On the Optimization of Conservation Law Models at a Junction with Inflow and Flow Distribution Controls

Abstract

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 L 1 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.

On the global controllability of scalar conservation laws with boundary and source controls

“…Control problems for conservation laws arise in many different applications including: vehicular traffic models [2,3,15,19], oil reservoir simulation and sedimentation models [10], supply chain [28,32], gas dynamics [25]. In practice a time dependent source control can be viewed as a control parameter acting on the flux function of the conservation law letting vary its flux capacity.…”

Attainable profiles for conservation laws with flux function spatially discontinuous at a single point

“…We point out that a further step in the research direction pursued in this paper is the characterization of the traces of admissible solutions at the the flux-discontinuity interface as well as the analysis of the reachable set when one fixes the initial data and considers such traces as control parameters (cfr. [3] within the network setting), which is the object of the forthcoming paper [5].…”

On the Global Controllability of Scalar Conservation Laws with Boundary and Source Controls

“…Control problems for conservation laws arise in many different applications, including vehicular traffic models [2,3,13,17], oil reservoir simulation and sedimentation models [8], supply chain [25,28], and gas dynamics [22]. In practice a time dependent source control can be viewed as a control parameter acting on the flux function of the conservation law letting vary its flux capacity.…”

“…2 , u \in [0, 1] . (5.7) Then, in connection with the set I 3,1 = [2 3 , 1], I 3,2 = [ 1 3 ,2 3 ] (see Figure5), by a direct computation we find[| f 3 | ] I3,1 = [| f 3 | ] I3,2 =On the other hand, we have \| f \prime \prime 3 \| C 0 ([0,1]) = | f \prime \prime 3 (0)| = 4, and thus[| f3| ] I 3,1 \| f \prime \prime 3 \| C 0 ([0,1]) = [| f3| ] I 3,2 \| f \prime \prime 3 \| C 0 ([0,1])\approx 0.055.…”