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Optimization of a Prey–Predator Model with Hysteresis and Convection

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Abstract

A nonconvex optimal control problem for a partial differential system is proposed and analyzed. Our nonlinear control system can be interpreted in the population dynamics framework when, along with diffusional, hysteretic and migratory effects in the evolution of biological species are accounted for. We prove the existence of a nearly optimal solution to the optimization problem through relaxation-type properties; we establish for the underlying minimizing cost functional and control system.

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References

  1. Aiki, T., Minchev, E.: A prey–predator model with hysteresis effect. SIAM J. Math. Anal. 36(6), 2020–2032 (2005)

    MathSciNet  MATH  Google Scholar 

  2. Alford, J., Balusek, C., Bowers, K., Hartnett, C.: A mathematical model of biocontrol of invasive aquatic weeds. Involve 5(4), 431–447 (2012)

    MathSciNet  MATH  Google Scholar 

  3. Alonso, D., Dobson, A., Pascual, M.: Evidence of critical transitions and coexistence of alternative states in nature: the case of malaria transmission, 73–79, Trends Math. Res. Perspect. CRM Barc., 11, Birkhäuser/Springer, Cham, (2019)

  4. Antil, H., Shirakawa, K., Yamazaki, N.: A class of parabolic systems associated with optimal controls of grain boundary motions. Adv. Math. Sci. Appl. 27(2), 299–336 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Aubert, G., Tahraoui, R.: Théorèmes d’Existence pour des Problèmes du Calcul des Variations du Type: \(\text{Inf}\int ^L_0 f(x, u^{\prime }(x))dx\) et \(\text{ Inf }\int ^L_0f(x, u(x), u^{\prime }(x))dx\). J. Differ. Equ. 33(1), 1–15 (1979)

  6. Bagagiolo, F., Benetton, M.: About an optimal visiting problem. Appl. Math. Optim. 65(1), 31–51 (2012)

    MathSciNet  MATH  Google Scholar 

  7. Balder, E.J.: Necessary and sufficient conditions for \(L_1\)-strong-weak lower semi-continuity of integral functionals. Nonlinear Anal. 11(12), 1399–1404 (1987)

  8. Bin, M., Liu, Z.: Relaxation in nonconvex optimal control for nonlinear evolution hemivariational inequalities. Nonlinear Anal. Real World Appl. 50, 613–632 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Bressan, A., Colombo, G.: Extensions and selections of maps with decomposable values. Studia Math. 90(1), 69–86 (1988)

    MathSciNet  MATH  Google Scholar 

  10. Brokate, M., Krejčí, P.: Optimal control of ODE systems involving a rate independent variational inequality. Discret. Contin. Dyn. Syst. Ser. B 18(2), 331–348 (2013)

    MathSciNet  MATH  Google Scholar 

  11. Brokate, M., Sprekels, J.: Hysteresis and Phase Transitions. Appl. Math. Sci., vol. 121. Springer, New York (1996)

    MATH  Google Scholar 

  12. Brokate, M., Fellner, K., Lang-Batsching, M.: Weak differentiability of the control-to-state mapping in a parabolic equation with hysteresis. NoDEA Nonlinear Differ. Equ. Appl. 26(6), Paper no. 46, 19 (2019)

  13. Cellina, A., Colombo, G.: On a classical problem of the calculus of variations without convexity assumptions. Ann. Inst. H. Poincaré C Anal. Non Linéaire 7(2), 97–106 (1990)

    MathSciNet  MATH  Google Scholar 

  14. Cesari, L.: Existence theorems for optimal solutions in Pontryagin and Lagrange problems. SIAM J. Control (Series A) 3, 475–498 (1965)

    MathSciNet  MATH  Google Scholar 

  15. Cesari, L.: Existence theorems for weak and usual optimal solutions in Lagrange problems with unilateral constraints. I and II. Trans. Am. Math. Soc. 124, 369–412 (1966)

    MATH  Google Scholar 

  16. Cesari, L.: An existence theorem without convexity conditions. SIAM J. Control 12, 319–331 (1974)

    MathSciNet  MATH  Google Scholar 

  17. Bin, C., Minchev, E., Timoshin, S.A., Xiaohan, J.: Control of a multi-component phase transition model with hysteresis. Appl. Math. Optim. 85(1), 1–20 (2022)

