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Time Optimal Control Problem for the Waste Water Biotreatment Model

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Abstract

This work is devoted to solving the time optimal control problem of a mathematical model describing the process of biological waste water treatment and is given as a three-dimensional nonlinear control system of differential equations. For analysis of this problem, the Pontryagin maximum principle is used and the corresponding two-point boundary value problem is formulated. In order to investigate the uniqueness of a solution to this problem, the properties of the corresponding attainable set and the multivalued mapping associated with it are studied. The basis of analysis of this set is its parametric description by moments of switching of piecewise constant controls. A scheme for the approximate solution of the time optimal control problem for the original system is proposed, and results of numerical calculations are presented.

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References

  1. Rojas J, Burke M, Chapwanya M, Doherty K, Hewitt I, Korobeinikov A, Meere M, McCarthy S, O’Brien M, Tuoi VTN, Winstenley H, Zhelev T. Modeling of autothermal thermophilic aerobic digestion. Math-Indust Case Stud J. 2010;2:34–63.

    Google Scholar 

  2. Gomez J, de Gracia M, Ayesa E, Garsia-Heras JL. Mathematical modeling of autothermal thermophilic aerobic digesters. Water Res. 2007;41(5):959–68.

    Article  Google Scholar 

  3. Moreno J. Optimal time control of sequencing batch reactors for industrial wastewater treatment. In: Decision and control. Proceedings of the 36th IEEE conference. vol. 1. 1997. p. 826–827.

  4. Moreno J. Optimal time control of bioreactors for the wastewater treatment. Optim Control Appl Methods. 1999;20(3):145–64.

    Article  Google Scholar 

  5. Betancur MJ, Moreno JA, Moreno-Andrade I, Buitron G. Control strategies for treating toxic wastewater using bioreactors. In: Proceedings of the 16th IFAC World Congress. vol. 16. Part 1. 2005.

  6. Moreno JA, Betancur MJ, Buitron G, Moreno-Andrade I. Event-driven time optimal control for a class of discontinuous bioreactors. Biotech Bioeng. 2006;94(4):803–14.

    Article  Google Scholar 

  7. Mazouni D, Hermand J, Rapaport A, Hammouri H. Optimal time switching control for multi-reaction batch process. Optim Control Appl Methods. 2010; 31(4):289–301.

    Article  MATH  Google Scholar 

  8. Lee EB, Markus L. Foundations of optimal control theory. New York: Wiley; 1970.

    Google Scholar 

  9. Chernous’ko FL. State estimation for dynamical systems. Boca Raton: CRS Press; 1994.

    Google Scholar 

  10. Aubin J-P, Cellina A. Differential inclusions: set-valued maps and viability theory. Berlin: Springer; 1984.

    Book  MATH  Google Scholar 

  11. Komarov VA. Estimates for the attainable set for differential inclusions. Math Notes. 1985;37(6):501–6.

    Article  MATH  MathSciNet  Google Scholar 

  12. Ovseevich AI, Chernous’ko FL. Two-sided estimates on the attainability domains of controlled systems. J Appl Math Mech. 1982;46(5):590–5.

    Article  MATH  MathSciNet  Google Scholar 

  13. Panasyuk AI, Panasyuk VI. Asymptotic turnpike optimization of control systems. Minsk: Nauka i Tehnika; 1986.

    Google Scholar 

  14. Panasyuk AI. Equations of attainable set dynamics. I. Integral funnel equations. J Optim Theory Appl. 1990;64(2):349–66.

    Article  MATH  MathSciNet  Google Scholar 

  15. Panasyuk AI. Equations of attainable set dynamics. II. Partial differential equations. J Optim Theory Appl. 1990;64(2):367–77.

    Article  MATH  MathSciNet  Google Scholar 

  16. Panasyuk AI, Panasyuk VI. An equation generated by a differential inclusion. Math Notes. 1980;27(3):213–8.

    Article  MATH  MathSciNet  Google Scholar 

  17. Grigorieva EV, Khailov EN. A nonlinear controlled system of differential equations describing the process of production and sales of a consumer good. Discret Contin Dyn S. 2003; supplement volume:359–64.

  18. Avvakumov SN, Kiselev YuN. Qualitative study and algorithms in the mathematical model of innovation diffusion. J Math Sci. 2003;116(6):3657–72.

    Article  MATH  MathSciNet  Google Scholar 

  19. Grigorieva EV, Khailov EN. On the attainability set for a nonlinear system in the plane. Mosc Univ Comput Math Cybern. 2001;25(4):27–32.

    MathSciNet  Google Scholar 

  20. Grigorieva EV, Khailov EN. Description of the attainability set of a nonlinear controlled system in the plane. Mosc Univ Comput Math Cybern. 2005;29(3):23–8.

