arXiv subjects

Daniel Wachsmuth

Publications and source records attributed to Daniel Wachsmuth.

35 records · Page 2Linked to original sources

arXiv2019

Optimal control of ODEs with state suprema

We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a novel LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.

math.OC↗

arXiv2018

A Lagrange Multiplier Method for Semilinear Elliptic State Constrained Optimal Control Problems

In this paper we apply an augmented Lagrange method to a class of semilinear elliptic optimal control problems with pointwise state constraints. We show strong convergence of subsequences of the primal variables to a local solution of the original problem as well as weak convergence of the adjoint states and weak* convergence of the multipliers associated to the state constraint. Moreover, we show existence of stationary points in arbitrary small neighborhoods of local solutions of the original problem. Additionally, various numerical results are presented.

math.OC↗

arXiv2018

Subgradients of Marginal Functions in Parametric Control Problems of Partial Differential Equations

The paper studies generalized differentiability properties of the marginal function of parametric optimal control problems of semilinear elliptic partial differential equations. We establish upper estimates for the regular and the limiting subgradients of the marginal function. With some additional assumptions, we show that the solution map of the perturbed optimal control problems has local upper Hölderian selections. This leads to a lower estimate for the regular subdifferential of the marginal function.

math.OC↗

arXiv2018

An Augmented Lagrangian Method for Optimization Problems in Banach Spaces

We propose a variant of the classical augmented Lagrangian method for constrained optimization problems in Banach spaces. Our theoretical framework does not require any convexity or second-order assumptions and allows the treatment of inequality constraints with infinite-dimensional image space. Moreover, we discuss the convergence properties of our algorithm with regard to feasibility, global optimality, and KKT conditions. Some numerical results are given to illustrate the practical viability of the method.

math.OC↗

arXiv2018

Iterative hard-thresholding applied to optimal control problems with $L^0(Ω)$ control cost

We investigate the hard-thresholding method applied to optimal control problems with $L^0(Ω)$ control cost, which penalizes the measure of the support of the control. As the underlying measure space is non-atomic, arguments of convergence proofs in $l^2$ or $\mathbb R^n$ cannot be applied. Nevertheless, we prove the surprising property that the values of the objective functional are lower semicontinuous along the iterates. That is, the function value in a weak limit point is less or equal than the lim-inf of the function values along the iterates. Under a compactness assumption, we can prove that weak limit points are strong limit points, which enables us to prove certain stationarity conditions for the limit points. Numerical experiments are carried out, which show the performance of the method. These indicates that the method is robust with respect to discretization. In addition, we show that solutions obtained by the thresholding algorithm are superior to solutions of $L^1(Ω)$-regularized problems.

math.OC↗

arXiv2018

Optimal control of an evolution equation with non-smooth dissipation

In the present work we study the optimal control of an evolution equation with non-smooth dissipation. The solution mapping of this system is non-smooth and hence the analysis is quite challenging. Our approach is to regularize the dissipation via approximation by a smooth function. We derive optimality conditions for the corresponding smooth optimal control problem. Then we drive the regularization parameter to zero and obtain necessary optimality conditions for the original non-smooth problem. However, in this process we lose regularity of the adjoint variables.

math.OC↗

arXiv2017

Full Stability for a Class of Control Problems of Semilinear Elliptic Partial Differential Equations

We investigate full Lipschitzian and full Hölderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where all the cost functional, the state equation, and the admissible control set of the control problems undergo perturbations. We establish explicit characterizations of both Lipschitzian and Hölderian full stability for the class of control problems. We show that for this class of control problems the two full stability properties are equivalent. In particular, the two properties are always equivalent in general when the admissible control set is an arbitrary fixed nonempty, closed, and convex set.

math.OC↗

arXiv2017

Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems

We consider bilinear optimal control problems, whose objective functionals do not depend on the controls. Hence, bang-bang solutions will appear. We investigate sufficient second-order conditions for bang-bang controls, which guarantee local quadratic growth of the objective functional in $L^1$. In addition, we prove that for controls that are not bang-bang, no such growth can be expected. Finally, we study the finite-element discretization, and prove error estimates of bang-bang controls in $L^1$-norms.