    MathSciNet  MATH  Google Scholar 

  18. Christof, C.: Sensitivity analysis and optimal control of obstacle-type evolution variational inequalities. SIAM J. Control. Optim. 57(1), 192–218 (2019)

    MathSciNet  MATH  Google Scholar 

  19. Clason, Ch., Rund, A., Kunisch, K.: Nonconvex penalization of switching control of partial differential equations. Syst. Control Lett. 106, 1–8 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Coletsos, J.: A relaxation approach to optimal control of Volterra integral equations. Eur. J. Control. 42, 25–31 (2018)

    MathSciNet  MATH  Google Scholar 

  21. De Angelis, T., Ferrari, G., Moriarty, J.: A nonconvex singular stochastic control problem and its related optimal stopping boundaries. SIAM J. Control. Optim. 53(3), 1199–1223 (2015)

    MathSciNet  MATH  Google Scholar 

  22. Debbouche, A., Nieto, J.J., Torres, D.F.M.: Optimal solutions to relaxation in multiple control problems of Sobolev type with nonlocal nonlinear fractional differential equations. J. Optim. Theory Appl. 174(1), 7–31 (2017)

    MathSciNet  MATH  Google Scholar 

  23. De Blasi, F.S., Pianigiani, G., Tolstonogov, A.A.: A Bogolyubov-type theorem with a nonconvex constraint in Banach spaces. SIAM J. Control. Optim. 43(2), 466–476 (2004)

    MathSciNet  MATH  Google Scholar 

  24. Fryszkowski, A.: Continuous selections for a class of nonconvex multivalued maps. Studia Math. 76, 163–174 (1983)

    MathSciNet  MATH  Google Scholar 

  25. Fryszkowski, A.: Fixed Point Theory for Decomposable Sets. Kluwer, Dordrecht (2004)

    MATH  Google Scholar 

  26. Gavioli, C., Krejčí, P.: Control and controllability of PDEs with hysteresis. Appl. Math. Optim. 84(1), 829–847 (2021)

    MathSciNet  MATH  Google Scholar 

  27. Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977)

    MATH  Google Scholar 

  28. Gudovich, A., Quincampoix, M.: Optimal control with hysteresis nonlinearity and multidimensional play operator. SIAM J. Control. Optim. 49(2), 788–807 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Kenmochi, N., Koyama, T., Meyer, G.H.: Parabolic PDEs with hysteresis and quasivariational inequalities. Nonlinear Anal. 34(5), 665–686 (1998)

    MathSciNet  MATH  Google Scholar 

  30. Krejčí, P.: Hysteresis, Convexity and Dissipation in Hyperbolic Equations. Gakuto Int. Ser. Math. Sci. Appl., vol. 8. Gakkōtosho, Tokyo (1996)

  31. Krejčí, P., Timoshin, S.A.: Coupled ODEs control system with unbounded hysteresis region. SIAM J. Control. Optim. 54(4), 1934–1949 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Krejčí, P., Timoshin, S.A., Tolstonogov, A.A.: Relaxation and optimisation of a phase-field control system with hysteresis. Int. J. Control 91(1), 85–100 (2018)

    MathSciNet  MATH  Google Scholar 

  33. Krejčí, P., Tolstonogov, A.A., Timoshin, S.A.: A control problem in phase transition modeling. NoDEA Nonlinear Differ. Equ. Appl. 22(4), 513–542 (2015)

    MathSciNet  MATH  Google Scholar 

  34. Křivan, V.: Behavioral refuges and predator–prey coexistence. J. Theoret. Biol. 339, 112–121 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Marcellini, P.: Some observations on the existence of the minimum of integrals of the calculus of variations without convexity hypotheses. Rend. Mat. (6) 13(2), 271–281 (1980)

    MathSciNet  Google Scholar 

  36. Minchev, E., Otani, M.: \(L^\infty \)-energy method for a parabolic system with convection and hysteresis effect. Commun. Pure Appl. Anal. 17(4), 1613–1632 (2018)

  37. Münch, Ch.: Optimal control of reaction–diffusion systems with hysteresis. ESAIM Control Optim. Calc. Var. 24(4), 1453–1488 (2018)

