## 49-XX CALCULUS OF VARIATIONS AND OPTIMAL CONTROL; OPTIMIZATION [See also 34H05, 34K35, 65Kxx, 90Cxx, 93-XX]

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#### Document Type

- Doctoral Thesis (3)
- Habilitation (1)
- Master's Thesis (1)
- Preprint (1)

#### Keywords

- Partial Differential Equations (2)
- Asymptotic Analysis (1)
- B-Spline (1)
- Beam models (1)
- Elasticity (1)
- Homogenization (1)
- Isogeometrische Analyse (1)
- NURBS (1)
- Numerical Analysis (1)
- Optimal Control (1)

#### Faculty / Organisational entity

Optimal control of partial differential equations is an important task in applied mathematics where it is used in order to optimize, for example, industrial or medical processes. In this thesis we investigate an optimal control problem with tracking type cost functional for the Cattaneo equation with distributed control, that is, \(\tau y_{tt} + y_t - \Delta y = u\). Our focus is on the theoretical and numerical analysis of the limit process \(\tau \to 0\) where we prove the convergence of solutions of the Cattaneo equation to solutions of the heat equation.
We start by deriving both the Cattaneo and the classical heat equation as well as introducing our notation and some functional analytic background. Afterwards, we prove the well-posedness of the Cattaneo equation for homogeneous Dirichlet boundary conditions, that is, we show the existence and uniqueness of a weak solution together with its continuous dependence on the data. We need this in the following, where we investigate the optimal control problem for the Cattaneo equation: We show the existence and uniqueness of a global minimizer for an optimal control problem with tracking type cost functional and the Cattaneo equation as a constraint. Subsequently, we do an asymptotic analysis for \(\tau \to 0\) for both the forward equation and the aforementioned optimal control problem and show that the solutions of these problems for the Cattaneo equation converge strongly to the ones for the heat equation. Finally, we investigate these problems numerically, where we examine the different behaviour of the models and also consider the limit \(\tau \to 0\), suggesting a linear convergence rate.

In this thesis we develop a shape optimization framework for isogeometric analysis in the optimize first–discretize then setting. For the discretization we use
isogeometric analysis (iga) to solve the state equation, and search optimal designs in a space of admissible b-spline or nurbs combinations. Thus a quite
general class of functions for representing optimal shapes is available. For the
gradient-descent method, the shape derivatives indicate both stopping criteria and search directions and are determined isogeometrically. The numerical treatment requires solvers for partial differential equations and optimization methods, which introduces numerical errors. The tight connection between iga and geometry representation offers new ways of refining the geometry and analysis discretization by the same means. Therefore, our main concern is to develop the optimize first framework for isogeometric shape optimization as ground work for both implementation and an error analysis. Numerical examples show that this ansatz is practical and case studies indicate that it allows local refinement.

The work consists of two parts.
In the first part an optimization problem of structures of linear elastic material with contact modeled by Robin-type boundary conditions is considered. The structures model textile-like materials and possess certain quasiperiodicity properties. The homogenization method is used to represent the structures by homogeneous elastic bodies and is essential for formulations of the effective stress and Poisson's ratio optimization problems. At the micro-level, the classical one-dimensional Euler-Bernoulli beam model extended with jump conditions at contact interfaces is used. The stress optimization problem is of a PDE-constrained optimization type, and the adjoint approach is exploited. Several numerical results are provided.
In the second part a non-linear model for simulation of textiles is proposed. The yarns are modeled by hyperelastic law and have no bending stiffness. The friction is modeled by the Capstan equation. The model is formulated as a problem with the rate-independent dissipation, and the basic continuity and convexity properties are investigated. The part ends with numerical experiments and a comparison of the results to a real measurement.

