@misc{CasasWachsmuthWachsmuth, author = {Casas, Eduardo and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-Order Analysis and Numerical Approximation for Bang-Bang Bilinear Control Problems}, series = {SIAM Journal on Control and Optimization (SICON)}, volume = {56}, journal = {SIAM Journal on Control and Optimization (SICON)}, number = {6}, issn = {0363-0129}, doi = {10.1137/17M1139953}, pages = {4203 -- 4227}, abstract = {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.}, language = {en} } @misc{CasasWachsmuthWachsmuth, author = {Casas, Eduardo and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Sufficient Second-Order Conditions for Bang-Bang Control Problems}, series = {SIAM Journal on Control and Optimization (SICON)}, volume = {55}, journal = {SIAM Journal on Control and Optimization (SICON)}, number = {5}, issn = {1095-7138}, doi = {10.1137/16M1099674}, pages = {3066 -- 3090}, abstract = {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)\$.}, language = {en} } @misc{StefanelliWachsmuthWachsmuth, author = {Stefanelli, Ulisse and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Optimal control of a rate-independent evolution equation via viscous regularization}, series = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)}, volume = {10}, journal = {Discrete and Continuous Dynamical Systems - Series S (DCDS-S)}, number = {6}, issn = {1937-1632}, doi = {10.3934/dcdss.2017076}, pages = {1467 -- 1485}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints}, series = {Control and Cybernetics}, volume = {40}, journal = {Control and Cybernetics}, number = {4}, issn = {0324-8569}, pages = {1125 -- 1158}, abstract = {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.}, language = {en} } @misc{GeigerWachsmuthWachsmuth, author = {Geiger, Tobias and Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Optimal control of ODEs with state suprema}, series = {Mathematical Control and Related Fields}, volume = {11}, journal = {Mathematical Control and Related Fields}, number = {3}, issn = {2156-8472}, doi = {10.3934/mcrf.2021012}, pages = {555 -- 578}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Gerd and Wachsmuth, Daniel}, title = {Convergence and Regularization Results for Optimal Control Problems with Sparsity Functional}, series = {ESAIM: Control, Optimisation and Calculus of Variations}, volume = {17}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, number = {3}, issn = {1292-8119}, doi = {10.1051/cocv/2010027}, pages = {858 -- 886}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-order conditions for non-uniformly convex integrands: quadratic growth in L^1}, series = {arXiv}, journal = {arXiv}, pages = {40}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {How not to discretize the control}, series = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, volume = {16}, journal = {Special Issue:Joint 87th Annual Meeting of the International Association of Applied Mathematics and Mechanics (GAMM) and Deutsche Mathematiker-Vereinigung (DMV)}, number = {1}, address = {Braunschweig}, doi = {10.1002/pamm.201610385}, pages = {793 -- 794}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Necessary conditions for convergence rates of regularizations of optimal control problems}, series = {System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers}, journal = {System Modeling and Optimization : 25th IFIP TC 7 Conference, CSMO 2011, Berlin, Germany, September 12-16, 2011, Revised Selected Papers}, pages = {145 -- 154}, abstract = {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.}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹}, series = {Journal of Nonsmooth Analysis and Optimizaton}, volume = {3}, journal = {Journal of Nonsmooth Analysis and Optimizaton}, issn = {2700-7448}, doi = {10.46298/jnsao-2022-8733}, pages = {1 -- 36}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces}, series = {arXiv}, journal = {arXiv}, pages = {1 -- 11}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces}, series = {Journal of Convex Analysis}, volume = {30}, journal = {Journal of Convex Analysis}, number = {4}, pages = {1319 -- 1328}, language = {en} } @misc{WachsmuthWachsmuth, author = {Wachsmuth, Daniel and Wachsmuth, Gerd}, title = {Continuous differentiability of the signum function and Newton's method for bang-bang control}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, address = {Ithaca, NY}, doi = {10.48550/arXiv.2509.24829}, pages = {1 -- 28}, abstract = {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 Fr{\´e}chet differentiable between appropriate function spaces. Numerical experiments show the applicability of the resulting method.}, language = {en} } @misc{WachsmuthWalter, author = {Wachsmuth, Gerd and Walter, Daniel}, title = {No-gap second-order conditions for minimization problems in spaces of measures}, series = {arXiv}, journal = {arXiv}, doi = {10.48550/arXiv.2403.12001}, pages = {36}, language = {en} } @misc{WachsmuthWalter, author = {Wachsmuth, Gerd and Walter, Daniel}, title = {Proximal gradient methods in Banach spaces}, series = {arXiv}, journal = {arXiv}, publisher = {arXiv}, doi = {10.48550/arXiv.2509.24685}, pages = {1 -- 39}, abstract = {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.}, language = {en} }