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Impulse Trajectory and Final Controllability of Parabolic-Hyperbolic Systems*

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Cybernetics and Systems Analysis Aims and scope

Abstract

The authors analyze the existence and uniqueness of the generalized solutions to boundary-value problems for equations of parabolic-hyperbolic type with generalized functions of finite order in their right-hand sides. The motivation is the analysis of the problems of trajectory and final controllability of systems described by these boundary-value problems and subjected to concentrated influences of impulse or point type. The systems can be considered “toy models” of the interaction of a solid body and a liquid. A priori inequalities in negative norms are obtained. The theorems of the existence and uniqueness of the generalized solutions and theorems of the trajectory and final controllability of systems with singular influences are proved.

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References

  1. A. G. Butkovskii and L. M. Pustyl’nikov, Theory of Dynamic Control of Distributed-Parameter Systems [in Russian], Nauka, Moscow (1980).

  2. S. I. Lyashko, D. A. Klyushin, and A. S. Trygub, Modeling and Optimization of Underground Mass Transfer [in Russian], Naukova Dumka, Kyiv (1998).

    Google Scholar 

  3. B. M. Miller and E. Ya. Rubinovich, Optimization of Dynamic Systems with Impulse Control [in Russian], Nauka, Moscow (2005).

    Google Scholar 

  4. S. I. Lyashko, D. A. Klyushin, D. A. Nomirovsky, and V. V. Semenov, “Identification of age-structured contamination sources in ground water,” in: R. Boucekkine, N. Hritonenko, and Y. Yatsenko (eds.), Optimal Control of Age-Structured Populations in Economy, Demography, and the Environment, Routledge, London–New York (2013), pp. 277–292.

    Google Scholar 

  5. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Ser. Grundlehren der mathematischen Wissenschaften, Vol. 170, Springer-Verlag, Berlin–Heidelberg (1971).

  6. J.-L. Lions, Control of Distributed Singular Systems, Ser. Modern Applied Mathematics, Gauthier-Villars (1985).

    Google Scholar 

  7. Yu. M. Berezanskii, Eigenfunction Expansion of Self-Adjoint Operators [in Russian], Naukova Dumka, Kyiv (1965).

    Google Scholar 

  8. V. P. Didenko, “On boundary-value problems for multidimensional hyperbolic equations with degeneracy,” Doklady AN SSSR, Vol. 205, No. 4, 763–766 (1972).

    MathSciNet  Google Scholar 

  9. I. I. Lyashko, V. P. Didenko, and O. E. Tsytrytskyi, Noise Filtration [in Russian], Naukova Dumka, Kyiv (1979).

    Google Scholar 

  10. S. I. Lyashko, Generalized Optimal Control of Linear Systems with Distributed Parameters, Kluwer Academic Publishers, Boston–Dordrecht–London (2002).

  11. V. V. Semenov, “Solvability of a parabolic transmission problem with the condition of a generalized proper lumped source,” Diff. Equations, Vol. 41, 878–886 (2005). https://doi.org/https://doi.org/10.1007/s10625-005-0227-x.

    Article  MATH  Google Scholar 

  12. A. V. Anikushyn and A. L. Hulianytskyi, “Generalized solvability of parabolic integro-differential equations,” Diff. Equations, Vol. 50, 98–109 (2014). https://doi.org/https://doi.org/10.1134/S0012266114010133.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. V. Anikushyn and D. A. Nomirovs’kyi, “Generalized solutions for linear operators with weakened a priori inequalities,” Ukr. Math. J., Vol. 62, 1175–1186 (2011). https://doi.org/10.1007/s11253-011-0435-x.

  14. I. B. Tymchyshyn and D. A. Nomirovskii, “Generalized solvability of a parabolic model describing transfer processes in domains with thin inclusions,” Diff. Equations, Vol. 57, 1053–1062 (2021). https://doi.org/https://doi.org/10.1134/S0012266121080097.

    Article  MathSciNet  MATH  Google Scholar 

  15. I. I. Lyashko, S. I. Lyashko, and V. V. Semenov, “Control of pseudo-hyperbolic systems by concentrated impacts,” J. Autom. Inform. Sci., Vol. 32, Iss. 12, 23–36 (2000). https://doi.org/10.1615/JAutomatInfScien.v32.i12.40.

  16. S. V. Denisov and V. V. Semenov, “Optimization and generalized solution in conjugation problems of parabolic systems,” J. Autom. Inform. Sci., Vol. 37, Iss. 1, 47–61 (2005). https://doi.org/10.1615/JAutomatInfScien.v37.i1.60.

