| Abstract: |
| In the last years, starting with the seminal papers by
Lighthill and Whitham \cite{LW} and Richards \cite{richards},
there was an increasing interest
in conservation laws for the modeling of traffic flow in road networks,
mainly justified by applications.
Particularly, the problems of reducing congestions, car accidents,
and pollution have been tackled with various approaches.
We present, in this talk, a way for controlling
traffic flow, through a control function acting
at the level of junctions \cite{a-c-c-g-1}.
We introduce the concept of solution,
we show that the solution
exists and that, in some cases, the input-output map is continuous.
It is also addressed the problem of minimization of
functionals describing traffic performance indexes \cite{a-c-c-g-2}.
Finally, in the case of $1$-$1$ junctions, the structure of optimal solutions
is presented \cite{a-c-c-g-3}.
\bibitem{a-c-c-g-1} F. Ancona, A. Cesaroni, G. M. Coclite, M. Garavello.
On the optimization of conservation law models at a junction
with inflow and flow distribution controls.
SIAM J. Control Optim. 56 (2018), no. 5, pp. 3370-3403.
\bibitem{a-c-c-g-2} F. Ancona, A. Cesaroni, G. M. Coclite, M. Garavello.
On optimization of traffic flow performance for conservation
laws on networks.
Minimax Theory Appl. 6 (2021), no. 2, pp. 205-226.
\bibitem{a-c-c-g-3} F. Ancona, A. Cesaroni, G. M. Coclite, M. Garavello.
On the structure of optimal solutions of conservation laws at a
junction with one incoming and one outgoing arc.
arXiv 2507.10090.
\bibitem{LW} M. J. Lighthill, G. B. Whitham.
On kinematic waves. II. A theory of traffic flow on long crowded
roads.
Proc. Roy. Soc. London. Ser. A. 229 (1955), pp. 317-345.
\bibitem{richards} P.I. Richards.
Shock waves on the highway.
Operations Res. 4 (1956), pp. 42-51. |
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