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On periodic solution to control problem with time-driven switching

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Abstract

In this contribution, we consider the optimal control problem for a switched dynamical system. While such systems can exhibit rather complex behavior in the case of only one switch, the most interesting problem corresponds to the case, when the system undergoes an infinite number of switches. We study the limiting behavior of optimal solutions under such assumption and show that there are three types of solutions, two of which correspond to cyclic evolution of the system state and control.

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Notes

  1. The case of \(\varTheta \) finite and multiple switches will be addressed in a subsequent publication.

  2. Note the exponential term.

  3. The upper boundary value is never reached for a cyclic trajectory \(\lambda (t)\) as it never crosses 0.

References

  1. Boltyanski, V.: The maximum principle for variable structure systems. Int. J. Control 77(17), 1445–1451 (2004). https://doi.org/10.1080/00207170412331319312

    Article  MathSciNet  MATH  Google Scholar 

  2. Bondarev, A., Greiner, A.: Catching-up and falling behind: effects of learning in an R&D differential game with spillovers. J. Econ. Dyn. Control 91, 134–156 (2018)

    Article  MathSciNet  Google Scholar 

  3. Colombo, A., Jeffrey, M.R.: Nondeterministic chaos, and the two-fold singularity in piecewise smooth flows. SIAM J. Appl. Dyn. Syst. 10(2), 423–451 (2011). https://doi.org/10.1137/100801846

    Article  MathSciNet  MATH  Google Scholar 

  4. Dawid, H., Kopel, M., Kort, P.M.: R&D competition versus R&D cooperation in oligopolistic markets with evolving structure. Int. J. Ind. Organ. 31(5), 527–537 (2013)

    Article  Google Scholar 

  5. Dockner, E., Jorgensen, S., Long, N., Sorger, G.: Differential Games in Economics and Management Sciences. Cambridge University Press, Cambridge (2000)

    Book  Google Scholar 

  6. Gromov, D., Castaños, F.: Sensitivity analysis of limit cycles in an alpha Stirling engine: a bifurcation-theory approach. SIAM J. Appl. Dyn. Syst. 19(3), 1865–1883 (2020). https://doi.org/10.1137/19M1299293

    Article  MathSciNet  MATH  Google Scholar 

  7. Gromov, D., Gromova, E.: On a class of hybrid differential games. Dyn. Games Appl. 7(2), 266–288 (2017). https://doi.org/10.1007/s13235-016-0185-3

    Article  MathSciNet  MATH  Google Scholar 

  8. Gromova, E.V., Tur, A.V., Barsuk, P.I.: A pollution control problem for the aluminum production in eastern siberia: differential game approach. In: Proceedings of IV Stability and Control Processes Conference, Lecture Notes in Computer Science. Springer (2021) (forthcoming)

  9. Klamerus-Iwan, A., Błońska, E., Lasota, J., Waligórski, P., Kalandyk, A.: Seasonal variability of leaf water capacity and wettability under the influence of pollution in different city zones. Atmos. Pollut. Res. 9(3), 455–463 (2018)

    Article  Google Scholar 

  10. Lunze, J., Lamnabhi-Lagarrigue, F. (eds.): Handbook of Hybrid Systems Control: Theory, Tools, Applications. Cambridge University Press, Cambridge (2009)

    MATH  Google Scholar 

  11. Marova, E., Gromova, E., Barsuk, P., Shagushina, A.: On the effect of the absorption coefficient in a differential game of pollution control. Mathematics 8(6), 961 (2020). https://doi.org/10.3390/math8060961

    Article  Google Scholar 

  12. Nieuwenhuijsen, M., Gomez-Perales, J., Colvile, R.: Levels of particulate air pollution, its elemental composition, determinants and health effects in metro systems. Atmos. Environ. 41(37), 7995–8006 (2007)

    Article  Google Scholar 

  13. Qin, W., Tan, X., Shi, X., Chen, J., Liu, X.: Dynamics and bifurcation analysis of a Filippov predator–prey ecosystem in a seasonally fluctuating environment. Int. J. Bifurc. Chaos 29(2), 1950020 (2019)

    Article  MathSciNet  Google Scholar 

  14. Reddy, P.V., Schumacher, J.M., Engwerda, J.: Analysis of optimal control problems for hybrid systems with one state variable. SIAM J. Control Optim. 58(6), 3262–3292 (2020). https://doi.org/10.1137/19M1272779

    Article  MathSciNet  MATH  Google Scholar 

  15. Seidl, A.: Zeno points in optimal control models with endogenous regime switching. J. Econ. Dyn. Control 100, 353–368 (2019)

    Article  MathSciNet  Google Scholar 

  16. Shaikh, M.S., Caines, P.E.: On the hybrid optimal control problem: theory and algorithms. IEEE Trans. Autom. Control 52(9), 1587–1603 (2007)

    Article  MathSciNet  Google Scholar 

  17. Wang, A., Xiao, Y., Smith, R.: Using non-smooth models to determine thresholds for microbial pest management. J. Math. Biol. 78(5), 1389–1424 (2019)

    Article  MathSciNet  Google Scholar 

  18. Wang, H., Shi, H., Li, Y., Yu, Y., Zhang, J.: Seasonal variations in leaf capturing of particulate matter, surface wettability and micromorphology in urban tree species. Front. Environ. Sci. Eng. 7(4), 579–588 (2013)

    Article  Google Scholar 

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Acknowledgements

The work of D. Gromov and E. Gromova on the computation of optimal solutions was supported by the Russian Science Foundation (Project No. 17-11-01093). The work of A. Bondarev was supported by the Key Program Special Fund of Xi’an Jiaotong-Liverpool University (Grant No. KSF-E-63).

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Authors and Affiliations

  1. Faculty of Applied Mathematics and Optimal Processes, Saint Petersburg State University, University Emb., 7/9, St Petersburg, Russia, 199034

    Dmitry Gromov

  2. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, Russia, 620990

    Dmitry Gromov & Ekaterina Gromova

  3. International Business School Suzhou, Xi’an Jiaotong-Liverpool University, Ren’ai Road 111, Suzhou, 215123, People’s Republic of China

    Anton Bondarev

  4. Department of Mathematics, National Research University Higher School of Economics, St. Petersburg Campus 16 Soyuza Pechatnikov Str., St Petersburg, Russia, 190121

    Ekaterina Gromova

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  1. Dmitry Gromov
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  2. Anton Bondarev
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  3. Ekaterina Gromova
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Correspondence to Ekaterina Gromova.

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Gromov, D., Bondarev, A. & Gromova, E. On periodic solution to control problem with time-driven switching. Optim Lett 16, 2019–2031 (2022). https://doi.org/10.1007/s11590-021-01749-6

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