  17. A. V. Anikushyn, “Generalized optimal control for systems described by linear integro-differential equations with nonnegative definite integral operators,” J. Autom. Inform. Sci., Vol. 46, Iss. 6, 58–67 (2014). https://doi.org/10.1615/JAutomatInfScien.v46.i6.60.

  18. S. I. Lyashko and V. V. Semenov, “Controllability of linear distributed systems in classes of generalized actions,” Cybern. Syst. Analysis, Vol. 37, No. 1, 13–32 (2001). https://doi.org/https://doi.org/10.1023/A:1016607831284.

    Article  MathSciNet  MATH  Google Scholar 

  19. S. I. Lyashko, D. A. Nomirovskii, and T. I. Sergienko, “Trajectory and final controllability in hyperbolic and pseudohyperbolic systems with generalized actions,” Cybern. Syst. Analysis, Vol. 37, No. 5, 756–763 (2001). https://doi.org/https://doi.org/10.1023/A:1013871026026.

    Article  MATH  Google Scholar 

  20. S. I. Lyashko, V. V. Semenov, and T. I. Sergienko, “Controllability and optimization of systems of pseudohyperbolic type,” Cybern. Syst. Analysis, Vol. 38, No. 4, 586–596 (2002). https://doi.org/https://doi.org/10.1023/A:1021162303764.

    Article  MathSciNet  MATH  Google Scholar 

  21. V. V. Semenov and N. V. Semenova, “A vector problem of optimal control in a Hilbert space,” Cybern. Syst. Analysis, Vol. 41, No. 2, 255–266 (2005). https://doi.org/https://doi.org/10.1007/s10559-005-0058-z.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. I. Lyashko, “Numerical solution of pseudoparabolic equations,” Cybern. Syst. Analysis, Vol. 31, No. 5, 718–722 (1995). https://doi.org/https://doi.org/10.1007/BF02366321.

    Article  MATH  Google Scholar 

  23. D. A. Nomirovskii, “Approximate method for solving the boundary value problem for a parabolic equation with inhomogeneous transmission conditions of nonideal contact type,” Comp. Math. and Math. Phys., Vol. 46, 995–1006 (2006). https://doi.org/https://doi.org/10.1134/S096554250606008X.

    Article  MathSciNet  Google Scholar 

  24. V. V. Semenov, “Modified extragradient method with Bregman divergence for variational inequalities,” J. Autom. Inform. Sci., Vol. 50, Iss. 8, 26–37 (2018). https://doi.org/10.1615/JAutomatInfScien.v50.i8.30.

  25. J.-L. Lions, “Controlabilite exacte des systemes distributes,” Comptes Rendue des Sciences de l’Acadenic des Sciences, Ser. 1, Vol. 302, No. 13, 471–475 (1986).

  26. L. Stupyalis, “Boundary-value problems for elliptic-hyperbolic equations,” Proc. V. A. Steklov Math. Inst., Vol. CXXV, 211–229 (1973).

  27. L. Stupyalis, “Initial–boundary-value problems for equations of mixed type,” Proc. V. A. Steklov Math. Inst., Vol. CXXVII, 115–145 (1975).

  28. J.-L. Lions and E. Zuazua, “Approximate controllability for a coupled hydro-elastic system,” ESAIM: Control, Optimisation and Calculus of Variations, Vol 1 (1), 1–15 (1995).

    MATH  Google Scholar 

  29. E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory, Ser. Lecture Notes in Physics, Vol. 127, Springer, Heidelberg (1980).

  30. J.-L. Lions, Some Methods of Solving Non-Linear Boundary-Value Problems, Dunod-Gauthier-Villars, Paris (1969).

    Google Scholar 

  31. O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  32. D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Y. I. Petunin, and V. V. Semenov, “Applications of the theory of generalized solvability of linear equations,” in: Generalized Solutions of Operator Equations and Extreme Elements, Springer Optimization and Its Applications, Vol. 55, Springer, New York (2012), pp. 29–101. https://doi.org/10.1007/978-1-4614-0619-8_4.

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Authors and Affiliations

  1. Taras Shevchenko National University of Kyiv, Kyiv, Ukraine

    V. V. Semenov & S. V. Denisov

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  1. V. V. Semenov
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  2. S. V. Denisov
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Correspondence to V. V. Semenov.

Additional information

The study was financially supported by the Ministry of Education and Science of Ukraine (project “Computational algorithms and optimization for artificial intelligence, medicine, and defense,” 0122U002026).

Translated from Kibernetyka ta Systemnyi Analiz, No. 3, May–June, 2023, pp. 71–82.

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Semenov, V.V., Denisov, S.V. Impulse Trajectory and Final Controllability of Parabolic-Hyperbolic Systems*. Cybern Syst Anal 59, 417–427 (2023). https://doi.org/10.1007/s10559-023-00576-0

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