Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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 beten on-modes and off-modes are the control variables of the considered optimization problem, where a general tracking-type functional is minimized.  investigate the differentiability of the reduced objective function, also in the presence of shocks.  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,  present an adjoint-based formula for the gradient of the reduced objective functional with respect to the switching times.
['Sebastian Pfaff', 'Stefan Ulbrich']
I am writing a research paper on a idea Optimal control of scalar conservation laws by on/off-switching
[{"from": "human", "value": "I am writing a research paper on a idea Optimal control of scalar conservation laws by on/off-switching"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  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 beten on-modes and off-modes are the control variables of the considered optimization problem, where a general tracking-type functional is minimized.  investigate the differentiability of the reduced objective function, also in the presence of shocks.  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,  present an adjoint-based formula for the gradient of the reduced objective functional with respect to the switching times."}]