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Optimization of Source Parameters in Multipoint Nonseparated Conditions for Linear Dynamical Systems

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Abstract

The problem of optimizing the right-hand sides of linear nonlocal multipoint conditions for a linear system of differential equations is considered. Necessary optimality conditions of the first order are obtained, which allow the use of efficient first-order methods for the numerical solution of the problem under consideration. Solutions of test problems are presented and analyzed.

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REFERENCES

  1. A. M. Nakhushev, Loaded Equations and Applications (Nauka, Moscow, 2012) [in Russian].

    MATH  Google Scholar 

  2. V. A. Nakhusheva, Differential Equations for Mathematical Models of Nonlocal Processes (Nauka, Moscow, 2006) [in Russian].

    MATH  Google Scholar 

  3. L. S. Pul’kina, “Boundary-value problems for a hyperbolic equation with nonlocal conditions of the I and II kind,” Russ. Math. 56 (4), 62–69 (2012).

    Article  Google Scholar 

  4. A. A. Alikhanov, “V.A. Steklov nonlocal boundary value problem of the second class for simple equations of mathematical physics,” Vestn. Sam. Gos. Tekh. Univ. Ser. Fiz.-Mat. Nauki 1 (30), 15–23 (2013).

    Google Scholar 

  5. K. R. Aida-zade and V. M. Abdullaev, “On the solution of boundary value problems with nonseparated multipoint and integral conditions,” Differ. Equations 49 (9), 1114–1125 (2013).

    Article  MathSciNet  Google Scholar 

  6. D. S. Dzhumabaev and A. E. Imanchiev, “The correct solvability of a linear multipoint boundary value problem,” Math. J. 5 (15), 30–38 (2005).

    MathSciNet  MATH  Google Scholar 

  7. A. T. Assanova, A. E. Imanchiyev, and Zh. M. Kadirbayeva, “On nonlocal problems for systems of Sobolev-type differential equations with a multipoint condition,” Izv. Vyssh. Uch. Zaved. Mat., No. 12, 3–15 (2019).

  8. D. Devadze and V. Beridze, “Optimality conditions and solution algorithms of optimal control problems for nonlocal boundary-value problems,” J. Math. Sci. 218, 731–736 (2016).

    Article  MathSciNet  Google Scholar 

  9. A. S. Antipin, “Terminal control of boundary models,” Comput. Math. Math. Phys. 54 (2), 257–285 (2014).

    Article  MathSciNet  Google Scholar 

  10. A. S. Antipin and E. V. Khoroshilova, “Feedback synthesis for a terminal control problem,” Comput. Math. Math. Phys. 58 (12), 1903–1918 (2018).

    Article  MathSciNet  Google Scholar 

  11. V. M. Abdullayev and K. R. Aida-zade, “Approach to the numerical solution of optimal control problems for loaded differential equations with nonlocal conditions,” Comput. Math. Math. Phys. 59 (5), 696–707 (2019).

    Article  MathSciNet  Google Scholar 

  12. D. S. Dzhumabaev, “On one approach to solve the linear boundary value problems for Fredholm integrodifferential equations,” J. Comput. Appl. Math. 294 (2), 342–357 (2016).

    Article  MathSciNet  Google Scholar 

  13. V. M. Abdullaev and K. R. Aida-zade, “On the numerical solution of loaded systems of ordinary differential equations,” Comput. Math. Math. Phys. 44 (9), 1505–1515 (2004).

    MathSciNet  Google Scholar 

  14. V. M. Abdullaev and K. R. Aida-zade, “Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations,” Comput. Math. Math. Phys. 54 (7), 1096–1109 (2014).

    Article  MathSciNet  Google Scholar 

  15. L. T. Ashchepkov, “Optimal control of a system with intermediate conditions,” J. Appl. Math. Mech. 45 (2), 153–158 (1981).

    Article  MathSciNet  Google Scholar 

  16. O. O. Vasil’eva and K. Mizukami, “Dynamical processes described by a boundary value problem: Necessary conditions of optimality and methods of solving,” J. Comput. Syst. Sci. Int. 39 (1), 90–95 (2000).

