Skip to main content
SpringerLink
Log in

One Time-Optimal Problem for a Set-Valued Linear Control System

  • Published:
Ukrainian Mathematical Journal Aims and scope

We consider a time-optimal problem for a set-valued linear control system in the case where a section of the solution of the system coincides with a target set. For this problem, we establish both the solvability conditions and the optimal time and optimal controls. The results are illustrated by model examples.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. F. S. de Blasi and F. Iervolino, “Equazioni differentiali con soluzioni a valore compatto convesso,” Boll. Unione Mat. Ital., 2, No. 4-5, 491–501 (1969).

    MATH  Google Scholar 

  2. V. A. Plotnikov, A.V. Plotnikov, and A. N. Vityuk, Differential Equations with Set-Valued Right-Hand Sides. Asymptotic Methods [in Russian], AstroPrint, Odessa (1999).

    Google Scholar 

  3. A.V. Plotnikov and N. V. Skripnik, Differential Equations with Clear and Fuzzy Set-Valued Right-Hand Sides. Asymptotic Methods [in Russian], AstroPrint, Odessa (2009).

    Google Scholar 

  4. V. Lakshmikantham, B. T. Granna, and D. J. Vasundhara, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publ., Cambridge (2006).

    Google Scholar 

  5. A. A. Martynyuk, Qualitative Analysis of Set-Valued Differential Equations, Springer Nature Switzerland AG, Birkhäuser, Cham (2019).

    Book  Google Scholar 

  6. A. A. Tolstonogov, Differential Inclusions in Banach Spaces [in Russian], Nauka, Novosibirsk (1986).

    Google Scholar 

  7. E. V. Ocheretnyuk and V. I. Slyn’ko, “Estimates of the area of solutions of pseudolinear differential equations with Hukuhara derivative in the space conv(R2),” Ukr. Mat. Zh., 69, No. 2, 189–214 (2017); English translation: Ukr. Math. J., 69, No. 2, 224–254 (2017).

  8. A.V. Plotnikov and N. V. Skripnik, “An existence and uniqueness theorem to the Cauchy problem for generalized set differential equations,” Dyn. Contin. Discrete Impuls. Syst. Ser. A, Math. Anal., 20, No. 4, 433–445 (2013).

    MathSciNet  MATH  Google Scholar 

  9. A.V. Plotnikov and N. V. Skripnik, “Conditions for the existence of local solutions of set-valued differential equations with generalized derivative,” Ukr. Mat. Zh., 65, No. 10, 1350–1362 (2013); English translation: Ukr. Math. J., 65, No. 10, 1498–1513 (2014).

  10. N. V. Skripnik, “Three-step averaging scheme for set-valued differential equations with generalized derivative,” Nelin. Kolyv., 20, No. 3, 391–400 (2017); English translation: J. Math. Sci., 236, No. 3, 333–342 (2019).

  11. A.V. Plotnikov, T. A. Komleva, and L. I. Plotnikova, “Averaging of a system of set-valued differential equations with the Hukuhara derivative,” J. Uncertain Systems, 13, No. 1, 3–13 (2019).

    Google Scholar 

  12. A.V. Plotnikov and A.V. Tumbrukaki, “Integro-differentiation equations with multivalued solutions,” Ukr. Mat. Zh., 52, No. 3, 359–367 (2000); English translation: Ukr. Math. J., 52, No. 3, 413–423 (2000).

  13. A.V. Plotnikov and A.V. Tumbrukaki, “Integro-differentiation inclusions with Hukuhara derivative,” Nelin. Kolyv., 8, No. 1, 80–88 (2005); English translation: Nonlin. Oscillat., 8, No. 1, 78–86 (2005).

  14. N. V. Skripnik, “Averaging of multivalued integral equations,” Nelin. Kolyv., 16, No. 3, 408–415 (2013); English translation: J. Math. Sci., 201, No. 3, 384–390 (2014).

  15. V. Babenko, “Numerical methods for solution of Volterra and Fredholm integral equations for functions with values in L-spaces,” Appl. Math. Comput., 291, 354–372 (2016).

    MathSciNet  MATH  Google Scholar 

  16. V. Babenko, “Calculus and nonlinear integral equations for functions with values in L-spaces,” Anal. Math., 45, No. 4, 727–755 (2019).

    Article  MathSciNet  Google Scholar 

  17. A.V. Plotnikov, T. A. Komleva, and I. V. Molchanyuk, “Existence and uniqueness theorem for set-valued Volterra–Hammerstein integral equations,” Asian-Europ. J. Math., 11, No. 3, 1850036 (2018).

    Article  MathSciNet  Google Scholar 

  18. A.V. Plotnikov and N. V. Skripnik, “Existence and uniqueness theorem for set integral equations,” J. Adv. Res. Dyn. Control Syst., 5, No. 2, 65–72 (2013).

