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Optimal synchronization problem for a multi-agent system
1. | Department of Mathematical Sciences, Rutgers University -Camden, 311 N. 5th Street Camden, NJ 08102, USA |
2. | Department of Computer Science, University of Verona, Strada Le Grazie 15, I-37134 Verona, Italy |
In this paper we investigate a time-optimal control problem in the space of positive and finite Borel measures on $\mathbb R^d$, motivated by applications in multi-agent systems. We provide a definition of admissible trajectory in the space of Borel measures in a particular non-isolated context, inspired by the so called optimal logistic problem, where the aim is to assign an initial amount of resources to a mass of agents, depending only on their initial position, in such a way that they can reach the given target with this minimum amount of supplies. We provide some approximation results connecting the microscopical description with the macroscopical one in the mass-preserving setting, we construct an optimal trajectory in the non isolated case and finally we are able to provide a Dynamic Programming Principle.
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[3] |
M. Bernot, V. Caselles and J. -M. Morel,
Optimal Transportation Networks -Models and Theory,
1955, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. |
[4] |
D. P. Bertsekas and S. E. Shreve,
Stochastic Optimal Control -the Discrete Time Case,
139, Mathematics in Science and Engineering, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. |
[5] |
M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, submitted, arXiv: 1610.03839v2. |
[6] |
R. Brockett and N. Khaneja, On the stochastic control of quantum ensembles, System theory: Modeling, analysis and control (Cambridge, MA, 1999), Kluwer Internat. Ser. Engrg.
Comput. Sci., Kluwer Acad. Publ., Boston, MA, 518 (2000), 75-96.
doi: 10.1007/978-1-4615-5223-9_6. |
[7] |
G. Buttazzo,
Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, 207 Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. |
[8] |
G. Buttazzo, C. Jimenez and E. Oudet,
An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48 (2009), 1961-1976.
doi: 10.1137/07070543X. |
[9] |
G. Cavagnari,
Regularity results for a time-optimal control problem in the space of probability measures, Mathematical Control and Related Fields, 7 (2017), 213-233.
doi: 10.3934/mcrf.2017007. |
[10] |
G. Cavagnari and A. Marigonda, Time-optimal control problem in the space of probability
measures, Large-scale scientific computing, Lecture Notes in Computer Science, Springer,
Cham, 9374 (2015), 109-116.
doi: 10.1007/978-3-319-26520-9. |
[11] |
G. Cavagnari, A. Marigonda, K. T. Nguyen and F. S. Priuli,
Generalized control systems in the space of probability measures, Set-Valued and Variational Analysis, 25 (2017), 1-29.
doi: 10.1007/s11228-017-0414-y. |
[12] |
G. Cavagnari, A. Marigonda and G. Orlandi, Hamilton-Jacobi-Bellman equation for a timeoptimal control problem in the space of probability measures, in System Modeling and Optimization. CSMO 2015 (eds. L. Bociu, J. -A. Désidéri and A. Habbal), IFIP Advances in
Information and Communication Technology, 494, Springer, Cham, 2016,200-208.
doi: 10.1007/978-3-319-55795-3_18. |
[13] |
G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures, submitted. |
[14] |
E. Cristiani, B. Piccoli and A. Tosin,
Multiscale Modeling of Pedestrian Dynamics,
12 MS & A. Modeling, Simulation and Applications, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[15] |
J. Dolbeault, B. Nazaret and G. Savaré,
A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.
doi: 10.1007/s00526-008-0182-5. |
[16] |
A. Isidori and C. I. Byrnes,
Output regulation of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 131-140.
doi: 10.1109/9.45168. |
[17] |
B. Oksendal,
Stochastic Differential Equations -an Introduction with Applications, 6th edition, Universitext, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-14394-6. |
[18] |
B. Oksendal and A. Sulem,
Applied Stochastic Control of Jump Diffusions, 2nd edition, Universitext, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-69826-5. |
[19] |
B. Piccoli and F. Rossi,
Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.
doi: 10.1007/s00205-013-0669-x. |
[20] |
B. Piccoli and F. Rossi, On properties of the Generalized Wasserstein distance, Archive for
Rational Mechanics and Analysis, 222 (2016), 1339-1365, arXiv: 1304.7014v3.
doi: 10.1007/s00205-016-1026-7. |
[21] |
B. Piccoli, F. Rossi and E. Trélat,
Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.
doi: 10.1137/140996501. |
[22] |
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. |
[23] |
R. Tempo, G. Calafiore and F. Dabbene,
Randomized Algorithms for Analysis and Control of Uncertain Systems -with Applications, Communications and Control Engineering Series, Springer-Verlag, London, 2013.
doi: 10.1007/978-1-4471-4610-0. |
[24] |
C. Villani,
Topics in Optimal Transportation,
58, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.
doi: 10.1007/b12016. |
[25] |
J. Yong and X. Y. Zhou,
Stochastic Controls -Hamiltonian Systems and HJB Equations,
43, Applications of Mathematics (New York), Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
show all references
References:
[1] |
L. Ambrosio, N. Fusco and D. Pallara,
Functions of Bounded Variation and Free Discontinuity Problems, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 2000. |
[2] |
L. Ambrosio, N. Gigli and G. Savaré,
Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2008. |
[3] |
M. Bernot, V. Caselles and J. -M. Morel,
Optimal Transportation Networks -Models and Theory,
1955, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2009. |
[4] |
D. P. Bertsekas and S. E. Shreve,
Stochastic Optimal Control -the Discrete Time Case,
139, Mathematics in Science and Engineering, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. |
[5] |
M. Bonafini, G. Orlandi and E. Oudet, Variational approximation of functionals defined on 1-dimensional connected sets: The planar case, submitted, arXiv: 1610.03839v2. |
[6] |
R. Brockett and N. Khaneja, On the stochastic control of quantum ensembles, System theory: Modeling, analysis and control (Cambridge, MA, 1999), Kluwer Internat. Ser. Engrg.
