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Control of the continuity equationwith a non local flow

Published online by Cambridge University Press:  24 March 2010

Rinaldo M. Colombo ,
Michael Herty  and
Magali Mercier
Rinaldo M. Colombo
Affiliation:
Department of Mathematics, Brescia University, Via Branze 38, 25133 Brescia, Italy. rinaldo@ing.unibs.it
Michael Herty
Affiliation:
RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany.
Magali Mercier
Affiliation:
Université de Lyon, Université Lyon 1, École Centrale de Lyon, INSA de Lyon, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.
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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.

Keywords

Optimal control of the continuity equationnon-local flows
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 2 , April 2011 , pp. 353 - 379
Copyright
© EDP Sciences, SMAI, 2010

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References

Agnew, C.E., Dynamic modeling and control of congestion-prone systems. Oper. Res. 24 (1976) 400–419. CrossRef
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.
Armbruster, D., Degond, P. and Ringhofer, C., A model for the dynamics of large queuing networks and supply chains. SIAM J. Appl. Math. 66 (2006) 896–920. CrossRef
Armbruster, D., Marthaler, D.E., Ringhofer, C., Kempf, K. and Jo, T.-C., A continuum model for a re-entrant factory. Oper. Res. 54 (2006) 933–950. CrossRef
Benzoni-Gavage, S., Colombo, R.M. and Gwiazda, P., Measure valued solutions to conservation laws motivated by traffic modelling. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462 (2006) 1791–1803. CrossRef
Bianchini, S., On the shift differentiability of the flow generated by a hyperbolic system of conservation laws. Discrete Contin. Dynam. Systems 6 (2000) 329–350. CrossRef
Bouchut, F. and James, F., One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32 (1998) 891–933. CrossRef
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).
Bressan, A. and Guerra, G., Shift-differentiability of the flow generated by a conservation law. Discrete Contin. Dynam. Systems 3 (1997) 35–58.
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.
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.
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).
Colombo, R.M. and Groli, A., On the optimization of the initial boundary value problem for a conservation law. J. Math. Analysis Appl. 291 (2004) 82–99. CrossRef
Colombo, R.M. and Rosini, M.D., Pedestrian flows and non-classical shocks. Math. Methods Appl. Sci. 28 (2005) 1553–1567. CrossRef
Colombo, R.M., Mercier, M. and Rosini, M.D., Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7 (2009) 37–65. CrossRef
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).
Coscia, V. and Canavesio, C., First-order macroscopic modelling of human crowd dynamics. Math. Models Methods Appl. Sci. 18 (2008) 1217–1247. CrossRef
Gugat, M., Herty, M., Klar, A. and Leugering, G., Conservation law constrained optimization based upon Front-Tracking. ESAIM: M2AN 40 (2006) 939–960. CrossRef
Hughes, R.L., A continuum theory for the flow of pedestrians. Transportation Res. Part B 36 (2002) 507–535. CrossRef
Karmarkar, U., Capacity loading and release planning in work-in-progess (wip) and lead-times. J. Mfg. Oper. Mgt. 2 (1989) 105–123.
Kružkov, S.N., 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).
Perthame, B. and Dalibard, A.-L., Existence of solutions of the hyperbolic Keller-Segel model. Trans. Amer. Math. Soc. 361 (2009) 2319–2335. CrossRef
Ulbrich, S., A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41 (2002) 740. CrossRef
Ulbrich, S., Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Contr. Lett. 48 (2003) 313–328. CrossRef
Yudovič, V.I., Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963) 1032–1066.