Optimal control problems involving hybrid binary-continuous control costs are challenging due to their lack of convexity and weak lower semicontinuity. Replacing such costs with their convex relaxation leads to a primal-dual optimality system that allows an explicit pointwise characterization and whose Moreau–Yosida regularization is amenable to a semismooth Newton method in function space. This approach is especially suited for computing switching controls for partial differential equations. In this case, the optimality gap between the original functional and its relaxation can be estimated and shown to be zero for controls with switching structure. Numerical examples illustrate the effectiveness of this approach.

DOI: 10.1051/cocv/2015017

Keywords: Optimal control, switching control, partial differential equations, nonsmooth optimization, convexification, semi-smooth Newton method

^{1}; Ito, Kazufumi

^{2}; Kunisch, Karl

^{3}

@article{COCV_2016__22_2_581_0, author = {Clason, Christian and Ito, Kazufumi and Kunisch, Karl}, title = {A convex analysis approach to optimal controls with switching structure for partial differential equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {581--609}, publisher = {EDP-Sciences}, volume = {22}, number = {2}, year = {2016}, doi = {10.1051/cocv/2015017}, mrnumber = {3491785}, zbl = {1338.49056}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2015017/} }

TY - JOUR AU - Clason, Christian AU - Ito, Kazufumi AU - Kunisch, Karl TI - A convex analysis approach to optimal controls with switching structure for partial differential equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2016 SP - 581 EP - 609 VL - 22 IS - 2 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2015017/ DO - 10.1051/cocv/2015017 LA - en ID - COCV_2016__22_2_581_0 ER -

%0 Journal Article %A Clason, Christian %A Ito, Kazufumi %A Kunisch, Karl %T A convex analysis approach to optimal controls with switching structure for partial differential equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2016 %P 581-609 %V 22 %N 2 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2015017/ %R 10.1051/cocv/2015017 %G en %F COCV_2016__22_2_581_0

Clason, Christian; Ito, Kazufumi; Kunisch, Karl. A convex analysis approach to optimal controls with switching structure for partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 22 (2016) no. 2, pp. 581-609. doi : 10.1051/cocv/2015017. http://archive.numdam.org/articles/10.1051/cocv/2015017/

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