math.OC↗

arXiv2017

Stability for Bang-Bang Control Problems of Partial Differential Equations

In this paper, we investigate solution stability for control problems of partial differential equations with the cost functional not involving the usual quadratic term for the control. We first establish a sufficient optimality condition for the optimal control problems with bang-bang controls. Then we obtain criteria for solution stability for the optimal control problems of bang-bang controls under linear perturbations. We prove Hölder stability of optimal controls in $L^1$.

math.OC↗

arXiv2017

On the switching behavior of sparse optimal controls for the one-dimensional heat equation

An optimal boundary control problem for the one-dimensional heat equation is considered. The objective functional includes a standard quadratic terminal observation, a Tikhonov regularization term with regularization parameter $ν$, and the $L^1$-norm of the control that accounts for sparsity. The switching structure of the optimal control is discussed for $ν\ge 0$. Under natural assumptions, it is shown that the set of switching points of the optimal control is countable with the final time as only possible accumulation point. The convergence of switching points is investigated for $ν\searrow 0$.

math.OC↗

arXiv2017

Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations

In this article, we consider the Tikhonov regularization of an optimal control problem of semilinear partial differential equations with box constraints on the control. We derive a-priori regularization error estimates for the control under suitable conditions. These conditions comprise second-order sufficient optimality conditions as well as regularity conditions on the control, which consists of a source condition and a condition on the active sets. In addition, we show that these conditions are necessary for convergence rates under certain conditions. We also consider sparse optimal control problems and derive regularization error estimates for them. Numerical experiments underline the theoretical findings.

math.OC↗

arXiv2017

Functional error estimators for the adaptive discretization of inverse problems

So-called functional error estimators provide a valuable tool for reliably estimating the discretization error for a sum of two convex functions. We apply this concept to Tikhonov regularization for the solution of inverse problems for partial differential equations, not only for quadratic Hilbert space regularization terms but also for nonsmooth Banach space penalties. Examples include the measure-space norm (i.e., sparsity regularization) or the indicator function of an $L^\infty$ ball (i.e., Ivanov regularization). The error estimators can be written in terms of residuals in the optimality system that can then be estimated by conventional techniques, thus leading to explicit estimators. This is illustrated by means of an elliptic inverse source problem with the above-mentioned penalties, and numerical results are provided for the case of sparsity regularization.

math.NA↗

arXiv2016

Optimal control of a rate-independent evolution equation via viscous regularization

We study the optimal control of a rate-independent system that is driven by a convex, quadratic energy. Since the associated solution mapping is non-smooth, the analysis of such control problems is challenging. In order to derive optimality conditions, we study the regularization of the problem via a smoothing of the dissipation potential and via the addition of some viscosity. The resulting regularized optimal control problem is analyzed. By driving the regularization parameter to zero, we obtain a necessary optimality condition for the original, non-smooth problem.

math.OC↗

arXiv2016

The regularity of the positive part of functions in $L^2(I; H^1(Ω)) \cap H^1(I; H^1(Ω)^*)$ with applications to parabolic equations

Let $u\in L^2(I; H^1(Ω))$ with $\partial_t u\in L^2(I; H^1(Ω)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\partial_t u^+ \not\in L^1(I; H^1(Ω)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations.

math.AP↗

arXiv2016

How not to discretize the control

In this short note, we address the discretization of optimal control problems with higher order polynomials. We develop a necessary and sufficient condition to ensure that weak limits of discrete feasible controls are feasible for the original problem. We show by means of a simple counterexample that a naive discretization by higher order polynomials can lead to non-feasible limits of sequences of discrete solutions.

math.NA↗

arXiv2016

An iterative Bregman regularization method for optimal control problems with inequality constraints

We study an iterative regularization method of optimal control problems 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 are obtained.

math.OC↗

arXiv2010

Polynomial integration on regions defined by a triangle and a conic

We present an efficient solution to the following problem, of relevance in a numerical optimization scheme: calculation of integrals of the type \[\iint_{T \cap \{f\ge0\}} ϕ_1ϕ_2 \, dx\,dy\] for quadratic polynomials $f,ϕ_1,ϕ_2$ on a plane triangle $T$. The naive approach would involve consideration of the many possible shapes of $T\cap\{f\geq0\}$ (possibly after a convenient transformation) and parameterizing its border, in order to integrate the variables separately. Our solution involves partitioning the triangle into smaller triangles on which integration is much simpler.

cs.SC↗