    Google Scholar 

  21. Grigorieva EV, Khailov EN. Attainable set of a nonlinear controlled microeconomic model. J Dyn Control Syst. 2005;11(2):157–76.

    Article  MATH  MathSciNet  Google Scholar 

  22. Grigorieva EV, Bondarenko NV, Khailov EN, Korobeinikov A. Three-dimensional nonlinear control model of wastewater biotreatment. Neural Parallel Sci Comput. 2012;20:23–36.

    MATH  MathSciNet  Google Scholar 

  23. Grigorieva EV, Bondarenko NV, Khailov EN, Korobeinikov A. Finite-dimensional methods for optimal control of autothermal thermophilic aerobic digestion. In: Show KY, Guo X, editors. Industrial waster. Croatia: InTech; 2012. p. 91–120.

  24. Kurzhanski AB, Valyi I. Ellipsoidal calculus for estimation and control. Boston: Birkhäuser; 1997.

    Book  MATH  Google Scholar 

  25. Varaiya P, Kurzhanski AB. Ellipsoidal methods for dynamics and control. Part 1. J Math Sci. 2006;139(5):6863–901.

    Article  MATH  MathSciNet  Google Scholar 

  26. Nikol’skii MS. Approximation of the attainability set for a controlled process. Math Notes. 1987;41(1):44–8.

    Article  Google Scholar 

  27. Guseinov KhG, Moiseev AN, Ushakov VN. On the approximation of reachable domains of control systems. J Appl Math Mech. 1998;62(2):169–75.

    Article  MathSciNet  Google Scholar 

  28. Ushakov VN, Matviychuk AR, Ushakov AV. Approximations of attainability sets and of integral funnels of differential inclusions. Bull Udmurt Univ Math Mech Comput Sci. 2011;(4):23–39.

  29. Colonius F, Szolnoki D. Algorithms for computing reachable sets and control sets. In: Proceedings of the IFAC symposium on nonlinear control systems (NOLCOS 2001), 4–6 July 2001, St. Petersburg, Russia. p. 756–761.

  30. Szolnoki D. Set-oriented methods for computing reachable sets and control sets. Discret Contin Dyn Syst Ser B. 2003;3(3):361–82.

    Article  MATH  MathSciNet  Google Scholar 

  31. Smirnov AI. Attainability analysis of the DICE model. IIASA, Interium report, IR-05-049, Laxenburg. 2005.

  32. Shevchenko GV. Numerical method for solving a nonlinear time optimal control problem with additive control. Comput Math Math Phys. 2007;47(11):1768–78.

    Article  MathSciNet  Google Scholar 

  33. Shevchenko GV. Numerical solution of a nonlinear time-optimal control problem. Comput Math Math Phys. 2011;51(4):537–49.

    Article  MathSciNet  Google Scholar 

  34. Gurman VI. The extension principle in control problems. Moscow: Nauka, Fizmatlit; 1997.

    MATH  Google Scholar 

  35. Kurzhanski AB. Differential equations in control synthesis problems. Differ Equ. 2005;41(1):10–21.

    Article  MathSciNet  Google Scholar 

  36. Kurzhanski AB. Comparison principle for equations of the Hamilton-Jacobi type in control theory. Proc Steklov Inst Math. 2006;253(supplement 1):185–95.

    Article  MathSciNet  Google Scholar 

  37. Gurman VI, Trushkova EA. Estimates for attainability sets of control systems. Differ Equ. 2009;45(11):1636–44.

    Article  MATH  MathSciNet  Google Scholar 

  38. Gornov AYu. The computational technologies for solving of optimal control problems. Novosibirsk: Nauka; 2009.

    Google Scholar 

  39. Chernous’ko FL, Kolmanovskii VB. Computational and approximate methods of optimal control. J Math Sci. 1979;12(3):310–53.

    Article  MATH  MathSciNet  Google Scholar 

  40. Tyatyushkin AI. Numerical methods and software tools of optimization of control systems. Novosibirsk: Nauka; 1992.

    Google Scholar 

  41. Tyatyushkin AI, Morzhin OV. Constructive methods of control optimization in nonlinear systems. Autom Remote Control. 2009;70(5):772–86.

    Article  MATH  MathSciNet  Google Scholar 

  42. Tyatyushkin AI, Morzhin OV. Numerical investigation of attainability sets of nonlinear controlled differential systems. Autom Remote Control. 2011;72(6):1291–300.

    Article  MATH  MathSciNet  Google Scholar 

  43. Sklyar GM, Ignatovich SYu. Moment approach to nonlinear time optimality. SIAM J Control Optim. 2000;38(6):1707–28.

    Article  MATH  MathSciNet  Google Scholar 

  44. Sklyar GM, Ignatovich SYu. Approximation of time optimal control problems via nonlinear power moment min-problems. SIAM J Control Optim. 2003;42(4):1325–46.