    MathSciNet  Google Scholar 

  19. N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Impulsive Differential Equations with Set-Valued and Discontinuous Right-Hand Sides [in Russian], Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2007).

    Google Scholar 

  20. N. A. Perestyuk, V. A. Plotnikov, A. M. Samoilenko, and N. V. Skripnik, Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities, Walter De Gruyter, Berlin (2011).

    Book  Google Scholar 

  21. N. A. Perestyuk and N. V. Skripnik, “Averaging of set-valued impulsive systems,” Ukr. Mat. Zh., 65, No. 1, 126–142 (2013); English translation: Ukr. Math. J., 65, No. 1, 140–157 (2013).

  22. N. V. Skripnik, “Averaging of impulsive differential inclusions with Hukuhara derivative,” Nelin. Kolyv., 10, No. 3, 416–432 (2007); English translation: Nonlin. Oscillat., 10, No. 3, 422–438 (2007).

  23. I. V. Atamas’ and V. I. Slyn’ko, “Stability of fixed points for a class of quasilinear cascades in the space conv(Rn),” Ukr. Mat. Zh., 69, No. 9, 1166–1179 (2017); English translation: Ukr. Math. J., 69, No. 9, 1354–1369 (2018).

  24. T. A. Komleva, L. I. Plotnikova, and A.V. Plotnikov, “A multivalued discrete system and its properties,” Ukr. Mat. Zh., 70, No. 11, 1519–1524 (2018); English translation: Ukr. Math. J., 70, No. 11, 1750–1757 (2019).

  25. T. A. Komleva, L. I. Plotnikova, and A.V. Plotnikov, “Partial averaging of discrete-time set-valued systems,” Stud. Univ. Babeş-Bolyai Math., 63, No. 4, 539–548 (2018).

    Article  MathSciNet  Google Scholar 

  26. T. A. Komleva and A.V. Plotnikov, “Differential inclusions with Hukuhara derivative,” Nelin. Kolyvannya, 10, No. 2, 229–246 (2007); English translation: Nonlin. Oscillations, 10, No. 2, 229–245 (2007).

  27. A.V. Plotnikov, “Averaging differential embeddings with Hukuhara derivative,” Ukr. Mat. Zh., 41, No. 1, 121–125 (1989); English translation: Ukr. Math. J., 41, No. 1, 112–115 (1989).

  28. N. V. Plotnikova, “Approximation of a bundle of solutions of linear differential inclusions,” Nelin. Kolyv., 9, No. 3, 386–400 (2006); English translation: Nonlin. Oscillat., 9, No. 3, 375–390 (2006).

  29. N. V. Skripnik, “Periodic solutions of linear impulsive differential inclusions,” Ukr. Mat. Zh., 60, No. 9, 1287–1296 (2008); English translation: Ukr. Math. J., 60, No. 9, 1498–1508 (2008).

  30. V. Lakshmikantham and R. N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London (2003).

    MATH  Google Scholar 

  31. N. A. Perestyuk and N. V. Skripnik, “Averaging of fuzzy systems,” Ukr. Mat. Zh., 70, No. 3, 412–428 (2018); English translation: Ukr. Math. J., 70, No. 3, 477–494 (2018).

  32. T. A. Komleva, A.V. Plotnikov, and N. V. Skripnik, “Differential equations with set-valued solutions,” Ukr. Mat. Zh., 60, No. 10, 1326–1337 (2008); English translation: Ukr. Math. J., 60, No. 10, 1540–1556 (2008).

  33. A.V. Plotnikov and T. A. Komleva, “Averaging of fuzzy differential equations on a finite interval,” Nelin. Kolyv., 14, No. 4, 516–527 (2011); English translation: Nonlin. Oscillat., 14, No. 4, 547–559 (2012).

  34. A.V. Plotnikov and T. A. Komleva, “The full averaging of fuzzy integrodifferential equations,” J. Adv. Res. Dyn. Control Syst., 4, No. 1, 48–59 (2012).

    MathSciNet  Google Scholar 

  35. A.V. Plotnikov and T. A. Komleva, “Averaging of the fuzzy differential equations,” J. Uncertain Systems, 6, No. 1, 30–37 (2012).

    MathSciNet  MATH  Google Scholar 

  36. A.V. Arsirii and A.V. Plotnikov, “Systems of control over set-valued trajectories with terminal quality criterion,” Ukr. Mat. Zh., 61, No. 8, 1142–1147 (2009); English translation: Ukr. Math. J., 61, No. 8, 1349–1356 (2009).

  37. T. A. Komleva, I. V. Molchanyuk, N. V. Skripnik, and A.V. Plotnikov, “One linear set-valued control problem,” Doslid. Mat. Mekh., 24, No. 2(34), 45–66 (2019).

