolic partial dierential equations of second order. It is an extension of Meidner and Vexler (2007), where optimal control problems of parabolic equations are analyzed. The state equation is formulated as a first order system in time and a posteriori error estimates are derived separating the in uences of time, space, and control discretization. Using this information the accuracy of the solution is improved by local mesh refinement. Numerical examples are presented. Finally, we analyze the conservation of energy of the homogeneous wave equation with respect to dynamically in time changing spatial meshes.

Remarks on the Internal Exponential Stabilization to a Nonstationary Solution for 1D Burgers Equations

The feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the control, a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examples are presented illustrating the stabilizing effect of the feedback control and suggesting that the existence of an estimate in the general case analogous to that in the particular one is plausible.

Local Minimization Algorithms for Dynamic Programming Equations

The numerical realization of the dynamic programming principle for continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls. This minimization is often performed by comparison over a finite number of elements of the control set. In this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively. The considered class of equations includes nonsmooth control problems with

Semi-Smooth Newton Methods for Optimal Control of the Dynamical Lamé System with Control Constraints

\ell_1

A minimum effort optimal control problem for the wave equation

-penalization which lead to sparse controls.

Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost

Optimal control problems governed by the dynamical Lamé system with additional constraints on the controls are analyzed. Different types of control action are considered: distributed, Neumann boundary and Dirichlet boundary control. To treat the inequality control constraints semi-smooth Newton methods are applied and their convergence is analyzed. Although semi-smooth Newton methods are widely studied in the context of PDE-constrained optimization little has been done in the context of the dynamical Lamé system. The novelty of the article is the proof that in case of distributed and Neumann boundary control the Newton method converges superlinearly. In case of Dirichlet control superlinear convergence is shown for a strongly damped Lamé system. The results are an extension of [14], where optimal control problems of the classical wave equation are considered. The control problems are discretized by finite elements and numerical examples are presented.

Semismooth Newton Methods for an Optimal Boundary Control Problem of Wave Equations

A minimum effort optimal control problem for the undamped wave equation is considered which involves L∞-control costs. Since the problem is non-differentiable a regularized problem is introduced. Uniqueness of the solution of the regularized problem is proven and the convergence of the regularized solutions is analyzed. Further, a semi-smooth Newton method is formulated to solve the regularized problems and its superlinear convergence is shown. Thereby special attention has to be paid to the well-posedness of the Newton iteration. Numerical examples confirm the theoretical results.

A priori error estimates for elliptic optimal control problems with a bilinear state equation

An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order HJB equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. A convergent semi-Lagrangian scheme is presented in detail.

Reduced-order minimum time control of advection-reaction-diffusion systems via dynamic programming.

In this paper optimal Dirichlet boundary control problems governed by the wave equation and the strongly damped wave equation with control constraints are analyzed. For treating inequality constraints semismooth Newton methods are discussed and their convergence properties are investigated. For numerical realization a space-time finite element discretization is introduced. Numerical examples illustrate the results.