рус eng
«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2023. Vol 46

Pontryagin’s Maximum Principle and Indirect Descent Method for Optimal Impulsive Control of Nonlocal Transport Equation

Author(s)
Maksim V. Staritsyn1, Nikolay I. Pogodaev1, Elena V. Goncharova1

1Matrosov Institute for System Dynamics and Control Theory of SB RAS, Irkutsk, Russian Federation

Abstract
We study a singular problem of optimal control of a nonlocal transport equation in the space of probability measures, in which the structure of the drivng vector field with respect to the control variable is somewhat equivalent to the affine one, while the set of controls is norm-unbounded and constrained in the integral sense only. We show that the problem at hand admits an impulse-trajectory relaxation in terms of discontinuous time reparameterization. This relaxation provides a correct statement of the variational problem in the class of control inputs constrained in both pointwise and integral senses. For the relaxed problem, we derive a new form of the Pontryagin’s maximum principle (PMP) with a separate adjoint system of linear balance laws on the space of signed measures. In contrast to the canonical formulation of the PMP in terms of the Hamiltonian equation on the cotangent bundle of the state space, our form allows one to formulate an indirect descent method for optimal impulsive control analogous to classical gradient descent. We expose a version of this method, namely, an algorithm of the steepest descent with an internal line search of the Lagrange multiplier associated with the integral bound on control. The algorithm is proven to monotonically converge to a PMP-extremal up to a subsequence.
About the Authors

Maksim V. Staritsyn, Cand. Sci. (Phys.–Math.), Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, starmax@icc.ru

Niokolay N. Pogodaev, Cand. Sci. (Phys.–Math.), Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, nickpogo@gmail.com

Elena V. Goncharova, Cand. Sci. (Phys.–Math.), Leading Researcher, Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033, Russian Federation, goncha@icc.ru

