We consider three type time optimal control problems, mainly with point target. Firstly we discuss maximum principle and bang-bang properties in the case of Schrödinger type equations. We next describe generalizations of the approach of Seidman and Mizel for parabolic problems. Finally we go to ODE systems with bilinear control which model low Reynolds number swimming.

The HJB equation provides optimal feedback solutions to control problems. Consequently these optimal solutions are stable with respect to system perturbation.
The price to pay for this advantage is the necesessity to cope with the curse of dimensionality. Different approaches are currently be developped to overcome this difficulty. In this talk I shall address a series expansion technique for a class of bilinear optimal control problems and on the generalized HJB equation for semi-linear parabolic equations.

We consider a minimization problem over the space of Radon measures, containing a smooth objective function and the total-variation norm as the cost term. Such problems are known to promote sparse solutions, which consist of a finite sum of Dirac-delta functions in many cases. For the algorithmic solution we consider a conditional gradient method, which iteratively inserts Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear $C/k$ rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the
finite-dimensional subproblems occurring in each step of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear $C\lambda^k$ convergence rate is obtained locally. Numerical experiments on the example of a sparse deconvolution problem confirm the theoretical findings and the practical efficiency of the method.

We present results about semismooth implicit functions defined on Banach spaces. In particular, we will discuss how these results can be applied to optimal control.

We apply the augmented Lagrangian method to the convex optimization problem of the instationary variational mean field games with diffusion. The system is first discretized with space-time tensor product piecewise polynomial bases. This leads to a sequence of linear problems posed on the space-time cylinder that are 2nd order in the temporal variable and 4th order in the spatial variable. To solve these (large) linear problems with the preconditioned conjugate gradients method we propose a parameter-robust preconditioner that is based on a temporal transformation coupled with a spatial multigrid. Numerical examples illustrate the method.

An accurate and efficient numerical scheme for solving a Liouville optimal control problem in the framework of the Pontryagin's maximum principle (PMP) is presented. The Liouville equation models the time-evolution of a density function that may represent a distribution of non-interacting particles or a probability density. In this work, the purpose of the control is to maximize the measure of a target set at a given final time. In order to solve this problem, a high-order accurate conservative and positive preserving discretization scheme is investigated and a novel iterative optimization method is formulated that solves the PMP optimality condition without requiring differentiability with respect to the control variable. Results of numerical experiments are presented that demonstrate the effectiveness of the proposed solution procedure.

In this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schrödinger one, with pointwise constraints on the bilinear control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly.

This contribution presents an iterative Bregman regularization method for solving optimal control problems of linear elliptic partial differential equations 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 for the control, state and adjoint state are presented. Numerical estimates indicate that the a priori estimate is sharp.

In this talk we apply an augmented Lagrange method to solve a pointwise state constrained elliptic control problem. Due to the low regularity of the Lagrange multipliers, pointwise state constraints cause several difficulties in their numerical treatments and are therefore still challenging. By applying an augmented Lagrange method, constraints that are causing difficulties can be eliminated from the set of explicit constraints by augmentation. Here we use augmentation of the state constraint only. Using augmented Lagrange algorithms, an approximation of the Lagrange multiplier corresponding to the state constraint is obtained by an iterative update that in general does not guarantee its $L^1$- boundedness, which is needed for proving convergence results. Consequently, we develop an appropriate update rule for the penalization parameter. We state an adapted version of the augmented Lagrange algorithm that yields strong convergence of the generated iterates of the state and the control as well as weak convergence of the adjoint states to limits that satisfy the initial problem. Moreover, exploiting the uniform boundedness of the approximated augmented Lagrange multipliers weak-* convergence to the Lagrange multiplier of the original problem is proven. The theoretical results will be illustrated by numerical examples.

We discuss a first-order stochastic swarm intelligence model in the spirit of consensus formation, namely a consensus-based optimization algorithm, which may be used for the global optimization of a function in multiple dimensions. The algorithm allows for passage to the mean-field limit resulting in a nonstandard, nonlocal, degenerate, parabolic PDE. Exploiting tools from PDE analysis we provide convergence results that help to understand the asymptotic behavior of the swarm intelligence model. In fact, under some assumptions it is possible to show the convergence of the algorithm arbitrary close to the global minimum of the objective function. Further, one can obtain a convergence rate. Numerical results underline the feasibility of the approach.

This talk is concerned with the optimal control of a semilinear partial differential equation that involves a non-differentiable Nemytskii operator. We characterize the Bouligand subdifferential (or, more precisely, the Bouligand subdifferentials) of the control-to-state map of the considered optimal control problem completely and use the resulting characterization to derive a necessary optimality condition that is stronger than Clarke stationarity. Our results allow us to identify Qi's subdifferential with an appropriately defined Bouligand subdifferential and, moreover, demonstrate that the behavior of an optimal control problem governed by a non-smooth partial differential equation changes significantly when the problem is discretized.

In [1], we present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Gamma-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.

[1] Florentine Fleißner, Gamma-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach, Submitted.
arXiv preprint:1603.02822 (2016)

This work is concerned with designing optimal order multigrid preconditioners for space-time distributed control of parabolic equations. The focus is on the reduced problem resulted from eliminating state and adjoint variables from the KKT system. Earlier numerical experiments have shown that our ability to design optimal order preconditioners depends strongly on the discretization of the parabolic equation, with several natural discretizations leading to suboptimal preconditioners. Using a continuous-in-space-discontinuous-in-time Galerkin discretization we obtain the desired optimality.

PDE-constrained shape optimization problems are characterized by target functionals that depend both on the shape of a domain (the control) and on the solution of a boundary value problem formulated on that domain (the state). In industrial applications, such optimization problems are formulated to improve the performance (expressed in terms of the target functional) of an initial domain.

PDE-constrained shape optimization problems can be solved numerically by meshing the initial domain and updating iteratively the coordinates of the nodes of this mesh. Steepest descent updates can be computed with approximate shape derivatives based on finite element approximations of the state.

In this work, we present a mathematical framework that generalizes standard moving mesh methods to higher-order discretizations of both computational domains and domain updates. We parametrize shapes by applying (discretized) deformation diffeomorphisms to the initial guess, and incorporate domain deformations into the PDE solver via isoparametric finite elements.

This approach allows for arbitrarily smooth representations of shapes, reduces approximation errors, and is compatible with standard finite element software.

We continue our presentation from last year's OCIP conference and
provide new findings in our aim to design a fully monolithic solution
algorithm for phase-field fracture propagation. Phase-field fracture
consists of two nonlinearly-coupled partial differential equations and it is well known
that the underlying energy functional is non-convex and requires
sophisticated numerical algorithms. Most of them are based on
partitioned approaches and require many subiterations until convergence.
Moreover, the algorithms
must account for crack irreversibility, i.e., an inequality constraint in time.
In this talk, we show computational findings which show the
efficiency of an inexact augmented Lagrangian iteration using
an error-oriented Newton method as inner solver (the
method presented last year). Furthermore, we found findings
in which the error-oriented approach did not converge quite
well and we design a modified Newton method that includes
fixed-point iterations. This method seems for certain
examples more robust than the error-oriented approach.
New computational results will substantiate our algorithmic
techniques.