Optimal control of scalar conservation laws by on/off-switching

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.

Keywords:

• optimal control
• scalar conservation law
• network

AMS Subject Classification:

• 49K20
• 35L65
• 49J50
• 35R05
• 35R02

Acknowledgments

We gratefully acknowledge discussions with G. Leugering and T. I. Seidman.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors gratefully acknowledge the support of the German Research Foundation (DFG) within the Priority Program 1253 ‘Optimization with Partial Differential Equations’ under grant UL158/8-1, cf. [47]. The second author was in parts supported by the DFG within the collaborative research center TRR 154 ‘Mathematical Modeling, Simulation and Optimization Using the Example of Gas Networks’.