Geodesic
A convex analysis approach to optimal controls with switching structure for partial differential equations
Clason, Christian1  ; Ito, Kazufumi2  ; Kunisch, Karl3 
1 Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany
2 Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA
3 Institute of Mathematics and Scientific Computing, Karl-Franzens-University of Graz, Heinrichstrasse 36, 8010 Graz, Austria
ESAIM: Control, Optimisation and Calculus of Variations, Tome 22 (2016) no. 2, pp. 581-609

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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.

Reçu le :
DOI : 10.1051/cocv/2015017
Classification : 49J20, 49K52, 49K20
Keywords: Optimal control, switching control, partial differential equations, nonsmooth optimization, convexification, semi-smooth Newton method

Clason, Christian 1 ; Ito, Kazufumi 2 ; Kunisch, Karl 3

1 Faculty of Mathematics, University Duisburg-Essen, 45117 Essen, Germany
2 Department of Mathematics, North Carolina State University, Raleigh, North Carolina, USA
3 Institute of Mathematics and Scientific Computing, Karl-Franzens-University of Graz, Heinrichstrasse 36, 8010 Graz, Austria 
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     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},
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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, Tome 22 (2016) no. 2, pp. 581-609. doi: 10.1051/cocv/2015017

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