On the structure of optimal solutions of conservation laws at a junction with one incoming and one outgoing arc

We consider a min-max problem for strictly concave conservation laws on a 1-1 network, with inflow controls acting at the junction. We investigate the minimization problem for a functional measuring the total variation of the flow of the solutions at the node, among those solutions that maximize the time integral of the flux. To formulate this problem we establish a regularity result showing that the total variation of the boundary-flux of the solution of an initial-boundary value problem is controlled by the total variation of the initial datum and of the flux of the boundary datum. In the case the initial datum is monotone, we show that the flux of the entropy weak solution at the node provides an optimal inflow control for this min-max problem. We also exhibit two prototype examples showing that, in the case where the initial datum is not monotone, the flux of the entropy weak solution is no more optimal.