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On the modeling of moving populations through set evolution equations
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 |
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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