For citation
Staritsyn M. V., Pogodaev N. I., Goncharova E. V. Pontryagin’s Maximum Principle and Indirect Descent Method for Optimal Impulsive Control of Nonlocal Transport Equation. The Bulletin of Irkutsk State University. Series Mathematics, 2023, vol. 46, pp. 66–84. (in Russian) https://doi.org/10.26516/1997-7670.2023.46.66
Keywords
nonlocal continuity equation, optimal control, impulsive control, Pontryagin’s maximum principle, numerical algorithms
UDC
517.977
MSC
49J20
DOI
https://doi.org/10.26516/1997-7670.2023.46.66
References
  1. Ambrosio L., Gigli N., Savare G. Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zurich. Birkhauser, Boston, 2005.
  2. Averboukh Y. Viability theorem for deterministic mean field type control systems. Set-Valued Var. Anal., 2018, vol. 26, no. 4, pp. 993–1008. https://doi.org/10.1007/s11228-018-0479-2
  3. Averboukh Y., Khlopin D. Pontryagin maximum principle for the deterministic mean field type optimal control problem via the lagrangian approach, 2022. https://doi.org/10.48550/arXiv.2207.01892
  4. Bongini M., Fornasier M., Rossi F., Solombrino F. Mean-field Pontryagin maximum principle. J. Optim. Theory Appl., 2017, vol. 175, pp. 1–38. https://doi.org/10.1007/s10957-017-1149-5
  5. Bonnet B. A Pontryagin maximum principle in Wasserstein spaces for constrained optimal control problems. ESAIM Control Optim. Calc. Var., 2019, vol. 25, no. 52. https://doi.org/10.1051/cocv/2019044
  6. Bonnet B., Cipriani C., Fornasier M., Huang H. A measure theoretical approach to the mean-field maximum principle for training NeurODEs. Nonlinear Anal., 2023, vol. 227, pp. 113–161. https://doi.org/10.1016/j.na.2022.113161
  7. Bonnet B., Frankowska H. Necessary optimality conditions for optimal control problems in Wasserstein spaces. Appl. Math. Optim., 2021, vol. 84, pp. 1281–1330. https://doi.org/10.1007/s00245-021-09772-w
  8. Brezis H. Functional analysis, Sobolev spaces and Partial differential equations. Universitext. Springer New York, 2010.
  9. Cardaliaguet P., Delarue F., Lasry J.-M., Lions P.-L. The master equation and the convergence problem in mean field games. Ann. Math. Stud., vol. 201, Princeton, NJ, Princeton University Press, 2019.
  10. Carrillo J.A., Fornasier M., Toscani G., Vecil F. Particle, kinetic, and hydrodynamic models of swarming. In Mathematical modeling of collective behavior in socio-economic and life sciences. Birkhauser, Boston, MA, 2010, pp. 297–336. https://doi.org/10.1007/978-0-8176-4946-3_12
  11. Cavagnari G., Marigonda A., Nguyen K. T., Priuli F. S. Generalized control systems in the space of probability measures. Set-Valued Var. Anal., 2018, vol. 26, no. 3, pp. 663–691. https://doi.org/10.1007/s11228-017-0414-y
  12. Chertovskih R., Pogodaev N., and Staritsyn M. Optimal control of nonlocal continuity equations: numerical solution, 2023. https://doi.org/10.48550/arXiv.2212.06608
  13. Colombo R.M., Herty M., and Mercier M. Control of the continuity equation with a non local flow. ESAIM Control Optim. Calc. Var., 2011, vol. 17, no. 2, pp. 353–379. https://doi.org/10.1051/cocv/2010007
  14. Conn A., Scheinberg K., and Vicente L. Introduction to DerivativeFree Optimization. SIAM, 2009. https://epubs.siam.org/doi/book/10.1137/1.9780898718768
  15. Cristiani E., Frasca P., Piccoli B. Effects of anisotropic interactions on the structure of animal groups. Journal of mathematical biology, 2011, vol. 62, pp. 569–88. https://doi.org/10.1007/s00285-010-0347-7
  16. Cucker F., Smale S. Emergent behavior in flocks. IEEE Trans. Autom. Control, 2007, vol. 52, no. 5, 852–862. https://doi.org/10.1109/TAC.2007.895842
  17. Dobrushin R.L. Vlasov equations. Funct. Anal. Its Appl., 1979, vol. 13, pp.115–123. https://doi.org/10.1007/BF01077243
  18. Duchon M., Malicky P. A Helly theorem for functions with values in metric spaces. Tatra Mountains Mathematical Publications, 2009, vol. 44, no. 12, pp. 159–168. https://doi.org/10.2478/tatra.v44i0.48
  19. Holden H., Risebro N.H. Front tracking for hyperbolic conservation laws. Berlin, Springer, 2015, vol. 152.
  20. Karamzin D.Y., de Oliveira V.A., Pereira F.L., Silva G.N. On the properness of an impulsive control extension of dynamic optimization problems. ESAIM Control Optim. Calc. Var., 2015, vol. 21, no. 3, pp. 857–875. https://doi.org/10.1051/cocv/2014053
  21. Kyritsi-Yiallourou S.T., Papageorgiou N.S. Handbook of Applied Analysis, vol. 19 of Advances in Mechanics and Mathematics. Springer US, Boston, MA, 2009.
  22. Marigonda A., Quincampoix M. Mayer control problem with probabilistic uncertainty on initial positions. J. Differ. Equ., 2018, vol. 264; no. 5, pp. 3212–3252. https://doi.org/10.1016/j.jde.2017.11.014
  23. McShane E.J., Warfield R.B. On Filippov’s Implicit Functions Lemma. Proceedings of the American Mathematical Society, 1967, vol. 18, no. 1, pp. 41–47. https://doi.org/10.2307/2035221
  24. Miller B.M., and Rubinovich E.Y. Impulsive control in continuous and discretecontinuous systems. New York, Kluwer Academic/Plenum Publishers, 2003.
  25. Piccoli B., Rossi F. Measure-theoretic models for crowd dynamics. In Modeling and Simulation in Science, Engineering and Technology. Springer, Basel, 2018, pp. 137–165. https://doi.org/10.1007/978-3-030-05129-7_6
  26. Pogodaev N., Staritsyn M. Impulsive control of nonlocal transport equations. J. Differ. Equ., 2020, vol. 269, no. 4, pp. 3585–3623, https://doi.org/10.1016/j.jde.2020.03.007
  27. Pogodaev, N. I., and Staritsyn, M. V. Nonlocal balance equations with parameters in the space of signed measures. Sbornik: Mathematics, vol. 213, no. 1, pp. 69–94. https://doi.org/10.4213/sm9516
  28. Srochko V.A. Iterative methods for solving optimal control problems. Moscow, Fizmatlit Publ., 2000. (in Russian)
  29. Weinan E., Han J., and Li Q. A mean-field optimal control formulation of deep learning. Res. Math. Sci., 2019, vol. 6. https://doi.org/10.1007/s40687-018-0172-y
  30. Staritsyn M., Pogodaev N. On a class of problems of optimal impulse control for a continuity equation. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 1, pp. 229–244. (in Russian) https://doi.org/10.21538/0134-4889-2019-25-1-229-244
  31. Staritsyn M.V., Maltugueva N.S., Pogodaev N.I., Sorokin S.P. Impulsive control of systems with network structure describing spread of political influence. The Bulletin of Irkutsk State University. Series Mathematics, 2018, vol. 25, pp. 126–143. (in Russian) https://doi.org/10.26516/1997-7670.2018.25.126

Full text (russian)