TY - GEN A1 - Casas, Eduardo A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems T2 - SIAM Journal on Control and Optimization (SICON) N2 - 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. KW - bang-bang control KW - bilinear controls KW - second-order conditions KW - sufficient optimality conditions KW - error analysis Y1 - 2018 U6 - https://doi.org/10.1137/17M1139953 SN - 0363-0129 SN - 1095-7138 VL - 56 IS - 6 SP - 4203 EP - 4227 ER - TY - GEN A1 - Casas, Eduardo A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Sufficient Second-Order Conditions for Bang-Bang Control Problems T2 - SIAM Journal on Control and Optimization (SICON) N2 - We provide sufficient optimality conditions for optimal control problems with bang-bang controls. Building on a structural assumption on the adjoint state, we additionally need a weak second-order condition. This second-order condition is formulated with functions from an extended critical cone, and it is equivalent to a formulation posed on measures supported on the set where the adjoint state vanishes. If our sufficient optimality condition is satisfied, we obtain a local quadratic growth condition in $L^1(\Omega)$. KW - bang-bang control KW - second-order conditions KW - sufficient optimality KW - critical cone Y1 - 2017 U6 - https://doi.org/10.1137/16M1099674 SN - 1095-7138 SN - 0363-0129 VL - 55 IS - 5 SP - 3066 EP - 3090 ER - TY - GEN A1 - Stefanelli, Ulisse A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Optimal control of a rate-independent evolution equation via viscous regularization T2 - Discrete and Continuous Dynamical Systems - Series S (DCDS-S) N2 - 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. KW - Rate-independent system KW - optimal control KW - necessary optimality conditions Y1 - 2017 U6 - https://doi.org/10.3934/dcdss.2017076 SN - 1937-1632 SN - 1937-1179 VL - 10 IS - 6 SP - 1467 EP - 1485 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints T2 - Control and Cybernetics N2 - In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L²(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assumption on the active sets. Moreover, we introduce a strategy to choose the regularization parameter in dependence of the noise level. We prove convergence of this parameter choice rule with optimal order. KW - source condition KW - discrepancy principle KW - non-smooth optimization KW - convex constraints KW - sparsity KW - regularization error estimates Y1 - 2011 SN - 0324-8569 VL - 40 IS - 4 SP - 1125 EP - 1158 ER - TY - GEN A1 - Geiger, Tobias A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Optimal control of ODEs with state suprema T2 - Mathematical Control and Related Fields N2 - 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 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. KW - Functional differential equations KW - differential equations with state suprema KW - optimality conditions KW - maximum principle KW - LogIntExp Y1 - 2021 U6 - https://doi.org/10.3934/mcrf.2021012 SN - 2156-8472 SN - 2156-8499 VL - 11 IS - 3 SP - 555 EP - 578 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Wachsmuth, Daniel T1 - Convergence and Regularization Results for Optimal Control Problems with Sparsity Functional T2 - ESAIM: Control, Optimisation and Calculus of Variations N2 - Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments. KW - Non-smooth optimization KW - sparsity KW - regularization error estimates KW - finite elements KW - discretization error estimates Y1 - 2011 U6 - https://doi.org/10.1051/cocv/2010027 SN - 1292-8119 SN - 1262-3377 VL - 17 IS - 3 SP - 858 EP - 886 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-order conditions for non-uniformly convex integrands: quadratic growth in L^1 T2 - arXiv N2 - We study no-gap second-order optimality conditions for a non-uniformly convex and non-smooth integral functional. The integral functional is extended to the space of measures. The obtained second-order derivatives contain integrals on lower-dimensional manifolds. The proofs utilize the convex pre-conjugate, which is an integral functional on the space of continuous functions. Application to non-smooth optimal control problems are given. KW - Second-order optimality conditions KW - twice epi-differentiability KW - bang-bang control KW - sparse control Y1 - 2021 UR - https://arxiv.org/pdf/2111.10238.pdf ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - How not to discretize the control T2 - Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker‐Vereinigung (DMV) N2 - 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. Y1 - 2016 UR - https://arxiv.org/abs/1603.07099 U6 - https://doi.org/10.1002/pamm.201610385 VL - 16 IS - 1 SP - 793 EP - 794 CY - Braunschweig ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Necessary conditions for convergence rates of regularizations of optimal control problems T2 - System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers N2 - We investigate the Tikhonov regularization of control constrained optimal control problems. We use a specialized source condition in combination with a condition on the active sets. In the case of high convergence rates, these conditions are necessary and sufficient. KW - optimal control problem, inequality constraints, Tikhonov regularization, source condition Y1 - 2012 UR - https://link.springer.com/chapter/10.1007/978-3-642-36062-6_15 SP - 145 EP - 154 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹ T2 - Journal of Nonsmooth Analysis and Optimizaton Y1 - 2022 U6 - https://doi.org/10.46298/jnsao-2022-8733 SN - 2700-7448 VL - 3 SP - 1 EP - 36 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - arXiv Y1 - 2022 UR - https://arxiv.org/abs/2204.00282 SP - 1 EP - 11 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces T2 - Journal of Convex Analysis Y1 - 2023 UR - https://www.heldermann.de/JCA/JCA30/JCA304/jca30059.htm VL - 30 IS - 4 SP - 1319 EP - 1328 ER - TY - GEN A1 - Wachsmuth, Daniel A1 - Wachsmuth, Gerd T1 - Continuous differentiability of the signum function and Newton's method for bang-bang control T2 - arXiv N2 - We investigate bang-bang control problems and the possibility to apply Newton's method to solve such kind of problems numerically. To this end, we show that the signum function is Fréchet differentiable between appropriate function spaces. Numerical experiments show the applicability of the resulting method. KW - Bang-bang controls KW - Newton method KW - Signum function Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2509.24829 SP - 1 EP - 28 PB - arXiv CY - Ithaca, NY ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Walter, Daniel T1 - No-gap second-order conditions for minimization problems in spaces of measures T2 - arXiv Y1 - 2024 UR - https://arxiv.org/abs/2403.12001 U6 - https://doi.org/10.48550/arXiv.2403.12001 ER - TY - GEN A1 - Wachsmuth, Gerd A1 - Walter, Daniel T1 - Proximal gradient methods in Banach spaces T2 - arXiv N2 - Proximal gradient methods are a popular tool for the solution of structured, nonsmooth minimization problems. In this work, we investigate an extension of the former to general Banach spaces and provide worst-case convergence rates for, both, convex and nonconvex, problem instances. Moreover, assuming additional regularity properties of stationary points, linear rates of convergence are derived. The theoretical results are illustrated for bang-bang type optimal control problems with partial differential equations which we study in the space of Radon measures. An efficient implementation of the resulting -proximal gradient method is given and its performance is compared to standard -proximal gradient as well as Frank-Wolfe methods. The paper is complemented by discussing the relationship among different regularity properties as well as by providing a novel characterization of the Polyak--Łojasiewicz--Kurdyka property via second-order conditions involving weak* second subderivatives. Y1 - 2025 U6 - https://doi.org/10.48550/arXiv.2509.24685 SP - 1 EP - 39 PB - arXiv ER -