    MathSciNet  MATH  Google Scholar 

  38. Pal, S., Bhattacharyya, J.: Catastrophic Transitions in Coral Reef Biome under Invasion and Overfishing, Mathematical Biology and Biological Physics, pp. 118–140. World Sci. Publ, Hackensack (2017)

    MATH  Google Scholar 

  39. Pimenov, A., Kelly, T.C., Korobeinikov, A., O’Callaghan, M.J., Rachinskii, D.: Memory and adaptive behavior in population dynamics: anti-predator behavior as a case study. J. Math. Biol. 74(6), 1533–1559 (2017)

    MathSciNet  MATH  Google Scholar 

  40. Přibylová, L., Berec, L.: Predator interference and stability of predator–prey dynamics. J. Math. Biol. 71(2), 301–323 (2015)

    MathSciNet  MATH  Google Scholar 

  41. Rapoport, L.B., Tormagov, T.A.: Relaxation methods for navigation satellites set optimization. Autom. Remote Control 81(9), 1711–1721 (2020)

    MathSciNet  MATH  Google Scholar 

  42. Sagara, N.: Relaxation and purification for nonconvex variational problems in dual Banach spaces: the minimization principle in saturated measure spaces. SIAM J. Control Optim. 55(5), 3154–3170 (2017)

    MathSciNet  MATH  Google Scholar 

  43. Timoshin, S.A., Aiki, T.: Control of biological models with hysteresis. Syst. Control Lett. 128, 41–45 (2019)

    MathSciNet  MATH  Google Scholar 

  44. Timoshin, S.A., Aiki, T.: Extreme solutions in control of moisture transport in concrete carbonation. Nonlinear Anal. Real World Appl. 47, 446–459 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Timoshin, S.A., Aiki, T.: Relaxation in population dynamics models with hysteresis. SIAM J. Control Optim. 59(1), 693–708 (2021)

    MathSciNet  MATH  Google Scholar 

  46. Tolstonogov, A.A.: Properties of solutions of a control system with hysteresis. J. Math. Sci. New York 196(3), 405–433 (2014)

    MathSciNet  MATH  Google Scholar 

  47. Tolstonogov, A.A.: Relaxation in nonconvex optimal control problems containing the difference of two subdifferentials. SIAM J. Control Optim. 54(1), 175–197 (2016)

    MathSciNet  MATH  Google Scholar 

  48. Tolstonogov, A.A.: Bogolyubov’s theorem for a controlled system related to a variational inequality. Izv. Math. 84(6), 1192–1223 (2020)

    MathSciNet  MATH  Google Scholar 

  49. Van Chuong, P.: A density theorem with an application in relaxation of non-convex-valued differential equations. J. Math. Anal. Appl. 124, 1–14 (1987)

    MathSciNet  MATH  Google Scholar 

  50. Visintin, A.: Differential Models of Hysteresis. Appl. Math. Sci., vol. 111. Springer, Berlin (1994)

    MATH  Google Scholar 

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Acknowledgements

The authors want to thank the anonymous referees for their valuable suggestions and remarks which helped to improve the manuscript.

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Authors and Affiliations

  1. Fujian Province University Key Laboratory of Computational Science, School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, China

    Chen Bin, Xiao Yu Liang & Sergey A. Timoshin

  2. Hikari Ltd, P.O. Box 85, 7000, Ruse, Bulgaria

    Emil Minchev

  3. Matrosov Institute for System Dynamics and Control Theory, Russian Academy of Sciences, Lermontov str. 134, 664033, Irkutsk, Russia

    Sergey A. Timoshin

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  1. Chen Bin
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  2. Xiao Yu Liang
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  3. Emil Minchev
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  4. Sergey A. Timoshin
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Correspondence to Sergey A. Timoshin.

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Communicated by Irena Lasiecka.

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Supported by National Natural Science Foundation of China (12071165 and 62076104) and Natural Science Foundation of Fujian Province (2020J01072 and 2023J01111108).

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Bin, C., Liang, X.Y., Minchev, E. et al. Optimization of a Prey–Predator Model with Hysteresis and Convection. J Optim Theory Appl 198, 347–371 (2023). https://doi.org/10.1007/s10957-023-02225-0

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