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The paper proposes a general framework to analyze control problems for conservation law models on a network. Namely, we consider a general class of junction distribution controls and inflow controls and we establish the compactness in L¹ of a class of flux-traces of solutions. We then derive the existence of solutions for two optimization problems: (I) the maximization of an integral functional depending on the flux-traces of solutions evaluated at points of the incoming and outgoing edges; (II) the minimization of the total variation of the optimal solutions of problem (I). Finally we provide an equivalent variational formulation of the min-max problem (II) and we discuss some numerical simulations for a junction with two incoming and two outgoing edges. [ABSTRACT FROM AUTHOR]

Abstract: Of concern is the nonlinear uniformly parabolic problem for and ; the last equation holds on the boundary ∂Ω. Here is a real, hermitian, uniformly positive definite matrix; with ; ; , is the Laplace–Beltrami operator on the boundary, and is the conormal derivative of u with respect to A: and everything is sufficiently regular. The solution of this wellposed problem depends continuously on the ingredients of the problem, namely, . This is shown using semigroup methods in [G.M. Coclite, A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian, Semigroup Forum 77 (1) (2008) 101–108]. Here we prove explicit stability estimates of the solution u with respect to the coefficients , β, γ, q, and the initial condition f. Moreover we cover the singular case of a problem with which is approximated by problems with positive q. [Copyright &y& Elsevier]

Abstract: Of concern is the uniformly parabolic problem for and . Here is a real, hermitian, uniformly positive definite matrix; with ; and is the conormal derivative of u with respect to A: and everything is sufficiently regular. The solution of this well-posed problem depends continuously on the ingredients of the problem, namely, . This is shown using semigroup methods in [G.M. Coclite, A. Favini, G.R. Goldstein, J.A. Goldstein, S. Romanelli, Continuous dependence on the boundary parameters for the Wentzell Laplacian, Semigroup Forum, in press]. More precisely, if we have a sequence of such problems with solutions , and if , , etc. in a suitable sense, then , the solution of the limiting problem. The abstract analysis associated with operator semigroup theory gives this conclusion, but no rate of convergence. Determining how fast the convergence of the solutions is requires detailed estimates. Such estimates are provided in this paper. [Copyright &y& Elsevier]