Shape and topology optimization problems for structural and fluid optimization problems are solved with a phase field method. In this approach a perimeter regularization will be approximated by a Cahn-Hiliard type energy. In the talk I will derive first order necessary conditions and discuss the sharp interface limit of the phase field method. Finally, I will present numerical computations which demonstrate that the approach can be applied for complex shape and topology optimization problems.

We study finite element approximations of Riesz representatives of shape gradients. First, we provide a general perspective on its error analysis. Then, we zero in on shape functionals constrained to elliptic boundary value problems and H1-representatives of shape gradients. We prove linear convergence in the energy norm for linear Lagrangian finite element approximations. This theoretical result is confirmed by several numerical experiments.

The velocity tracking problem for the evolutionary Navier-Stokes equations is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are discussed. The fully-discrete scheme is based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space. Provided that the time and space discretization parameters, $\tau$ and $h$ respectively, satisfy $\tau \leq Ch^2$, then $L^2$ error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.

We consider a parabolic optimal control problem:
\begin{equation} \label{eq:intro-obj}
\min_{q,u}J(q,u):= \frac{1}{2} \int_0^T \|u(t) - \widehat{u}(t)\|_{L^2(\Omega)}^2 dt
+ \frac{\alpha}{2} \int_0^T |q(t)|^2 dt
\end{equation}
subject to the second order parabolic equation
\begin{subequations} \label{eq:intro-state}
\begin{align}
u_t(t,x)-\Delta u(t,x) &= q(t)\delta_{x_0}, & (t,x) &\in I\times\Omega,\; \\
u(t,x) &= 0, & (t,x) &\in I\times\partial\Omega, \\
u(0,x) &= 0, & x &\in \Omega
\end{align}
\end{subequations}
and subject to pointwise control constraints
\begin{equation}\label{eq:control_constraints}
q_a \le q(t) \le q_b \quad \text{a.\,e. in } I.
\end{equation}
Here $I=[0,T]$, $\Omega$ is a convex polygonal or polyhedral domain, $x_0\in \Omega$ fixed, and $\delta_{x_0}$ is the Dirac delta function. The parameter $\alpha$ is assumed to be positive and the desired state $\widehat u$ fulfills $\widehat u \in L^2(I;L^\infty(\Omega))$. The control bounds $q_a,q_b \in \mathbb{R}\cup\{\pm \infty\}$ fulfill $q_a <q_b$.

To approximate the problem numerically we use the standard continuous piecewise linear approximation in space and the first order discontinuous Galerkin method in time. Despite low regularity of the state equation, we establish almost optimal $h^2+k$ convergence rate in 2D and $h+\sqrt{k}$ in 3D for the control in $L^2$ norm. I will explain the key regularity estimates and sharp a priori fully discrete global and local error estimates in $L^2([0,T]; L^\infty(\Omega))$ norms for parabolic problems. These new error estimates are essential in our analysis and rather technical, especially in 3D, and require a new technique which I will explain. Using these sharp results we improve almost twice the previously obtained error estimates in \cite{key-1}. The 2D result were published in \cite{key-2}, but 3D results are new.

\begin{thebibliography}{1}
\bibitem[1]{key-1}W. Gong, M. Hinze, Z. Zhou. A priori error analysis for finite element approximation of
parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52 (2014), 97\textendash 119.

\bibitem[2]{key-2}D. Leykekhman, B. Vexler. SOptimal a priori error estimates of parabolic optimal control
problems with pointwise control. SIAM J. Numer. Anal. 51 (2013), 2797\textendash 2821.

\end{thebibliography}

Optimal control problems for semi-linear parabolic equations with control costs involving the total bounded variation semi-norm are analysed. This choice of control cost favors optimal controls which are piecewise constant and it penalizes the number of jumps. It is an appropriate choice if a simple structure of the optimal controls is desired, which, however, is still sufficiently flexible so that good tracking properties can be maintained. Well-posedness of the optimal controls, necessary and sufficient optimality conditions, and sparsity properties of the derivatives are obtained. Convergence of a finite element approximation is analysed and numerical examples illustrating structural properties of the optimal controls are given.

This talk continues the presentation of Eduardo Casas. We explain how to obtain a full discretization of the optimal control problem under consideration and provide an algorithm to solve it numerically. We construct examples with known optimal controls and present numerical results for these and other examples to illustrate the structural properties of optimal controls as well as the theoretical results presented by Eduardo Casas.

