Consider a scalar conservation law with discontinuous flux (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.
Keywords: Conservation laws, discontinuous flux, attainable profiles, controllability
@article{COCV_2020__26_1_A124_0,
author = {Ancona, Fabio and Chiri, Maria Teresa},
title = {Attainable profiles for conservation laws with flux function spatially discontinuous at a single point},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2020},
publisher = {EDP Sciences},
volume = {26},
doi = {10.1051/cocv/2020044},
mrnumber = {4188826},
zbl = {1458.35263},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020044/}
}
TY - JOUR AU - Ancona, Fabio AU - Chiri, Maria Teresa TI - Attainable profiles for conservation laws with flux function spatially discontinuous at a single point JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020044/ DO - 10.1051/cocv/2020044 LA - en ID - COCV_2020__26_1_A124_0 ER -
%0 Journal Article %A Ancona, Fabio %A Chiri, Maria Teresa %T Attainable profiles for conservation laws with flux function spatially discontinuous at a single point %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020044/ %R 10.1051/cocv/2020044 %G en %F COCV_2020__26_1_A124_0
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
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