Divided Optimal Control for Parabolic-Hyperbolic Equation with Non-local Pointed Boundary Conditions and Quadratic Quality Criterion

  • Volodymyr O. Kapustyan
  • Ivan O. PyshnograievEmail author
Part of the Understanding Complex Systems book series (UCS)


We obtain necessary and sufficient conditions for finding the divided optimal control for parabolic-hyperbolic equations with non-local boundary conditions and general quadratic criterion in the special norm. The initial data, which guarantee the classical solvability of the problem, was drown out. The unique solvability of problem is established, systems kernels are estimated, and the convergence of solutions of the problem is proved.


  1. 1.
    Kseniia, I., Ivan, P.: A composite indicator of K-society measurement. In: Proceedings of the 11th International Conference on ICT in Education, Research and Industrial Applications: Integration, Harmonization and Knowledge Transfer, pp. 161–171 (2015)Google Scholar
  2. 2.
    Zgurovsky, M., Boldak, A., Yefremov, K., Pyshnograiev, I.: Modeling and investigating the behavior of complex socio-economic systems. In: Conference Proceedings of 2017 IEEE First Ukraine Conference on Electrical and Computer Engineering (UKRCON), pp. 1113–1116 (2017)Google Scholar
  3. 3.
    Kapustyan, V.O., Pyshnograiev, I.O.: Problem of optimal control for parabolic-hyperbolic equations with nonlocal point boundary conditions and semidefinite quality criterion. Ukrainskyi Matematychnyi Zhurnal. 67(8), 1068–1081 (2016)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Kapustyan, V.O., Kapustian, O.A., Mazur, O.K.: Problem of optimal control for the Poisson equation with nonlocal boundary conditions. J. Math. Sci. 201, 325–334 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kapustyan, V.O., Kapustyan, O.V., Kapustian, O.A., Mazur, O.K.: The optimal control problem for parabolic equation with nonlocal boundary conditions in circular sector. In: Continuous and Distributed Systems II, pp. 297–314. Springer, Cham (2015)Google Scholar
  6. 6.
    Infante, G.: Positive solutions of nonlocal boundary value problems with singularities. Discrete Contin. Dyn. Syst. 3, 377384 (2009)MathSciNetGoogle Scholar
  7. 7.
    Martin-Vaquero, J., Wade, B.A.: On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions. Appl. Numer. Math. 36, 3411–3418 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Nakhusheva, Z.A.: Nonlocal problem for the Lavrentev-Bitsadze equation and its analogs in the theory of equations of mixed parabolic-hyperbolic type. Differ. Equ. 49(10), 1299–1306 (2013)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Repin, O.A., Kumykova, S.K.: A nonlocal problem for a mixed-type equation whose order degenerates along the line of change of type. Russ. Math. 57(9), 49–56 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kapustyan, V.O., Pyshnograiev, I.O.: Approximate Optimal Control for Parabolic Hyperbolic Equations with Nonlocal Boundary Conditions and General Quadratic Quality Criterion. Advances in Dynamical Systems and Control, pp. 387–401. Springer, Cham (2016)Google Scholar
  11. 11.
    Kapustyan, V.O., Pyshnograiev, I.O.: The conditions of existence and uniqueness of the solution of a parabolic-hyperbolic equation with nonlocal boundary conditions [Ukrainian]. Science News NTUU “KPI” 4, 72–86 (2012)Google Scholar
  12. 12.
    Egorov, A.I.: Optimal Control for Heating and Diffusing Processes. Nauka, Moscow (1978)Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Volodymyr O. Kapustyan
    • 1
  • Ivan O. Pyshnograiev
    • 2
    Email author
  1. 1.Igor Sikorsky Kyiv Polytechnic InstituteKyivUkraine
  2. 2.World Data Center for Geoinformatics and Sustainable DevelopmentNational Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”KyivUkraine

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