    Article  MATH  MathSciNet  Google Scholar 

  45. Moiseev NN. Numerical methods in optimal systems theory. Moscow: Nauka; 1971.

    Google Scholar 

  46. Moiseev NN. Elements of optimal systems theory. Moscow: Nauka; 1975.

    MATH  Google Scholar 

  47. Fedorenko RP. Approximate solution of optimal control problems. Moscow: Nauka; 1978.

    MATH  Google Scholar 

  48. Evtushenko YuG. Methods of solving extremal problems and their application in optimization systems. Moscow: Nauka; 1982.

    MATH  Google Scholar 

  49. Subrahmanyam MB. A computational method for solution of time optimal control problems by Newton’s method. Int J Control. 1986;44(5):1233–43.

    Article  MATH  Google Scholar 

  50. Mohler RR. Bilinear control processes. New York: Academic Press; 1973.

    MATH  Google Scholar 

  51. Mohler RR. Nonlinear systems: vol. 2 Applications to bilinear control. Englewood Cliffs: Prentice Hall; 1991.

    Google Scholar 

  52. Meier E-B, Bryson AE Jr. Efficient algorithm for time optimal control of a two-link manipulator. J Guid Control Dyn. 1990;13(5):859–66.

    Article  MATH  MathSciNet  Google Scholar 

  53. Kaya CY, Noakes JL. Computations and time optimal controls. Optim Control Appl Methods. 1996;17(3):171–85.

    Article  MATH  MathSciNet  Google Scholar 

  54. Kaya CY, Noakes JL. Computational method for time optimal switching control. J Optim Theory Appl. 2003;117(1):69–92.

    Article  MATH  MathSciNet  Google Scholar 

  55. Wong KH, Clements DJ, Teo KL. Optimal control computation for nonlinear time-lag systems. J Optim Theory Appl. 1985;47(1):91–107.

    Article  MATH  MathSciNet  Google Scholar 

  56. Teo KL, Goh CJ, Wong KH. Unified computational approach to optimal control problems. Essex: Longman Scientific and Technical; 1991.

    MATH  Google Scholar 

  57. Krasnov KS, Vorob’ev NK, Godnev IN, et al. Physical chemestry. vol. 2. Moscow: Vysshaya Shkola; 1995.

    Google Scholar 

  58. Conti R. Sulla prolungabilitá delle soluzioni di un sistema di equazioni differenziali ordinarie. Boll Unione Math Ital. 1956;11:510–4.

    MATH  MathSciNet  Google Scholar 

  59. Brauer F. Some simple epidemic models. Math Biosci Eng. 2006;3(1):1–15.

    Article  MATH  MathSciNet  Google Scholar 

  60. Aseev SM, Kryazhimskii AV. The Pontryagin maximum principle and optimal economic growth problems. Proc Steklov Inst Math. 2007;257:1–272.

    Article  MATH  MathSciNet  Google Scholar 

  61. Fleming WH, Rishel RW. Deterministic and stochastic optimal control. Berlin: Springer; 1975.

    Book  MATH  Google Scholar 

  62. Pontryagin LS, Boltyanskii VG, Gamkrelidze RV, Mishchenko EF. Mathematical theory of optimal processes. New York: Wiley; 1962.

    MATH  Google Scholar 

  63. Bressan A, Piccoli B. Introduction to the mathematical theory of control. AIMS; 2007.

  64. Kuratovski K. Topology. vol. 1 and vol. 2. New York: Academic Press; 1966 and 1968.

  65. Blagodatskikh VI, Filippov AF. Differential inclusions and optimal control. Proc Steklov Inst Math. 1986;169:199–259.

    MATH  Google Scholar 

  66. Hall DW, Spencer GL. Elementary topology. New York: Wiley; 1955.

    MATH  Google Scholar 

  67. Vasil’ev FP. Optimization methods. Moscow: Factorial Press; 2002.

    Google Scholar 

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Acknowledgments

We would like to thank Dr. Ellen Perlow for valuable comments and style recommendations.

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Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Texas Woman’s University, Denton, TX, 76204, USA

    E. V. Grigorieva

  2. Department of Computational Mathematics and Cybernetics, Moscow State Lomonosov University, Moscow, 119992, Russia

    N. V. Bondarenko & E. N. Khailov

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  1. E. V. Grigorieva
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  3. E. N. Khailov
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Correspondence to E. V. Grigorieva.

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Grigorieva, E.V., Bondarenko, N.V. & Khailov, E.N. Time Optimal Control Problem for the Waste Water Biotreatment Model. J Dyn Control Syst 21, 3–24 (2015). https://doi.org/10.1007/s10883-014-9214-y

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  • DOI: https://doi.org/10.1007/s10883-014-9214-y

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