    Google Scholar 

  38. T. A. Komleva, L. I. Plotnikova, A.V. Plotnikov, and N. V. Skripnik, “Averaging in fuzzy controlled systems,” Nelin. Kolyv., 14, No. 3, 325–332 (2011); English translation: Nonlin. Oscillat., 14, No. 3, 342–349 (2012).

  39. A.V. Plotnikov, “Controlled quasidifferential equations and some their properties,” Differents. Uravn., 34, No. 10, 1332–1336 (1998).

    MATH  Google Scholar 

  40. V. A. Plotnikov and O. D. Kichmarenko, “Averaging of controlled equations with Hukuhara derivative,” Nelin. Kolyv., 9, No. 3, 376–385 (2006); English translation: Nonlin. Oscillat., 9, No. 3, 365–374 (2006).

  41. Y. Feng and L. Hua, “On the quasi-controllability of continuous-time dynamic fuzzy control systems,” Chaos, Solitons, Fractals, 30, No. 1, 177–188 (2006).

    Article  MathSciNet  Google Scholar 

  42. R. Jafari, S. Razvarz, A. Gegov, and W. Yu, “Fuzzy control of uncertain nonlinear systems with numerical techniques: A survey,” in: Z. Ju, L. Yang, C. Yang, A. Gegov, and D. Zhou (editors), Advances in Computational Intelligence Systems, Conference Proceedings UKCI 2019, 1043, Springer, Cham (2020), pp. 3–14.

    Google Scholar 

  43. R. Jafari and W. Yu, “Fuzzy control for uncertainty nonlinear systems with dual fuzzy equations,” J. Intell. Fuzzy Syst., 29, 1229–1240 (2015).

    Article  MathSciNet  Google Scholar 

  44. S. Melliani, A. El Allaoui, and S. Chadli, “Controlled fuzzy evolution equations,” in: S. Melliani and O. Castillo (editors), Recent Advances in Intuition. Fuzzy Logic Systems, Springer, Cham (2019), pp. 113–126.

    Chapter  Google Scholar 

  45. M. Najariyan and M. H. Farahi, “Optimal control of fuzzy controlled system with fuzzy initial conditions,” Iran. J. Fuzzy Syst., No. 10, 21–35 (2013).

    MathSciNet  MATH  Google Scholar 

  46. M. Najariyan, M. H. Farahi, and M. Alavian, “Optimal control of HIV infection by using fuzzy dynamical systems,” J. Math. Comput. Sci., No. 2, 639–649 (2011).

    Article  Google Scholar 

  47. A.V. Plotnikov and T. A. Komleva, “The averaging of control linear fuzzy 2𝜋-periodic differential equations,” Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, 18, No. 6, 833–847 (2011).

    MathSciNet  MATH  Google Scholar 

  48. N. D. Phu, N. V. Hoa, and H. Vu, “On comparisons of set solutions for fuzzy control integro-differential systems,” J. Adv. Res. Appl. Math., 4, No. 1, 84–101 (2012).

    Article  MathSciNet  Google Scholar 

  49. N. D. Phu and T. T. Tung, “Some properties of sheaf solutions of sheaf set control problems,” Nonlin. Anal., Hybrid Syst., 67, 1309–1315 (2007).

    Article  Google Scholar 

  50. N. D. Phu and T. T. Tung, “Existence of solutions of set control differential equations,” Sci. Technol. Devel., 10, No. 6, 5–14 (2007).

    Google Scholar 

  51. L. T. Quang, N. D. Phu, N. V. Hoa, and H. Vu, “On maximal and minimal solutions for set integro-differential equations with feedback control,” Nonlin. Stud., 20, No. 1, 39–56 (2013).

    MathSciNet  MATH  Google Scholar 

  52. W. Witayakiattilerd, “Nonlinear fuzzy differential equation with time delay and optimal control problem,” Abstr. Appl. Anal., 2015, Article ID 659072 (2015), 14 p.

    MathSciNet  MATH  Google Scholar 

  53. W. Yu and R. Jafari, “Modelling and control of uncertain nonlinear systems with fuzzy equations and Z-number,” Wiley-IEEE Press (2019).

  54. M. Hukuhara, “Integration des applications mesurables dont la valeur est un compact convexe,” Funkcial. Ekvac., No. 10, 205–223 (1967).

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Odessa State Academy of Building and Architecture, Odessa, Ukraine

    T. O. Komleva & A.V. Plotnikov

Authors
  1. T. O. Komleva
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A.V. Plotnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. O. Komleva.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 8, pp. 1082–1094, August, 2020. Ukrainian DOI: 10.37863/umzh.v72i8.2300.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Komleva, T.O., Plotnikov, A. One Time-Optimal Problem for a Set-Valued Linear Control System. Ukr Math J 72, 1251–1266 (2021). https://doi.org/10.1007/s11253-020-01854-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11253-020-01854-2

Search

Navigation