Attainable profiles for conservation laws with flux function spatially discontinuous at a single point
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 124.

Consider a scalar conservation law with discontinuous flux (1):

u t +f(x,u) x =0,f(x,u)=f l (u)ifx<0,f r (u)ifx>0,(1)

where u = u(x, t) is the state variable and f$$, f$$ are strictly convex maps. We study the Cauchy problem for (1) from the point of view of control theory regarding the initial datum as a control. Letting $$ denote the solution of the Cauchy problem for (1), with initial datum $$, that satisfy at x = 0 the interface entropy condition associated to a connection (A, B) (see Adimurthi, S. Mishra and G.D. Veerappa Gowda, J. Hyperbolic Differ. Equ. 2 (2005) 783–837), we analyze the family of profiles that can be attained by (1) at a given time T > 0:

$$

We provide a full characterization of $$ as a class of functions in $$ that satisfy suitable Oleǐnik-type inequalities, and that admit one-sided limits at x = 0 which satisfy specific conditions related to the interface entropy criterion. Relying on this characterisation, we establish the $$-compactness of the set of attainable profiles when the initial data $$ vary in a given class of uniformly bounded functions, taking values in closed convex sets. We also discuss some applicationsof these results to optimization problems arising in traffic flow.

DOI : 10.1051/cocv/2020044
Classification : 35L65, 35F25, 35Q93
Mots-clés : Conservation laws, discontinuous flux, attainable profiles, controllability 
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     title = {Attainable profiles for conservation laws with flux function spatially discontinuous at a single point},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
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Ancona, Fabio; Chiri, Maria Teresa. Attainable profiles for conservation laws with flux function spatially discontinuous at a single point. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 124. doi : 10.1051/cocv/2020044. http://www.numdam.org/articles/10.1051/cocv/2020044/

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