@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{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Analysis of Spatio-Temporally Sparse Optimal Control Problems of Semilinear Parabolic Equations}, series = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, volume = {23}, journal = {ESAIM: Control, Optimisation and Calculus of Variations (COCV)}, number = {1}, issn = {1292-8119}, doi = {10.1051/cocv/2015048}, pages = {263 -- 295}, abstract = {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.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Approximation of Sparse Controls in Semilinear Equations by Piecewise Linear Functions Numerische Mathematik}, series = {Numerische Mathematik}, volume = {122}, journal = {Numerische Mathematik}, number = {4}, issn = {0945-3245}, doi = {10.1007/s00211-012-0475-7}, pages = {645 -- 669}, abstract = {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.}, language = {en} } @misc{CasasHerzogWachsmuth, author = {Casas, Eduardo and Herzog, Roland and Wachsmuth, Gerd}, title = {Optimality Conditions and Error Analysis of Semilinear Elliptic Control Problems with L¹ Cost Functional}, series = {SIAM Journal on Optimization}, volume = {22}, journal = {SIAM Journal on Optimization}, number = {3}, issn = {1052-6234}, doi = {10.1137/110834366}, pages = {795 -- 820}, abstract = {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.}, language = {en} }