On the modeling of moving populations through set evolution equations

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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

Abstract
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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 & Continuous Dynamical Systems - A, 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. doi: 10.4171/IFB/131. Google Scholar
[2]	J.-P. Aubin, Mutational equations in metric spaces,, Set-Valued Anal., 1 (1993), 3. doi: 10.1007/BF01039289. Google Scholar
[3]	J.-P. Aubin, Mutational and Morphological Analysis,, Systems & Control: Foundations & Applications, (1999). doi: 10.1007/978-1-4612-1576-9. Google Scholar
[4]	J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory,, Grundlehren der mathematischen Wissenschaften, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar
[5]	J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Systems & Control: Foundations & Applications, (1990). Google Scholar
[6]	G. Beer, Topologies on Closed and Closed Convex Sets,, Mathematics and Its Applications, (1993). Google Scholar
[7]	W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91. doi: 10.1007/s00607-007-0240-4. Google Scholar
[8]	A. Bressan and D. Zhang, Control problems for a class of set valued evolutions,, Set-Valued Var. Anal., 20 (2012), 581. doi: 10.1007/s11228-012-0204-5. Google Scholar
[9]	P. Cannarsa and P. Cardaliaguet, Perimeter estimates for reachable sets of control systems,, J. Convex Anal., 13 (2006), 253. Google Scholar
[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. doi: 10.1051/cocv:2006002. Google Scholar
[11]	P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,, Progress in Nonlinear Differential Equations and their Applications, (2004). Google Scholar
[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. doi: 10.1016/j.jde.2011.11.003. Google Scholar
[13]	G. Colombo and K. T. Nguyen, On the structure of the minimum time function,, SIAM J. Control Optim., 48 (2010), 4776. doi: 10.1137/090774549. Google Scholar
[14]	R. M. Colombo and G. Guerra, Balance laws as quasidifferential equations in metric spaces,, in Hyperbolic Problems: Theory, (2009), 527. doi: 10.1090/psapm/067.2/2605248. Google Scholar
[15]	R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications,, Discrete Contin. Dyn. Syst., 23 (2009), 733. doi: 10.3934/dcds.2009.23.733. Google Scholar
[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. doi: 10.1007/s00332-011-9107-0. Google Scholar
[17]	R. M. Colombo, T. Lorenz and N. Pogodaev, Fixed grid method for a class of set evolution equations,, in preparation., (). Google Scholar
[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. doi: 10.1137/110854321. Google Scholar
[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. doi: 10.1137/12087791X. Google Scholar
[20]	M. C. Delfour and J.-P. Zolésio, Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization,, Second edition, (2011). doi: 10.1137/1.9780898719826. Google Scholar
[21]	H. Federer, Geometric Measure Theory,, Die Grundlehren der mathematischen Wissenschaften, (1969). Google Scholar
[22]	M. Giaquinta and G. Modica, Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables,, Springer, (2011). Google Scholar
[23]	J. Grimm and W. Grimm, Deutsche Sagen,, Second edition, (1865). Google Scholar
[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. doi: 10.1016/j.jde.2010.02.010. Google Scholar
[25]	T. Lorenz, Boundary regularity of reachable sets of control systems,, Systems Control Lett., 54 (2005), 919. doi: 10.1016/j.sysconle.2005.02.006. Google Scholar
[26]	T. Lorenz, Morphological control problems with state constraints,, SIAM J. Control Optim., 48 (2010), 5510. doi: 10.1137/090752183. Google Scholar
[27]	T. Lorenz, Mutational Analysis. A Joint Framework for Cauchy Problems in and Beyond Vector Spaces,, Lecture Notes in Mathematics, (1996). doi: 10.1007/978-3-642-12471-6. Google Scholar
[28]	S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations,, J. Math. Pures Appl. (9), 87 (2007), 601. doi: 10.1016/j.matpur.2007.04.001. Google Scholar
[29]	C. Nour, R. J. Stern and J. Takche, The union of uniform closed balls conjecture,, Control Cybernet., 38 (2009), 1525. Google Scholar
[30]	C. Nour, R. J. Stern and J. Takche, Validity of the union of uniform closed balls conjecture,, J. Convex Anal., 18 (2011), 589. Google Scholar
[31]	C. Nour and J. Takche, On the union of closed balls property,, J. Optim. Theory Appl., 155 (2012), 376. doi: 10.1007/s10957-012-0068-8. Google Scholar
[32]	B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source,, , (2012). doi: 10.1007/s00205-013-0669-x. Google Scholar
[33]	B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707. doi: 10.1007/s00205-010-0366-y. Google Scholar

