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.
Mots clés : 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}, pages = {353--379}, publisher = {EDP-Sciences}, volume = {17}, number = {2}, year = {2011}, doi = {10.1051/cocv/2010007}, mrnumber = {2801323}, zbl = {1232.35176}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2010007/} }
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 PY - 2011 SP - 353 EP - 379 VL - 17 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010007/ DO - 10.1051/cocv/2010007 LA - en ID - COCV_2011__17_2_353_0 ER -
%0 Journal Article %A Colombo, Rinaldo M. %A Herty, Michael %A Mercier, Magali %T Control of the continuity equation with a non local flow %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 353-379 %V 17 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2010007/ %R 10.1051/cocv/2010007 %G en %F 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, Tome 17 (2011) no. 2, pp. 353-379. doi : 10.1051/cocv/2010007. http://www.numdam.org/articles/10.1051/cocv/2010007/
[1] Dynamic modeling and control of congestion-prone systems. Oper. Res. 24 (1976) 400-419. | MR | Zbl
,[2] 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 | Zbl
,[3] A model for the dynamics of large queuing networks and supply chains. SIAM J. Appl. Math. 66 (2006) 896-920. | MR | Zbl
, and ,[4] A continuum model for a re-entrant factory. Oper. Res. 54 (2006) 933-950. | Zbl
, , , and ,[5] 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 | Zbl
, and ,[6] On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete Contin. Dynam. Systems 6 (2000) 329-350. | MR | Zbl
,[7] One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32 (1998) 891-933. | MR | Zbl
and ,[8] 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 | Zbl
and ,[9] Shift-differentiability of the flow generated by a conservation law. Discrete Contin. Dynam. Systems 3 (1997) 35-58. | MR | Zbl
and ,[10] 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 | Zbl
and ,[11] Optimality conditions for solutions to hyperbolic balance laws, in Control methods in PDE-dynamical systems, Contemp. Math. 426, AMS, USA (2007) 129-152. | MR
and ,[12] A eulerian approach to the analysis of rendez-vous algorithms, in Proceedings of the IFAC World Congress (2008).
, and ,[13] On the optimization of the initial boundary value problem for a conservation law. J. Math. Analysis Appl. 291 (2004) 82-99. | MR | Zbl
and ,[14] Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28 (2005) 1553-1567. | MR | Zbl
and ,[15] Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7 (2009) 37-65. | MR | Zbl
, and ,[16] 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 | Zbl
, , and ,[17] First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci. 18 (2008) 1217-1247. | MR | Zbl
and ,[18] Conservation law constrained optimization based upon Front-Tracking. ESAIM: M2AN 40 (2006) 939-960. | Numdam | MR | Zbl
, , and ,[19] A continuum theory for the flow of pedestrians. Transportation Res. Part B 36 (2002) 507-535.
,[20] Capacity loading and release planning in work-in-progess (wip) and lead-times. J. Mfg. Oper. Mgt. 2 (1989) 105-123.
,[21] First order quasilinear equations with several independent variables. Mat. Sb. (N.S.) 81 (1970) 228-255. | MR | Zbl
,[22] Control of continuum models of production systems. IEEE Trans. Automat. Contr. (to appear).
, , and ,[23] Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc. 361 (2009) 2319-2335. | MR | Zbl
and ,[24] A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740. | MR | Zbl
,[25] Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Contr. Lett. 48 (2003) 313-328. | MR | Zbl
,[26] Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963) 1032-1066. | MR | Zbl
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