# Control of the continuity equation with a non local flow

Rinaldo M. Colombo; Michael Herty; Magali Mercier

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 2, page 353-379
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topColombo, Rinaldo M., Herty, Michael, and Mercier, Magali. "Control of the continuity equation with a non local flow." ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011): 353-379. <http://eudml.org/doc/276335>.

@article{Colombo2011,

abstract = {
This paper focuses on the analytical properties of the
solutions to the continuity equation with non local flow. Our
driving examples are a supply chain model and an equation for the
description of pedestrian flows. To this aim, we prove the well
posedness of weak entropy solutions in a class of equations
comprising these models. Then, under further regularity conditions,
we prove the differentiability of solutions with respect to the
initial datum and characterize this derivative. A necessary
condition for the optimality of suitable integral functionals then
follows.
},

author = {Colombo, Rinaldo M., Herty, Michael, Mercier, Magali},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal control of the continuity equation; non-local flows; optimal control of the continuity equation; entropy solutions; pedestrian traffic model},

language = {eng},

month = {5},

number = {2},

pages = {353-379},

publisher = {EDP Sciences},

title = {Control of the continuity equation with a non local flow},

url = {http://eudml.org/doc/276335},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Colombo, Rinaldo M.

AU - Herty, Michael

AU - Mercier, Magali

TI - Control of the continuity equation with a non local flow

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2011/5//

PB - EDP Sciences

VL - 17

IS - 2

SP - 353

EP - 379

AB -
This paper focuses on the analytical properties of the
solutions to the continuity equation with non local flow. Our
driving examples are a supply chain model and an equation for the
description of pedestrian flows. To this aim, we prove the well
posedness of weak entropy solutions in a class of equations
comprising these models. Then, under further regularity conditions,
we prove the differentiability of solutions with respect to the
initial datum and characterize this derivative. A necessary
condition for the optimality of suitable integral functionals then
follows.

LA - eng

KW - Optimal control of the continuity equation; non-local flows; optimal control of the continuity equation; entropy solutions; pedestrian traffic model

UR - http://eudml.org/doc/276335

ER -

## References

top- C.E. Agnew, Dynamic modeling and control of congestion-prone systems. Oper. Res.24 (1976) 400–419. Zbl0336.90011
- L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of variations and nonlinear partial differential equations, Lecture Notes in Math.1927, Springer, Berlin, Germany (2008) 1–41.
- D. Armbruster, P. Degond and C. Ringhofer, A model for the dynamics of large queuing networks and supply chains. SIAM J. Appl. Math.66 (2006) 896–920. Zbl1107.90002
- D. Armbruster, D.E. Marthaler, C. Ringhofer, K. Kempf and T.-C. Jo, A continuum model for a re-entrant factory. Oper. Res.54 (2006) 933–950. Zbl1167.90477
- S. Benzoni-Gavage, R.M. Colombo and P. Gwiazda, Measure valued solutions to conservation laws motivated by traffic modelling. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.462 (2006) 1791–1803. Zbl1149.35386
- S. Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete Contin. Dynam. Systems6 (2000) 329–350. Zbl1018.35051
- F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal.32 (1998) 891–933. Zbl0989.35130
- F. Bouchut and F. James, Differentiability with respect to initial data for a scalar conservation law, in Hyperbolic problems: theory, numerics, applications, Internat. Ser. Numer. Math., Birkhäuser, Basel, Switzerland (1999). Zbl0928.35097
- A. Bressan and G. Guerra, Shift-differentiability of the flow generated by a conservation law. Discrete Contin. Dynam. Systems3 (1997) 35–58. Zbl0948.35077
- A. Bressan and M. Lewicka, Shift differentials of maps in BV spaces, in Nonlinear theory of generalized functions (Vienna, 1997), Res. Notes Math.401, Chapman & Hall/CRC, Boca Raton, USA (1999) 47–61. Zbl0935.46025
- A. Bressan and W. Shen, Optimality conditions for solutions to hyperbolic balance laws, in Control methods in PDE-dynamical systems, Contemp. Math.426, AMS, USA (2007) 129–152. Zbl1354.49002
- C. Canuto, F. Fagnani and P. Tilli, A eulerian approach to the analysis of rendez-vous algorithms, in Proceedings of the IFAC World Congress (2008).
- R.M. Colombo and A. Groli, On the optimization of the initial boundary value problem for a conservation law. J. Math. Analysis Appl.291 (2004) 82–99. Zbl1041.35037
- R.M. Colombo and M.D. Rosini, Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci.28 (2005) 1553–1567. Zbl1108.90016
- R.M. Colombo, M. Mercier and M.D. Rosini, Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci.7 (2009) 37–65. Zbl1183.35197
- R.M. Colombo, G. Facchi, G. Maternini and M.D. Rosini, On the continuum modeling of crowds, in Hyperbolic Problems: Theory, Numerics, Applications67, Proceedings of Symposia in Applied Mathematics, E. Tadmor, J.-G. Liu and A.E. Tzavaras Eds., American Mathematical Society, Providence, USA (2009). Zbl1190.35149
- V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci.18 (2008) 1217–1247. Zbl1171.91018
- M. Gugat, M. Herty, A. Klar and G. Leugering, Conservation law constrained optimization based upon Front-Tracking. ESAIM: M2AN40 (2006) 939–960. Zbl1116.65079
- R.L. Hughes, A continuum theory for the flow of pedestrians. Transportation Res. Part B36 (2002) 507–535.
- U. Karmarkar, Capacity loading and release planning in work-in-progess (wip) and lead-times. J. Mfg. Oper. Mgt.2 (1989) 105–123.
- S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.)81 (1970) 228–255.
- M. Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems. IEEE Trans. Automat. Contr. (to appear).
- B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc.361 (2009) 2319–2335. Zbl1180.35343
- S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim.41 (2002) 740. Zbl1019.49026
- S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Contr. Lett.48 (2003) 313–328. Zbl1157.49306
- V.I. Yudovič, Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz.3 (1963) 1032–1066.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.