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 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - Optimal control problems with semilinear parabolic state equations are considered. The objective features one out of three different terms promoting various spatio-temporal sparsity patterns of the control variable. For each problem, first-order necessary optimality conditions, as well as second-order necessary and sufficient optimality conditions are proved. The analysis includes the case in which the objective does not contain the squared norm of the control. KW - Optimal control KW - directional sparsity KW - second-order optimality conditions KW - semilinear parabolic equations Y1 - 2017 U6 - https://doi.org/10.1051/cocv/2015048 SN - 1292-8119 SN - 1262-3377 VL - 23 IS - 1 SP - 263 EP - 295 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Approximation of Sparse Controls in Semilinear Equations by Piecewise Linear Functions Numerische Mathematik T2 - Numerische Mathematik N2 - Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. A priori finite element error estimates for piecewise linear discretizations for the control and the state are proved. These are obtained by a new technique based on an appropriate discretization of the objective function. Numerical experiments confirm the convergence rates. Y1 - 2012 U6 - https://doi.org/10.1007/s00211-012-0475-7 SN - 0945-3245 SN - 0029-599X VL - 122 IS - 4 SP - 645 EP - 669 ER - TY - GEN A1 - Casas, Eduardo A1 - Herzog, Roland A1 - Wachsmuth, Gerd T1 - Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L¹ Cost Functional T2 - SIAM Journal on Optimization N2 - Semilinear elliptic optimal control problems involving the L¹ norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discretization of the problem in the sense of [M. Hinze, Comput. Optim. Appl., 30 (2005), pp. 45–61] are also obtained. Numerical experiments confirm the convergence rates. KW - optimal control of partial differential equations KW - nondifferentiable objective KW - sparse controls KW - finite element discretization KW - a priori error estimates Y1 - 2012 U6 - https://doi.org/10.1137/110834366 SN - 1052-6234 SN - 1095-7189 VL - 22 IS - 3 SP - 795 EP - 820 ER -