Test rig optimization
(2014)

Designing good test rigs for fatigue life tests is a common task in the auto-
motive industry. The problem to find an optimal test rig configuration and
actuator load signals can be formulated as a mathematical program. We in-
troduce a new optimization model that includes multi-criteria, discrete and
continuous aspects. At the same time we manage to avoid the necessity to
deal with the rainflow-counting (RFC) method. RFC is an algorithm, which
extracts load cycles from an irregular time signal. As a mathematical func-
tion it is non-convex and non-differentiable and, hence, makes optimization
of the test rig intractable.
The block structure of the load signals is assumed from the beginning.
It highly reduces complexity of the problem without decreasing the feasible
set. Also, we optimize with respect to the actuators’ positions, which makes
it possible to take torques into account and thus extend the feasible set. As
a result, the new model gives significantly better results, compared with the
other approaches in the test rig optimization.
Under certain conditions, the non-convex test rig problem is a union of
convex problems on cones. Numerical methods for optimization usually need
constraints and a starting point. We describe an algorithm that detects each
cone and its interior point in a polynomial time.
The test rig problem belongs to the class of bilevel programs. For every
instance of the state vector, the sum of functions has to be maximized. We
propose a new branch and bound technique that uses local maxima of every
summand.

This papers deals with the minimization of seminorms \(\|L\cdot\|\) on \(\mathbb R^n\) under the constraint of a bounded I-divergence \(D(b,H\cdot)\). The I-divergence is also known as Kullback-Leibler divergence and appears in many models in imaging science, in particular when dealing with Poisson data. Typically, \(H\) represents here, e.g., a linear blur operator and \(L\) is some discrete derivative operator. Our preference for the constrained approach over
the corresponding penalized version is based on the fact that the I-divergence of data
corrupted, e.g., by Poisson noise or multiplicative Gamma noise can be estimated by statistical methods. Our minimization technique rests upon relations between constrained and penalized convex problems and resembles the idea of Morozov's discrepancy principle.
More precisely, we propose first-order primal-dual algorithms which reduce the problem to the solution of certain proximal minimization problems in each iteration step. The most interesting of these proximal minimization problems is an I-divergence constrained least squares problem. We solve this problem by connecting it to the corresponding I-divergence
penalized least squares problem with an appropriately chosen regularization parameter. Therefore, our algorithm produces not only a sequence of vectors which converges to a minimizer of the constrained problem but also a sequence of parameters which convergences to a regularization parameter so that the penalized problem has the same solution as our constrained one. In other words, the solution of this penalized problem fulfills the I-divergence constraint. We provide the proofs which are necessary to understand
our approach and demonstrate the performance of our algorithms for different
image restoration examples.

The primary objective of this work is the development of robust, accurate and efficient simulation methods for the optimal control of mechanical systems, in particular of constrained mechanical systems as they appear in the context of multibody dynamics. The focus is on the development of new numerical methods that meet the demand of structure preservation, i.e. the approximate numerical solution inherits certain characteristic properties from the real dynamical process.
This task includes three main challenges. First of all, a kinematic description of multibody systems is required that treats rigid bodies and spatially discretised elastic structures in a uniform way and takes their interconnection by joints into account. This kinematic description must not be subject to singularities when the system performs large nonlinear dynamics. Here, a holonomically constrained formulation that completely circumvents the use of rotational parameters has proved to perform very well. The arising constrained equations of motion are suitable for an easy temporal discretisation in a structure preserving way. In the temporal discrete setting, the equations can be reduced to minimal dimension by elimination of the constraint forces. Structure preserving integration is the second important ingredient. Computational methods that are designed to inherit system specific characteristics – like consistency in energy, momentum maps or symplecticity – often show superior numerical performance regarding stability and accuracy compared to standard methods. In addition to that, they provide a more meaningful picture of the behaviour of the systems they approximate. The third step is to take the previ- ously addressed points into the context of optimal control, where differential equation and inequality constrained optimisation problems with boundary values arise. To obtain meaningful results from optimal control simulations, wherein energy expenditure or the control effort of a motion are often part of the optimisation goal, it is crucial to approxi- mate the underlying dynamics in a structure preserving way, i.e. in a way that does not numerically, thus artificially, dissipate energy and in which momentum maps change only and exactly according to the applied loads.
The excellent numerical performance of the newly developed simulation method for optimal control problems is demonstrated by various examples dealing with robotic systems and a biomotion problem. Furthermore, the method is extended to uncertain systems where the goal is to minimise a probability of failure upper bound and to problems with contacts arising for example in bipedal walking.