Control of the continuity equation with a non local flow
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, p. 353-379

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.

DOI : https://doi.org/10.1051/cocv/2010007
Classification:  35L65,  49K20,  93C20
Keywords: optimal control of the continuity equation, non-local flows
@article{COCV_2011__17_2_353_0,
     author = {Colombo, Rinaldo M. and Herty, Michael and Mercier, Magali},
     title = {Control of the continuity equation with a non local flow},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {2},
     year = {2011},
     pages = {353-379},
     doi = {10.1051/cocv/2010007},
     zbl = {1232.35176},
     mrnumber = {2801323},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_2_353_0}
}
Colombo, Rinaldo M.; Herty, Michael; Mercier, Magali. Control of the continuity equation with a non local flow. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 2, pp. 353-379. doi : 10.1051/cocv/2010007. http://www.numdam.org/item/COCV_2011__17_2_353_0/

[1] C.E. Agnew, Dynamic modeling and control of congestion-prone systems. Oper. Res. 24 (1976) 400-419. | MR 432275 | Zbl 0336.90011

[2] 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. | MR 2408257 | Zbl 1159.35041

[3] 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. | MR 2216725 | Zbl 1107.90002

[4] 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. | Zbl 1167.90477

[5] 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. | MR 2272056 | Zbl 1149.35386

[6] S. Bianchini, On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete Contin. Dynam. Systems 6 (2000) 329-350. | MR 1739381 | Zbl 1018.35051

[7] F. Bouchut and F. James, One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32 (1998) 891-933. | MR 1618393 | Zbl 0989.35130

[8] 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). | MR 1715739 | Zbl 0928.35097

[9] A. Bressan and G. Guerra, Shift-differentiability of the flow generated by a conservation law. Discrete Contin. Dynam. Systems 3 (1997) 35-58. | MR 1422538 | Zbl 0948.35077

[10] 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. | MR 1699858 | Zbl 0935.46025

[11] 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. | MR 2311524 | Zbl pre05194890

[12] 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).

[13] 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. | MR 2034059 | Zbl 1041.35037

[14] R.M. Colombo and M.D. Rosini, Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28 (2005) 1553-1567. | MR 2158218 | Zbl 1108.90016

[15] 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. | MR 2512832 | Zbl 1183.35197

[16] R.M. Colombo, G. Facchi, G. Maternini and M.D. Rosini, On the continuum modeling of crowds, in Hyperbolic Problems: Theory, Numerics, Applications 67, Proceedings of Symposia in Applied Mathematics, E. Tadmor, J.-G. Liu and A.E. Tzavaras Eds., American Mathematical Society, Providence, USA (2009). | MR 2605247 | Zbl 1190.35149

[17] V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci. 18 (2008) 1217-1247. | MR 2438214 | Zbl 1171.91018

[18] M. Gugat, M. Herty, A. Klar and G. Leugering, Conservation law constrained optimization based upon Front-Tracking. ESAIM: M2AN 40 (2006) 939-960. | Numdam | MR 2293253 | Zbl 1116.65079

[19] R.L. Hughes, A continuum theory for the flow of pedestrians. Transportation Res. Part B 36 (2002) 507-535.

[20] U. Karmarkar, Capacity loading and release planning in work-in-progess (wip) and lead-times. J. Mfg. Oper. Mgt. 2 (1989) 105-123.

[21] S.N. Kružkov, First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255. | MR 267257 | Zbl 0215.16203

[22] M. Marca, D. Armbruster, M. Herty and C. Ringhofer, Control of continuum models of production systems. IEEE Trans. Automat. Contr. (to appear).

[23] B. Perthame and A.-L. Dalibard, Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc. 361 (2009) 2319-2335. | MR 2471920 | Zbl 1180.35343

[24] S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740. | MR 1939870 | Zbl 1019.49026

[25] S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Contr. Lett. 48 (2003) 313-328. | MR 2020647 | Zbl 1157.49306

[26] V.I. Yudovič, Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963) 1032-1066. | MR 158189 | Zbl 0147.44303