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The paper proposes a general framework to analyze control problems forconservation law models on a network. Namely we consider a general class ofjunction distribution controls and inflow controls and we establish thecompactness in $L^1$ of a class of flux-traces of solutions. We then derive theexistence of solutions for two optimization problems: (I) the maximization ofan integral functional depending on the flux-traces of solutions evaluated atpoints of the incoming and outgoing edges; (II) the minimization of the totalvariation of the optimal solutions of problem (I). Finally we provide anequivalent variational formulation of the min-max problem (II) and we discusssome numerical simulations for a junction with two incoming and two outgoingedges.

On the optimization of conservation law models at a junction with inflow and flow distribution controls