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. doi: 10.4171/IFB/131. Google Scholar
[2]	J.-P. Aubin, Mutational equations in metric spaces,, Set-Valued Anal., 1 (1993), 3. doi: 10.1007/BF01039289. Google Scholar
[3]	J.-P. Aubin, Mutational and Morphological Analysis,, Systems & Control: Foundations & Applications, (1999). doi: 10.1007/978-1-4612-1576-9. Google Scholar
[4]	J.-P. Aubin and A. Cellina, Differential Inclusions. Set-valued Maps and Viability Theory,, Grundlehren der mathematischen Wissenschaften, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar
[5]	J.-P. Aubin and H. Frankowska, Set-valued Analysis,, Systems & Control: Foundations & Applications, (1990). Google Scholar
[6]	G. Beer, Topologies on Closed and Closed Convex Sets,, Mathematics and Its Applications, (1993). Google Scholar
[7]	W.-J. Beyn and J. Rieger, Numerical fixed grid methods for differential inclusions,, Computing, 81 (2007), 91. doi: 10.1007/s00607-007-0240-4. Google Scholar
[8]	A. Bressan and D. Zhang, Control problems for a class of set valued evolutions,, Set-Valued Var. Anal., 20 (2012), 581. doi: 10.1007/s11228-012-0204-5. Google Scholar
[9]	P. Cannarsa and P. Cardaliaguet, Perimeter estimates for reachable sets of control systems,, J. Convex Anal., 13 (2006), 253. Google Scholar
[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. doi: 10.1051/cocv:2006002. Google Scholar
[11]	P. Cannarsa and C. Sinestrari, Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control,, Progress in Nonlinear Differential Equations and their Applications, (2004). Google Scholar
[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. doi: 10.1016/j.jde.2011.11.003. Google Scholar
[13]	G. Colombo and K. T. Nguyen, On the structure of the minimum time function,, SIAM J. Control Optim., 48 (2010), 4776. doi: 10.1137/090774549. Google Scholar
[14]	R. M. Colombo and G. Guerra, Balance laws as quasidifferential equations in metric spaces,, in Hyperbolic Problems: Theory, (2009), 527. doi: 10.1090/psapm/067.2/2605248. Google Scholar
[15]	R. M. Colombo and G. Guerra, Differential equations in metric spaces with applications,, Discrete Contin. Dyn. Syst., 23 (2009), 733. doi: 10.3934/dcds.2009.23.733. Google Scholar
[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. doi: 10.1007/s00332-011-9107-0. Google Scholar
[17]	R. M. Colombo, T. Lorenz and N. Pogodaev, Fixed grid method for a class of set evolution equations,, in preparation., (). Google Scholar
[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. doi: 10.1137/110854321. Google Scholar
[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. doi: 10.1137/12087791X. Google Scholar
[20]	M. C. Delfour and J.-P. Zolésio, Shapes and Geometries. Metrics, Analysis, Differential Calculus, and Optimization,, Second edition, (2011). doi: 10.1137/1.9780898719826. Google Scholar
[21]	H. Federer, Geometric Measure Theory,, Die Grundlehren der mathematischen Wissenschaften, (1969). Google Scholar
[22]	M. Giaquinta and G. Modica, Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables,, Springer, (2011). Google Scholar
[23]	J. Grimm and W. Grimm, Deutsche Sagen,, Second edition, (1865). Google Scholar
[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. doi: 10.1016/j.jde.2010.02.010. Google Scholar
[25]	T. Lorenz, Boundary regularity of reachable sets of control systems,, Systems Control Lett., 54 (2005), 919. doi: 10.1016/j.sysconle.2005.02.006. Google Scholar
[26]	T. Lorenz, Morphological control problems with state constraints,, SIAM J. Control Optim., 48 (2010), 5510. doi: 10.1137/090752183. Google Scholar
[27]	T. Lorenz, Mutational Analysis. A Joint Framework for Cauchy Problems in and Beyond Vector Spaces,, Lecture Notes in Mathematics, (1996). doi: 10.1007/978-3-642-12471-6. Google Scholar
[28]	S. Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations,, J. Math. Pures Appl. (9), 87 (2007), 601. doi: 10.1016/j.matpur.2007.04.001. Google Scholar
[29]	C. Nour, R. J. Stern and J. Takche, The union of uniform closed balls conjecture,, Control Cybernet., 38 (2009), 1525. Google Scholar
[30]	C. Nour, R. J. Stern and J. Takche, Validity of the union of uniform closed balls conjecture,, J. Convex Anal., 18 (2011), 589. Google Scholar
[31]	C. Nour and J. Takche, On the union of closed balls property,, J. Optim. Theory Appl., 155 (2012), 376. doi: 10.1007/s10957-012-0068-8. Google Scholar
[32]	B. Piccoli and F. Rossi, Generalized Wasserstein distance and its application to transport equations with source,, , (2012). doi: 10.1007/s00205-013-0669-x. Google Scholar
[33]	B. Piccoli and A. Tosin, Time-evolving measures and macroscopic modeling of pedestrian flow,, Arch. Ration. Mech. Anal., 199 (2011), 707. doi: 10.1007/s00205-010-0366-y. Google Scholar

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