Daniel Wachsmuth

On Time Optimal Control of the Wave Equation and Its Numerical Realization as Parametric Optimization Problem

Time optimal control of the wave equation is analyzed on the basis of a regularized formulation which is considered as a bilevel optimization problem. For the lower level problems, which are constrained optimal control problems for the wave equation, a detailed sensitivity analysis is carried out. Further a semismooth Newton method is analyzed and proved to converge locally superlinearly. Numerical examples are provided.

Path-following for optimal control of stationary variational inequalities

Moreau-Yosida based approximation techniques for optimal control of variational inequalities are investigated. Properties of the path generated by solutions to the regularized equations are analyzed. Combined with a semi-smooth Newton method for the regularized problems these lead to an efficient numerical technique.

Optimal Control of Planar Flow of Incompressible Non-Newtonian Fluids

We consider an optimal control problem for the evolutionary flow of incompressible non-Newtonian fluids in a two-dimensional domain. The existence of optimal controls is proven. Furthermore, we investigate first-order necessary as well as second-order sufficient optimality conditions. The analysis relies on new results providing solutions with bounded gradients for the flow equations.

Numerical Verification of Optimality Conditions

A class of optimal control problems for a semilinear elliptic partial differential equation with control constraints is considered. It is well known that sufficient second-order conditions ensure the stability of optimal solutions, and the convergence of numerical methods. Otherwise, such conditions are very difficult to verify (analytically or numerically). We will propose a new approach as follows: Starting with a numerical solution for a fixed mesh we will show the existence of a local minimizer of the continuous problem. Moreover, we will prove that this minimizer satisfies the sufficient second-order conditions.

An iterative Bregman regularization method for optimal control problems with inequality constraints

We study an iterative regularization method of optimal control problems with control constraints. The regularization method is based on generalized Bregman distances. We provide convergence results under a combination of a source condition and a regularity condition on the active sets. We do not assume attainability of the desired state. Furthermore, a priori regularization error estimates are obtained.

Boundary concentrated finite elements for optimal boundary control problems of elliptic PDEs

We investigate the discretization of optimal boundary control problems for elliptic equations on two-dimensional polygonal domains by the boundary concentrated finite element method. We prove that the discretization error

Numerical Study of the Optimization of Separation Control

\|u^{*}-u_{h}^{*}\|_{L^{2}(\Gamma)}

Newton Methods for the Optimal Control of Closed Quantum Spin Systems

decreases like N−1, where N is the total number of unknowns. This makes the proposed method favorable in comparison to the h-version of the finite element method, where the discretization error behaves like N−3/4 for uniform meshes. Moreover, we present an algorithm that solves the discretized problem in almost optimal complexity. The paper is complemented with numerical results.

Necessary Conditions for Convergence Rates of Regularizations of Optimal Control Problems

The concept of active flow control is applied to the steady flow around a NACA4412 and to the unsteady flow around a generic high-lift configuration in order to delay separation. To the former steady suction upstream of the detachment position is applied. In a series of computations the suction angle β is varied and the main flow features are analyzed. A gradient descent method and an adjoint-based method are successfully used to optimize β. For the unsteady case periodic blowing and suction is employed to control the separation. Various calculations are conducted to obtain the dependency of the lift on the amplitude and frequency of the perturbation and the amplitude is optimized with the gradient descent method.

How to Check Numerically the Sufficient Optimality Conditions for Infinite-dimensional Optimization Problems

An efficient and robust computational framework for solving closed quantum spin optimal-control and exact-controllability problems with control constraints is presented. Closed spin systems are of fundamental importance in modern quantum technologies such as nuclear magnetic resonance (NMR) spectroscopy, quantum imaging, and quantum computing. These systems are modeled by the Liouville--von Neumann master (LvNM) equation describing the time evolution of the density operator representing the state of the system. A unifying setting is provided to discuss optimal-control and exact-controllability results. Different controllability results for the LvNM model are given, and necessary optimality conditions for the LvNM control problems are analyzed. Existence and regularity of optimal controls are proved. The computational framework is based on matrix-free reduced-Hessian semismooth Krylov--Newton schemes for solving optimal-control problems of the LvNM equation in a real vector space rotating-frame representat...