TY - GEN A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - Optimal control in first-order Sobolev spaces with inequality constraints T2 - Computational Optimization and Applications N2 - In this paper, an elliptic optimal control problem with controls from H¹(Ω) which have to satisfy standard box constraints is considered. Thus, Lagrange multipliers associated with the box constraints are, in general, elements of H¹(Ω)* as long as the lower and upper bound belong to H¹(Ω) as well. If these bounds possess less regularity, the overall existence of a Lagrange multiplier is not even guaranteed. In order to avoid the direct solution of a not necessarily available KKT system, a penalty method is suggested which finds the minimizer of the control-constrained problem. Its convergence properties are analyzed. Furthermore, some numerical strategies for the computation of optimal solutions are suggested and illustrated. KW - Control constraints KW - Optimal control KW - Optimality conditions KW - Penalty method KW - Semismooth Newton method Y1 - 2019 U6 - https://doi.org/10.1007/s10589-018-0053-8 SN - 0926-6003 SN - 1573-2894 VL - 72 IS - 3 SP - 797 EP - 826 ER - TY - GEN A1 - Clason, Christian A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - Optimal control problems with control complementarity constraints: existence results, optimality conditions, and a penalty method T2 - Optimization Methods and Software N2 - A special class of optimal control problems with complementarity constraints on the control functions is studied. It is shown that such problems possess optimal solutions whenever the underlying control space is a first-order Sobolev space. After deriving necessary optimality conditions of strong stationarity-type, a penalty method based on the Fischer–Burmeister function is suggested and its theoretical properties are analyzed. Finally, the numerical treatment of the problem is discussed and results of computational experiments are presented. KW - Fischer–Burmeister function KW - mathematical problems with complementarity constraints KW - optimal control KW - optimality conditions KW - penalty method Y1 - 2020 U6 - https://doi.org/10.1080/10556788.2019.1604705 SN - 1055-6788 SN - 1029-4937 VL - 35 IS - 1 SP - 142 EP - 170 ER - TY - GEN A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - On an optimal control problem with gradient constraints T2 - Optimization N2 - Usually, control functions in control-constrained optimal control are chosen from a Lebesgue space. This choice, however, makes it impossible to postulate additional conditions on the control function's slope which is practically relevant in some situations. In order to overcome this disadvantage, a natural assumption would be to demand at least first-order Sobolev regularity for control functions. The present paper is devoted to the study of an elliptic optimal control problem whose control function is chosen from a Sobolev space and has to satisfy additional equality constraints on its weak gradient. Noting that the associated Karush–Kuhn–Tucker conditions do not provide a necessary optimality condition for the underlying optimal control problem in general, one cannot simply solve the problem of interest by considering the system of first-order optimality conditions. Instead a penalization procedure with strong convergence properties for the computational solution is suggested and its computational implementation is studied in detail. Particularly, some essential difficulties arising from the gradient constraints which do not appear in standard optimal control are discussed. KW - Control gradient constraints KW - enforcement phenomena in FEM KW - optimal control KW - optimality conditions KW - penalty method Y1 - 2020 U6 - https://doi.org/10.1080/02331934.2019.1604707 SN - 0233-1934 SN - 1029-4945 VL - 69 IS - 3 SP - 519 EP - 551 ER - TY - GEN A1 - Clason, Christian A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - A penalization scheme for the numerical solution of optimal control problems with control complementarity constraints T2 - PAMM Proceedings in Applied Mathematics and Mechanics N2 - We suggest a simple penalty method for the numerical solution of optimal control problems with control complementarity constraints which is based on the famous Fischer–Burmeister function. The distinct advantage of this approach is the resulting smoothness of the penalty term and the unconstrainedness of the associated penalized surrogate problems. A numerical example from elliptic PDE control is presented as well. Y1 - 2019 U6 - https://doi.org/10.1002/pamm.201900122 SN - 1617-7061 VL - 19 IS - 1 SP - 1 EP - 2 PB - Wiley-VCH GmbH CY - Weinheim ER - TY - GEN A1 - Deng, Yu A1 - Mehlitz, Patrick A1 - Prüfert, Uwe T1 - Coupled versus decoupled penalization of control complementarity constraints T2 - ESAIM: Control, Optimisation and Calculus of Variations (COCV) N2 - This paper deals with the numerical solution of optimal control problems with control complementarity constraints. For that purpose, we suggest the use of several penalty methods which differ with respect to the handling of the complementarity constraint which is either penalized as a whole with the aid of NCP-functions or decoupled in such a way that non-negativity constraints as well as the equilibrium condition are penalized individually. We first present general global and local convergence results which cover several different penalty schemes before two decoupled methods which are based on a classical ℓ1- and ℓ2-penalty term, respectively, are investigated in more detail. Afterwards, the numerical implementation of these penalty methods is discussed. Based on some examples, where the optimal boundary control of a parabolic partial differential equation is considered, some quantitative properties of the resulting algorithms are compared. KW - Complementarity constraints KW - optimal control KW - parabolic PDE KW - penalty method Y1 - 2021 U6 - https://doi.org/10.1051/cocv/2021022 SN - 1292-8119 SN - 1262-3377 VL - 27 SP - 1 EP - 31 ER -