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January  2015, 35(1): 73-98. doi: 10.3934/dcds.2015.35.73

On the modeling of moving populations through set evolution equations

Rinaldo M. Colombo 1, , Thomas Lorenz 2, and  Nikolay I. Pogodaev 3,

1. 

INdAM Unit, University of Brescia, Via Branze 38, 25123 Brescia, Italy
2. 

RheinMain University of Applied Sciences, Kurt-Schumacher-Ring 18, 65197 Wiesbaden, Germany
3. 

Institute for System Dynamics and Control Theory, 134 Lermontova st., 664033 Irkutsk, Russian Federation

Received  December 2013 Revised  March 2014 Published  August 2014

We introduce a class of set evolution equations that can be used to describe population's movements as well as various instances of individual-population interactions. Optimal control/management problems can be formalized and tackled in this framework. A rigorous analytical structure is established and the basic well posedness results are obtained. Several examples show possible applications and their numerical integrations display possible qualitative behaviors of solutions.
Keywords: evolution of measures, Set evolution equations, confinement problems, differential inclusions, agents-population interactions..
Mathematics Subject Classification: 34A60, 93B03, 34G2.
Citation: Rinaldo M. Colombo, Thomas Lorenz, Nikolay I. Pogodaev. On the modeling of moving populations through set evolution equations. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 73-98. doi: 10.3934/dcds.2015.35.73
References:
[1]

O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces Free Bound., 7 (2005), 415-434. doi: 10.4171/IFB/131.

[2]

J.-P. Aubin, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), 3-46. doi: 10.1007/BF01039289.

[3]

J.-P. Aubin, Mutational and Morphological Analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1576-9.

[4]

J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory, Grundlehren der mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.

[5]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston Inc., Boston, MA, 1990.

[6]

G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications, 268, Kluwer Academic Publishers Group, Dordrecht, 1993.

[7]

W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions, Computing, 81 (2007), 91-106. doi: 10.1007/s00607-007-0240-4.

[8]

A. Bressan and D. Zhang, Control problems for a class of set valued evolutions, Set-Valued Var. Anal., 20 (2012), 581-601. doi: 10.1007/s11228-012-0204-5.

[9]

P. Cannarsa and P. Cardaliaguet, Perimeter estimates for reachable sets of control systems, J. Convex Anal., 13 (2006), 253-267.

[10]

P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM Control Optim. Calc. Var., 12 (2006), 350-370. doi: 10.1051/cocv:2006002.

[11]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.

[12]

J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003.

[13]

G. Colombo and K. T. Nguyen, On the structure of the minimum time function, SIAM J. Control Optim., 48 (2010), 4776-4814. doi: 10.1137/090774549.

[14]

R. M. Colombo and G. Guerra, Balance laws as quasidifferential equations in metric spaces, in Hyperbolic Problems: Theory, Numerics and Applications (eds. A. E. T. Eitan Tadmor and Jian-Guo Liu), Proc. Sympos. Appl. Math. Amer. Math. Soc., 67, Providence, RI, 2009, 527-536. doi: 10.1090/psapm/067.2/2605248.

[15]

R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753. doi: 10.3934/dcds.2009.23.733.

[16]

R. M. Colombo and M. Lécureux-Mercier, An analytical framework to describe the interactions between individuals and a continuum, J. Nonlinear Sci., 22 (2012), 39-61. doi: 10.1007/s00332-011-9107-0.

[17]

R. M. Colombo, T. Lorenz and N. Pogodaev, Fixed grid method for a class of set evolution equations,, in preparation., (). 

[18]

R. M. Colombo and N. Pogodaev, Confinement strategies in a model for the interaction between individuals and a continuum, SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770. doi: 10.1137/110854321.

[19]

R. M. Colombo and N. Pogodaev, On the control of moving sets: Positive and negative confinement results, SIAM J. Control Optim., 51 (2013), 380-401. doi: 10.1137/12087791X.

[20]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization, Second edition, {Advances in Design and Control}, 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719826.

[21]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.

[22]

M. Giaquinta and G. Modica, Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables, Springer, 2011.

[23]

J. Grimm and W. Grimm, Deutsche Sagen, Second edition, Nicolaische Verlagsbuchhandlung, Berlin, 1865.

[24]

P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010.

[25]

T. Lorenz, Boundary regularity of reachable sets of control systems, Systems Control Lett., 54 (2005), 919-924. doi: 10.1016/j.sysconle.2005.02.006.

[26]

T. Lorenz, Morphological control problems with state constraints, SIAM J. Control Optim., 48 (2010), 5510-5546. doi: 10.1137/090752183.

[27]

T. Lorenz, Mutational Analysis. A Joint Framework for Cauchy Problems in and Beyond Vector Spaces, Lecture Notes in Mathematics, 1996, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12471-6.

[28]

S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, J. Math. Pures Appl. (9), 87 (2007), 601-626. doi: 10.1016/j.matpur.2007.04.001.

[29]

C. Nour, R. J. Stern and J. Takche, The union of uniform closed balls conjecture, Control Cybernet., 38 (2009), 1525-1534.

[30]

C. Nour, R. J. Stern and J. Takche, Validity of the union of uniform closed balls conjecture, J. Convex Anal., 18 (2011), 589-600.

