The variational optimality condition in the problem of minimizing the finite state norm by a composite system of hyperbolic and ordinary differential equations
DOI:
https://doi.org/10.21638/11701/spbu10.2023.410Abstract
An optimal control problem for a system of linear first-order hyperbolic equations is studied. The boundary conditions are determined from controlled systems of ordinary differential equations. A nonclassical exact formulae for the increment of a linear performance index (a finite state norm) is suggested. Based on this result, a variational optimality condition is proved. The original optimal control problems for a hyperbolic system is reduced to the problem for systems of ordinary differential equations.
Keywords:
hyperbolic system, controlled boundary conditions, norm minimization, variational optimality condition, problem reduction
Downloads
References
Апонин Ю. М., Апонина Е. А., Кузнецов Ю. А. Математическое моделирование пространственно-временной динамики возрастной структуры популяции растений // Математическая биология и биоинформатика. 2006. Т. 1. № 1. С. 1–16.
Vazquez J. L., Zuazua E. Large time behavior for a simplified 1D model of fluid-solid interaction // Comm. Partial Differential Equations. 2003. Vol. 28. N 9–10. P. 1705–1738. https://doi.org/10.1081/PDE-120024530
Faugeras B., Blum J., Heumann H., Boulbe C. Optimal control of a coupled partial and ordinary differential equations system for the assimilation of polarimetry Stokes vector measurements in tokamak free-boundary equilibrium reconstruction with application to ITER // Computer Physics Communications. 2017. Vol. 217. P. 43–57.
Weihua Ruan. A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model // Journal of Mathematical Analysis and Applications. 2008. Vol. 343. Iss. 2. P. 778–796. https://doi.org/10.1016/j.jmaa.2008.01.064
Arguchintsev A. V. Solution of the problem of the optimal control of initial-boundary conditions of a hyperbolic system based on exact increment formulas // Russian Mathematics. 2002. Vol. 46. N 12. P. 21–27.
Габасов Р., Кириллова Ф. М. Особые оптимальные управления. М.: Наука, 1973. 256 с.
Рождественский Б. Л., Яненко Н. Н. Системы квазилинейных уравнений и их приложения к газовой динамике. М.: Наука, 1978. 592 с.
Potapov M. M. A generalized solution of a mixed problem for a first-order semilinear hyperbolic system // Differential Equations. 1983. Vol. 19. N 10. P. 1826–1828.
Срочко В. А., Аксенюшкина Е. В. Параметризация некоторых задач управления линейными системами // Известия Иркутского государственного университета. Сер. Математика. 2019. Т. 30. С. 83–98. https://doi.org/10.26516/1997-7670.2019.30.83
Срочко В. А., Аксенюшкина Е. В., Антоник В. Г. Решение линейно-квадратичной задачи оптимального управления на основе конечномерных моделей // Известия Иркутского государственного университета. Сер. Математика. 2021. Т. 37. С. 3–16. https://doi.org/10.26516/1997-7670.2021.37.3
Popkov A. S. Optimal program control in the class of quadratic splines for linear systems // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2020. Т. 16. Вып. 4. С. 462–470.
Дривотин О. И. О численном решении задачи оптимального управления на основе метода, использующего вторую вариацию траектории // Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления. 2019. Т. 15. Вып. 2. С. 283–295. https://doi.org/10.21638/11702/spbu10.2019.211
References
Aponin Yu. M., Aponina E. А., Kuznetsov Yu. A. Matematicheskoe modelirovanie prostranstvenno-vremennoi dinamiki vozrastnoi struktury populyatsii rastenii [Mathematical modeling of space-time dynamics of the age structure of the plants population]. Mathematical Biology and Bioinformatics, 2006, vol. 1, no. 1, pp. 1–16. (In Russian)
Vazquez J. L., Zuazua E. Large time behavior for a simplified 1D model of fluid-solid interaction. Comm. Partial Differential Equations, 2003, vol. 28, no. 9–10, pp. 1705–1738. https://doi.org/10.1081/PDE-120024530
Faugeras B., Blum J., Heumann H., Boulbe C. Optimal control of a coupled partial and ordinary differential equations system for the assimilation of polarimetry Stokes vector measurements in tokamak free-boundary equilibrium reconstruction with application to ITER. Computer Physics Communications, 2017, vol. 217, pp. 43–57.
Weihua Ruan. A coupled system of ODEs and quasilinear hyperbolic PDEs arising in a multiscale blood flow model. Journal of Mathematical Analysis and Applications, 2008, vol. 343, iss. 2, pp. 778–796. https://doi.org/10.1016/j.jmaa.2008.01.064
Arguchintsev A. V. Solution of the problem of the optimal control of initial-boundary conditions of a hyperbolic system based on exact increment formulas. Russian Mathematics, 2002, vol. 46, no. 12, pp. 21–27.
Gabasov R., Kirillova F. M. Osobye optimal’nye upravleniya [ Special optimal controls]. Moscow, Nauka Publ., 1973, 256 p. (In Russian)
Rozhdestvenskiyi B. L., Yanenko N. N. Systemi kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike [ Systems of quasilinear equations and their applications to gas dynamics]. Moscow, Nauka Publ., 1978, 592 p. (In Russian)
Potapov M. M. A generalized solution of a mixed problem for a first-order semilinear hyperbolic system. Differential Equations, 1983, vol. 19, no. 10, pp. 1826–1828.
Srochko V. A., Aksenyushkina E. V. Parametrizatsiya nekotorykh zadach upravleniya lineynymi sistemami [Parameterization of some linear systems control problems]. The Bulletin of Irkutsk State University. Series Mathematics, 2019, vol. 30, pp. 83–98. https://doi.org/10.26516/1997-7670.2019.30.83 (In Russian)
Srochko V. A., Aksenyushkina E. V., Antonik V. G. Reshenie lineino-kvadratichnoi zadachi optimal'nogo upravlenia na osnove konechnomernykh modelei [Resolution of a linear-quadratic optimal control problem based on finite-dimensional models]. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 37, pp. 3–16.
Popkov A. S. Optimal program control in the class of quadratic splines for linear systems. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2020, vol. 16, iss. 4, pp. 462–470.
Drivotin O. I. O chislennom reshenii zadach optimal'nogo upravleniya na osnove metoda, ispol'zuyuschego vtoruyu variatsiyu traektorii [On numerical solution of the optimal control problem based on a method using the second variation of a trajectory]. Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes, 2019, vol. 15, iss. 2, pp. 283–295. https://doi.org/10.21638/11702/spbu10.2019.211 (In Russian)
Downloads
Published
How to Cite
Issue
Section
License
Articles of "Vestnik of Saint Petersburg University. Applied Mathematics. Computer Science. Control Processes" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
