Optimal control of scalar conservation laws by on/off-switching. (4th July 2017)

• Record Type: Journal Article
• Authors: Pfaff, Sebastian; Ulbrich, Stefan
• Is Part Of: Optimization methods and software. Volume 32:Number 4(2017)
• Page Start: 904
• Page End: 939
• Publication Date: 2017-07-04
• Subjects: optimal control -- scalar conservation law -- network; 49K20 -- 35L65 -- 49J50 -- 35R05 -- 35R02; Mathematical optimization -- Periodicals; Algorithms -- Periodicals; 519.7
• DOI: 10.1080/10556788.2016.1236796
• Languages: English
• ISSNs: 1055-6788

Abstract

Abstract : This paper studies the differentiability properties of the control-to-state mapping for entropy solutions to a scalar hyperbolic conservation law on with respect to the switching times of an on/off-control. The switching times between on-modes and off-modes are the control variables of the considered optimization problem, where a general tracking-type functional is minimized. We investigate the differentiability of the reduced objective function, also in the presence of shocks. We show that the state at some observation time depends differentiably on the switching times in a generalized sense that implies total differentiability for the composition with a tracking functional. Furthermore, we present an adjoint-based formula for the gradient of the reduced objective functional with respect to the switching times.