# Optimization with Partial Differential Equations

The solution of partial differential equations has been in the focus of numerical analysis for many decades as many processes in engineering and science are described by a PDE or by systems of PDEs. In the past, the solution of the PDE, i.e., forward problem, has been at the heart of many research projects. Recently with advances in algorithms and computing technology, the solution of so-called optimal control problems with respect to PDE constraints has become increasingly popular. The goal in this field is to find the parameters/setup for a particular PDE or systems of PDEs such that an objective function is minimized. The CSC group focuses on two categories of optimal control problems.

### Open loop Control and Efficient Solution of Linear Saddle Point Systems

The first category is the so-called open loop control problem where the system input is computed purely on the basis of a-priori knowledge. Here, we focus on the efficient solution of the optimization problem with state-of-the-art numerical techniques. The task of solving optimal control problems is challenging as most realistic problems are posed in three dimensions and the discretization, often done with finite elements, results in many degrees of freedom. At the heart of the problem typically lies the solution of the first order optimality conditions or Karush–Kuhn–Tucker (KKT) conditions and the corresponding linear system in saddle point form. For realistic scenarios it is not possible to use techniques based on factorizations to solve this problem. Hence iterative Krylov subspace solvers have to be employed. These are usually only feasible if some form of preconditioning is applied; a technique that solves a modified, simpler problem. We derived preconditioners that take the structure of the KKT system and the underlying infinite-dimensional problem into account. For steady and unsteady problems we constructed efficient approximations to the Schur-complement of the saddle point problem, which involves the forward and the adjoint discretized PDE. In the case of unsteady problems we showed that multilevel techniques approximating only one matrix provide sufficient approximations for the Schur-complement and that independence with respect to the mesh-parameter is given. For a variety of problems, such as the (time-periodic) heat equation or the unsteady Stokes equation, we illustrated that only crude approximations to the discretized PDE suffice to obtain accurate results to the optimal control problem.

Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well.

## Internal Projects

## Concluded Projects

### Closed Loop Control and Solution of Large Scale Matrix Equations

The second category of control problems studied within the research group is the one of regulator problems. These problems are described by the minimization of an objective function over a possibly infinite time horizon incorporating a feedback loop, i.e., the incorporation of certain output (observed quantities) of the current system behavior into the control input. Again, one is interested in the minimization of a functional subject to unsteady, possibly nonlinear, PDEs. In general the PDEs are rewritten as an infinite dimensional evolution equation, or Cauchy problem. In the case of a linear PDE the optimization problem can then be posed as a linear quadratic regulator (LQR) problem that we solve via a system theoretic approach leading to algebraic or differential Riccati equations. For nonlinear PDEs this approach must be embedded in a model predictive control (MPC) scheme, where local linearizations around working points (or working trajectories) are employed. In any case after semi-discretization using the method of lines, a very large system of ordinary differential equations (ODEs) and related very large matrix equations are obtained. Our special focus in these approaches is to prove the numerical feasibility of such matrix equation based approaches in the PDE context.

Matrix equations of the one kind or the other are a central tool in all kinds of applications. In optimal control the linear quadratic control problem features a feedback solution that is determined via the solution of an algebraic Riccati equation. The system Gramians of a linear time invariant dynamical system are the solutions of two adjoint Lyapunov equations. Riccati, Lyapunov and Sylvester equations play an important role in different model order reduction techniques for continuous time linear dynamical systems. They all have discrete time counterparts as e.g. the well known Stein equations. Krylov subspace and eigenvalue methods can be related to certain Sylvester equations, as well.