# Forschung

**Research topics**

**Optimal control and parameter identification with ODEs, DAEs and PDEs: Necessary conditions and numerical methods**

The research interests address the development and analysis of discretization methods for the numerical solution of control and state constrained optimal control problems and parameter identification problems subject to ordinary differential equations (ODEs), differential-algebraic equations (DAEs), and partial differential equations (PDEs). We are interested in both, theoretical topics and implementational issues. Specific theoretical questions address the derivation of necessary and sufficient conditions for optimal control problems subject to mixed control-state constraints and pure state constraints. Closely connected with necessary and sufficient conditions are the postoptimal estimation of adjoints and the convergence of discretized optimal control problems. An extension towards optimal control problems subject to partial differential-algebraic equations (PDAEs) is subject of our current research and provides a huge field of research for the next years.**Semismooth Newton methods**

In addition to established methods like the Lagrange-Newton method or SQP methods for the solution of optimal control problems, we investigate alternative methods. A numerically interesting approach is based on a semismooth Newton's method. This method attempts to satisfy the first order necessary conditions numerically. The idea is to transform the necessary conditions by use of so-called NCP functions into a nonlinear and nonsmooth equation, which can be solved by a nonsmooth version of Newton's method. This method converges under suitable assumptions at a superlinear or even quadratic rate.**Nonlinear programming**

The main focus is on the development of software products for small and dense as well as large-scale and sparse nonlinear optimization problems. This software is used to solve discretized optimal control problems arising in the aerospace industry for the computation of low-thrust trajectories, ascent trajectories, and satellite constellations. Several globalization strategies based on linesearch methods for merit functions and filter methods have been implemented and tested.**Mixed-integer optimal control**

For instance, the task of choosing an optimal sequence of gears in a simulation of test-drives leads to optimal control problems with mixed-integer optimization variables. Although necessary conditions for such optimal control problems are given by Pontryagin's minimum principle, it is often cumbersome to use the minimum principle to construct a solution. Alternatively, direct discretization approaches using variable time transformations are favored.**Realtime optimization and realtime optimal control**

For time critical problems algorithms are needed which are capable of providing approximations for optimal solutions in realtime. The method of choice is to perform a parametric sensitivity analysis of either the discretized or the continuous optimal control problem and to construct an approximation of the optimal perturbed control by linearization. The theoretical basis for real-time approximations is that of solution differentiability of parametric optimization problems.**Scientific computing and applications in engineering sciences and industry**

The developed methods for the solution of optimal control problems and optimization problems are applied in practice. Herein, a particular focus is on the exploitation of special structures, e.g. for mechanical multibody systems, and on the development of especially adapted solution methods, e.g. moving horizon techniques resp. model-predictive control algorithms. For instance, a moving horizon approach was implemented successfully at Volkswagen AG, Germany, for the computation of reference trajectories for an automatically driving car. Further interesting and demanding applications are the computation of reference trajectories for industrial robots or trajectory optimization in aerospace applications.