    Google Scholar 

  17. O. O. Vasil’eva and K. Mizukami, “Optimal control for boundary value problems,” Russ. Math. 38 (12), 31–39 (1994).

    MathSciNet  MATH  Google Scholar 

  18. Y. A. Sharifov and N. B. Mammadova, “Optimal control problem described by impulsive differential equations with nonlocal boundary conditions,” Differ. Equations 50 (3), 403–411 (2014).

    Article  MathSciNet  Google Scholar 

  19. A. T. Assanova, “Solvability of a nonlocal problem for a hyperbolic equation with integral conditions,” Electron. J. Differ. Equations 170, 1–12 (2017).

    MathSciNet  MATH  Google Scholar 

  20. C. J. de la Vallée-Poussin, “Sur l'équation différentielle linéaire du second ordre: Determination d’une intégrale par deux valeurs assignées: Extension aux équations d’orde n,” J. Math. Pures Appl. 8 (9), 125–144 (1929).

    MATH  Google Scholar 

  21. Ya. D. Tamarkin, On Certain General Problems in the Theory of Linear Ordinary Differential Equations and on Series Expansions of Arbitrary Functions (Petrograd, 1917) [in Russian].

    Google Scholar 

  22. I. T. Kiguradze, “Boundary value problems for system of ordinary differential equations,” Itogi Nauki Tekh. Sovrem. Probl. Mat. Nov. Dostizheniya 30, 3–103 (1987).

    MathSciNet  Google Scholar 

  23. A. S. Antipin and E. V. Khoroshilova, “Controlled dynamic model with boundary-value problem of minimizing a sensitivity function,” Optim. Lett. 13 (3), 451–473 (2019).

    Article  MathSciNet  Google Scholar 

  24. K. R. Aida-zade and E. R. Ashrafova, “Numerical leak detection in a pipeline network of complex structure with unsteady flow,” Comput. Math. Math. Phys. 57 (12), 1919–1934 (2017).

    Article  MathSciNet  Google Scholar 

  25. V. M. Abdullayev and K. R. Aida-zade, “Numerical solution of the problem of determining the number and locations of state observation points in feedback control of a heating process,” Comput. Math. Math. Phys. 58 (1), 78–89 (2018).

    Article  MathSciNet  Google Scholar 

  26. A. G. Butkovskii, Methods for Control of Distributed Parameter Systems (Nauka, Moscow, 1975) [in Russian].

    Google Scholar 

  27. F. P. Vasil’ev, Optimization Methods (Faktorial, Moscow, 2002) [in Russian].

    Google Scholar 

  28. B. T. Polyak, Introduction to Optimization (Nauka, Moscow, 1983) [in Russian.

  29. Yu. G. Evtushenko, Methods for Solving Optimization Problems and Their Applications in Optimization Systems (Nauka, Moscow, 1982) [in Russian].

    MATH  Google Scholar 

  30. K. Moszyński, “A method of solving the boundary value problem for a system of linear ordinary differential equation,” Algorytmy Varshava 11 (3), 25–43 (1964).

    MathSciNet  MATH  Google Scholar 

  31. A. A. Abramov, “A variation of the 'dispersion' method,” USSR Comput. Math. Math. Phys. 1, 368–371 (1961).

    Article  Google Scholar 

  32. A. A. Abramov, N. G. Burago, et al., “Software package for solving linear two-point boundary value problems,” Reports on Computer Software (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1982) [in Russian].

    Google Scholar 

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Correspondence to V. M. Abdullayev or K. R. Aida-zade.

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Translated by I. Ruzanova

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Abdullayev, V.M., Aida-zade, K.R. Optimization of Source Parameters in Multipoint Nonseparated Conditions for Linear Dynamical Systems. Comput. Math. and Math. Phys. 61, 512–526 (2021). https://doi.org/10.1134/S0965542521020020

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