We consider an optimal control problem governed by a phase-field fracture model. One challenge of this model problem is a non-differentiable irreversibility condition on the fracture growth, which we relax using a penalization approach. Consequently, certain boundedness properties are not a priori clear and need to be established for the regularized problem when discussing existence of a solution to the penalized fracture model. After discussing the fracture model, existence of at least one solution for the regularized optimal control problem, as well as first order optimality conditions are presented.

In this talk, we consider an optimal control problem governed by an ODE. In
order to induce the sparsity of the optimal control, we use a grouped-sparsity
approach. We derive stability estimates of the unregularized problems w.r.t.
perturbations, and this will be used to estimate the discretization error. The
theoretical results will be illustrated by numerical examples.

In this talk we discuss convergence results for finite element discretized Dirichlet control problems in polygonal domains. We investigate unconstrained as well as control constrained problems. In both cases we discretize the state and the control by piecewise linear and continuous functions. The error estimates, which we obtain, mainly depend on the size of the interior angles but also on the presence of control constraints and the structure of the underlying mesh. For instance, considering non-convex domains, the convergence rates of the discrete optimal controls in the unconstrained case can even be worse than in the control constrained case.

This talk deals with the computation of the numerical solution of boundary value problems with Neumann boundary conditions and optimal control problems with Neumann control in polygonal domains using the finite element method. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in case of smooth domains. As a remedy one can use local mesh refinement near the corners. In order to prove optimal error estimates regularity results weighted Sobolev spaces are exploited. In such a case the convergence rate of $|\ln h|^{3/2} h^2$ using piecewise linear ansatz functions can be shown for the state variable as well as the adjoint variable in the domain and the control variable on the boundary. Similar results for boundary value problems with Dirichlet boundary conditions and optimal control problems with distributed control were obtained by Th. Apel, A. R\"osch and D. Sirch (2009).

A class of infinite horizon optimal control problems involving $L^p$-type cost functionals with $0<p\leq 1$ is investigated. The existence of optimal controls is discussed for both the convex case with $p=1$ and the nonconvex case with $0<p<1$, and the sparse structural properties of the optimal controls promoted by the $L^p$-type penalties are analyzed. A dynamic programming approach is proposed to solve the problems, and numerical examples are presented to illustrate the sparse optimal controls obtained by solving Hamilton-Jacobi-Bellman equations.

We study an optimal control problem of a new model for mechanical damage processes. In general the corresponding control-to-state mapping is not differentiable. Therefore we focus on effects arising from this property. We present a new numerical approach for the simulation of the system and the optimal control problem and provide test results which illustrate our approach.

We consider rate-independent evolutions ( = hysteresis operators) with scalar input and construct Newton as well as Bouligand derivatives. We then apply this to systems where such operators and differential equations are coupled.

We discuss an optimal control problem, where the state equation is given by a reaction-diffusion system, with a reaction term that includes a hysteresis operator.

The memory effect of hysteresis complicates the proof of existence and uniqueness theorems for the state equation.
Due to the non-smoothness of the system, it is also not clear, whether or in which sense the solution operator is differentiable.

It turns out that the state equation has unique solutions and is Hadamard differentiable.
Nevertheless, some additional work has to be done to derive first order optimality conditions in form of an adjoint system.

Our main focus lies on deriving the adjoint system for the special case, when the hysteresis is given by a scalar stop operator.
We either consider a distributed control, or one which acts on the Neumann boundary part of the domain.

The minimization of a nonlinear function under convex constraints by pseudo-timestepping methods is considered. These methods arise from a time discretization of a (sub-)gradient flow. We show that in many cases the time stepping methods are equivalent to a variable metric projection type (VMPT) method, which is a generalization of the projected gradient method with variable metric in Banach space. Due to this equivalence global convergence of the time stepping methods can be shown and a sensible stopping criterion is now available. Moreover, a time adaptivity strategy can be deduced, which is based on the Armijo condition. The convergence analysis allows the time step to tend to infinity. As a numerical example we consider a phase field model, where the considered gradient flows are of Allen-Cahn and Cahn-Hilliard type. It turns out that the pseudo-timestepping is about 100 times faster using the new adaptive strategy compared to a fixed time step. In the experiment the time step indeed tends to infinity without destroying global convergence. A more elaborate choice of the variable metric including second order information leads to even better performance.