Comput. Sci., Kluwer Acad. Publ., Boston, MA, 518 (2000), 75-96.
doi: 10.1007/978-1-4615-5223-9_6. |
[7] |
G. Buttazzo,
Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations, 207 Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. |
[8] |
G. Buttazzo, C. Jimenez and E. Oudet,
An optimization problem for mass transportation with congested dynamics, SIAM J. Control Optim., 48 (2009), 1961-1976.
doi: 10.1137/07070543X. |
[9] |
G. Cavagnari,
Regularity results for a time-optimal control problem in the space of probability measures, Mathematical Control and Related Fields, 7 (2017), 213-233.
doi: 10.3934/mcrf.2017007. |
[10] |
G. Cavagnari and A. Marigonda, Time-optimal control problem in the space of probability
measures, Large-scale scientific computing, Lecture Notes in Computer Science, Springer,
Cham, 9374 (2015), 109-116.
doi: 10.1007/978-3-319-26520-9. |
[11] |
G. Cavagnari, A. Marigonda, K. T. Nguyen and F. S. Priuli,
Generalized control systems in the space of probability measures, Set-Valued and Variational Analysis, 25 (2017), 1-29.
doi: 10.1007/s11228-017-0414-y. |
[12] |
G. Cavagnari, A. Marigonda and G. Orlandi, Hamilton-Jacobi-Bellman equation for a timeoptimal control problem in the space of probability measures, in System Modeling and Optimization. CSMO 2015 (eds. L. Bociu, J. -A. Désidéri and A. Habbal), IFIP Advances in
Information and Communication Technology, 494, Springer, Cham, 2016,200-208.
doi: 10.1007/978-3-319-55795-3_18. |
[13] |
G. Cavagnari, A. Marigonda and B. Piccoli, Averaged time-optimal control problem in the space of positive Borel measures, submitted. |
[14] |
E. Cristiani, B. Piccoli and A. Tosin,
Multiscale Modeling of Pedestrian Dynamics,
12 MS & A. Modeling, Simulation and Applications, Springer, Cham, 2014.
doi: 10.1007/978-3-319-06620-2. |
[15] |
J. Dolbeault, B. Nazaret and G. Savaré,
A new class of transport distances between measures, Calc. Var. Partial Differential Equations, 34 (2009), 193-231.
doi: 10.1007/s00526-008-0182-5. |
[16] |
A. Isidori and C. I. Byrnes,
Output regulation of nonlinear systems, IEEE Trans. Automat. Control, 35 (1990), 131-140.
doi: 10.1109/9.45168. |
[17] |
B. Oksendal,
Stochastic Differential Equations -an Introduction with Applications, 6th edition, Universitext, Springer-Verlag, Berlin, 2003.
doi: 10.1007/978-3-642-14394-6. |
[18] |
B. Oksendal and A. Sulem,
Applied Stochastic Control of Jump Diffusions, 2nd edition, Universitext, Springer, Berlin, 2007.
doi: 10.1007/978-3-540-69826-5. |
[19] |
B. Piccoli and F. Rossi,
Generalized Wasserstein distance and its application to transport equations with source, Arch. Ration. Mech. Anal., 211 (2014), 335-358.
doi: 10.1007/s00205-013-0669-x. |
[20] |
B. Piccoli and F. Rossi, On properties of the Generalized Wasserstein distance, Archive for
Rational Mechanics and Analysis, 222 (2016), 1339-1365, arXiv: 1304.7014v3.
doi: 10.1007/s00205-016-1026-7. |
[21] |
B. Piccoli, F. Rossi and E. Trélat,
Control to flocking of the kinetic Cucker-Smale model, SIAM J. Math. Anal., 47 (2015), 4685-4719.
doi: 10.1137/140996501. |
[22] |
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. |
[23] |
R. Tempo, G. Calafiore and F. Dabbene,
Randomized Algorithms for Analysis and Control of Uncertain Systems -with Applications, Communications and Control Engineering Series, Springer-Verlag, London, 2013.
doi: 10.1007/978-1-4471-4610-0. |
[24] |
C. Villani,
Topics in Optimal Transportation,
58, Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2003.
doi: 10.1007/b12016. |
[25] |
J. Yong and X. Y. Zhou,
Stochastic Controls -Hamiltonian Systems and HJB Equations,
43, Applications of Mathematics (New York), Springer-Verlag, New York, 1999.
doi: 10.1007/978-1-4612-1466-3. |
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