[31]

C. Nour and J. Takche, On the union of closed balls property, J. Optim. Theory Appl., 155 (2012), 376-389. doi: 10.1007/s10957-012-0068-8.

[32]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, arXiv:1206.3219v1, 2012. doi: 10.1007/s00205-013-0669-x.

[33]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

show all references

References:
[1]

O. Alvarez, P. Cardaliaguet and R. Monneau, Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces Free Bound., 7 (2005), 415-434. doi: 10.4171/IFB/131.

[2]

J.-P. Aubin, Mutational equations in metric spaces, Set-Valued Anal., 1 (1993), 3-46. doi: 10.1007/BF01039289.

[3]

J.-P. Aubin, Mutational and Morphological Analysis, Systems & Control: Foundations & Applications, Birkhäuser Boston Inc., Boston, MA, 1999. doi: 10.1007/978-1-4612-1576-9.

[4]

J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory, Grundlehren der mathematischen Wissenschaften, 264, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.

[5]

J.-P. Aubin and H. Frankowska, Set-valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston Inc., Boston, MA, 1990.

[6]

G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications, 268, Kluwer Academic Publishers Group, Dordrecht, 1993.

[7]

W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions, Computing, 81 (2007), 91-106. doi: 10.1007/s00607-007-0240-4.

[8]

A. Bressan and D. Zhang, Control problems for a class of set valued evolutions, Set-Valued Var. Anal., 20 (2012), 581-601. doi: 10.1007/s11228-012-0204-5.

[9]

P. Cannarsa and P. Cardaliaguet, Perimeter estimates for reachable sets of control systems, J. Convex Anal., 13 (2006), 253-267.

[10]

P. Cannarsa and H. Frankowska, Interior sphere property of attainable sets and time optimal control problems, ESAIM Control Optim. Calc. Var., 12 (2006), 350-370. doi: 10.1051/cocv:2006002.

[11]

P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control, Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston Inc., Boston, MA, 2004.

[12]

J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differential Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003.

[13]

G. Colombo and K. T. Nguyen, On the structure of the minimum time function, SIAM J. Control Optim., 48 (2010), 4776-4814. doi: 10.1137/090774549.

[14]

R. M. Colombo and G. Guerra, Balance laws as quasidifferential equations in metric spaces, in Hyperbolic Problems: Theory, Numerics and Applications (eds. A. E. T. Eitan Tadmor and Jian-Guo Liu), Proc. Sympos. Appl. Math. Amer. Math. Soc., 67, Providence, RI, 2009, 527-536. doi: 10.1090/psapm/067.2/2605248.

[15]

R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications, Discrete Contin. Dyn. Syst., 23 (2009), 733-753. doi: 10.3934/dcds.2009.23.733.

[16]

R. M. Colombo and M. Lécureux-Mercier, An analytical framework to describe the interactions between individuals and a continuum, J. Nonlinear Sci., 22 (2012), 39-61. doi: 10.1007/s00332-011-9107-0.

[17]

R. M. Colombo, T. Lorenz and N. Pogodaev, Fixed grid method for a class of set evolution equations,, in preparation., (). 

[18]

R. M. Colombo and N. Pogodaev, Confinement strategies in a model for the interaction between individuals and a continuum, SIAM J. Appl. Dyn. Syst., 11 (2012), 741-770. doi: 10.1137/110854321.

[19]

R. M. Colombo and N. Pogodaev, On the control of moving sets: Positive and negative confinement results, SIAM J. Control Optim., 51 (2013), 380-401. doi: 10.1137/12087791X.

[20]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization, Second edition, {Advances in Design and Control}, 22, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. doi: 10.1137/1.9780898719826.

[21]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.

[22]

M. Giaquinta and G. Modica, Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables, Springer, 2011.

[23]

J. Grimm and W. Grimm, Deutsche Sagen, Second edition, Nicolaische Verlagsbuchhandlung, Berlin, 1865.

[24]

P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010.

[25]

T. Lorenz, Boundary regularity of reachable sets of control systems, Systems Control Lett., 54 (2005), 919-924. doi: 10.1016/j.sysconle.2005.02.006.

[26]

T. Lorenz, Morphological control problems with state constraints, SIAM J. Control Optim., 48 (2010), 5510-5546. doi: 10.1137/090752183.

[27]

T. Lorenz, Mutational Analysis. A Joint Framework for Cauchy Problems in and Beyond Vector Spaces, Lecture Notes in Mathematics, 1996, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-12471-6.

[28]

S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, J. Math. Pures Appl. (9), 87 (2007), 601-626. doi: 10.1016/j.matpur.2007.04.001.

[29]

C. Nour, R. J. Stern and J. Takche, The union of uniform closed balls conjecture, Control Cybernet., 38 (2009), 1525-1534.

[30]

C. Nour, R. J. Stern and J. Takche, Validity of the union of uniform closed balls conjecture, J. Convex Anal., 18 (2011), 589-600.

[31]

C. Nour and J. Takche, On the union of closed balls property, J. Optim. Theory Appl., 155 (2012), 376-389. doi: 10.1007/s10957-012-0068-8.

[32]

B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source, arXiv:1206.3219v1, 2012. doi: 10.1007/s00205-013-0669-x.

[33]

B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow, Arch. Ration. Mech. Anal., 199 (2011), 707-738. doi: 10.1007/s00205-010-0366-y.

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