M. A. Artemov, E. S. Baranovskii
AN EXTREMUM PROBLEM FOR A LINEAR INTEGRO-DIFFERENTIAL SYSTEM DESCRIBING CREEPING FLOWS OF A VISCOELASTIC FLUID
DOI: 10.17804/2410-9908.2021.2.052-063 We consider an optimal control problem for an integro-differential system (with a quadratic cost functional) modeling a three-dimensional creeping flow of an incompressible viscoelastic fluid in a bounded domain with impermeable solid walls. The fluid flow is controlled by the time-dependent external force. The concept of the control operator is proposed. We prove a theorem on the existence of a unique optimal control under the assumption that the set of admissible controls is convex and that it is closed in a suitable function space. Moreover, we obtain a variational inequality for the optimal control. The proof of this theorem is based on the application of the Faedo–Galerkin approximation scheme taking into account energy estimates of approximate solutions and using the lemma on the existence and uniqueness of the metric projection of a point onto a closed convex set in a real Hilbert space.
Keywords: viscoelastic fluid, creeping flow, integro-differential equations, control operator, optimal control, existence theorem, variational inequality Bibliography:
- Fursikov A.V. Optimal Control of Distributed Systems. Providence, RI, AMS, 2000.
- Litvinov W.G. Optimization in Elliptic Problems with Application to Mechanics of Deformable Body and Fluid Mechanics. Basel, Springer, 2000.
- Alekseev G.V., Tereshko D.A. Analiz i optimizatsiya v gidrodinamike vyazkoi zhidkosti [Analysis and optimization in viscous fluid hydrodynamics]. Vladivostok, Dalnauka Publ., 2008. (In Russian).
- Saut J.-C. Lectures on the mathematical theory of viscoelastic fluids. Lectures on the analysis of nonlinear partial differential equations, part 3, Int. Press, Somerville, 2013, pp. 325–393.
- Temam R. Navier–Stokes Equations. Theory and Numerical Analysis, Amsterdam, New York, Oxford, North-Holland Publishing Co., 1977.
- Oskolkov A.P., Shadiev R. Towards a theory of global solvability on [0,∞) of initial-boundary value problems for the equations of motion of Oldroyd and Kelvin–Voight fluids. Journal of Mathematical Sciences, 1994, vol. 68, pp. 240–253. DOI: 10.1007/BF01249338.
- Oskolkov A.P. Smooth global solutions of initial boundary-value problems for the equations of Oldroyd fluids and of their ε-approximations. Journal of Mathematical Sciences, 1998, vol. 89, 1750–1763. DOI: 10.1007/BF02355375.
- Doubova A., Fernandez-Cara E. On the control of viscoelastic Jeffreys fluids. Systems & Control Letters, 2012, vol. 61, pp. 573–579. DOI: 10.1016/j.sysconle.2012.02.003.
- Hatzikiriakos S. G. Wall slip of molten polymers. Progress in Polymer Science, 2012, vol. 37, pp. 624–643. DOI: 10.1016/j.progpolymsci.2011.09.004.
- Ullah H., Siddiqui A.M., Sun H., Haroon T. Slip effects on creeping flow of slightly non-Newtonian fluid in a uniformly porous slit. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2019, vol. 41, article ID 412. DOI: 10.1007/s40430-019-1917-2.
- Burmasheva N.V., Prosviryakov E.Yu. An exact solution for describing the unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Temperature field investigation. Diagnostics, Resource and Mechanics of materials and structures, 2020, iss. 1, pp. 6–23. DOI: 10.17804/2410-9908.2020.1.006-023.
- Burmasheva N.V., Prosviryakov E.Yu. Exact solution for describing a unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Pressure field investigation. Diagnostics, Resource and Mechanics of materials and structures, 2020, iss. 2, pp. 61–75. DOI: 10.17804/2410-9908.2020.2.061-075. Available at: https://dream-journal.org/DREAM_Issue_2_2020_Burmasheva_N.V._et_al._061_075.pdf
- Kinderlehrer D., Stampacchia G. An Introduction to Variational Inequalities and Their Applications. New York, Academic Press, 1980.
М. А. Артемов, Е. С. Барановский
ЭКСТРЕМАЛЬНАЯ ЗАДАЧА ДЛЯ ЛИНЕЙНОЙ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОЙ СИСТЕМЫ, ОПИСЫВАЮЩЕЙ ПОЛЗУЩЕЕ ТЕЧЕНИЕ ВЯЗКОУПРУГОЙ ЖИДКОСТИ
Рассматривается экстремальная задача управления для интегро-дифференциальной системы (с квадратичным целевым функционалом), моделирующей трехмерное ползущее течение несжимаемой вязкоупругой жидкости в ограниченной области с непроницаемыми твердыми стенками. Параметром управления служит зависящая от времени внешняя сила. Вводится понятие оператора управления. Доказана теорема о существовании единственного оптимального управления в предположении, что множество допустимых управлений выпукло и замкнуто в соответствующем функциональном пространстве. Кроме того, получено вариационное неравенство, которому удовлетворяет оптимальное управление. Доказательство теоремы основано на применении метода Фаэдо–Галеркина с учетом энергетических оценок приближенных решений и использовании леммы о существовании и единственности метрической проекции точки на выпуклое замкнутое множество в вещественном гильбертовом пространстве.
Ключевые слова: вязкоупругая жидкость, ползущее течение, интегро-дифференциальные уравнения, оператор управления, оптимальное управление, теорема существования, вариационные неравенства Библиография:
- Fursikov A. V. Optimal Control of Distributed Systems, Rus. transl. – Providence, RI, AMS, 2000.
- Litvinov W. G. Optimization in Elliptic Problems with Application to Mechanics of Deformable Body and Fluid Mechanics. – Basel, Springer, 2000.
- Алексеев Г. В., Терешко Д. А. Анализ и оптимизация в гидродинамике вязкой жидкости. – Дальнаука, Владивосток, 2008.
- Saut J.-C. Lectures on the mathematical theory of viscoelastic fluids. Lectures on the analysis of nonlinear partial differential equations. Part 3. – Int. Press, Somerville, 2013. – P. 325–393.
- Temam R. Navier–Stokes Equations. Theory and Numerical Analysis. – Amsterdam, New York, Oxford : North-Holland Publishing Co., 1977.
- Oskolkov A. P., Shadiev R. Towards a theory of global solvability on [0,∞) of initial-boundary value problems for the equations of motion of Oldroyd and Kelvin–Voight fluids // Journal of Mathematical Sciences. – 1994. – Vol. 68. – P. 240–253. – DOI: 10.1007/BF01249338.
- Oskolkov A. P. Smooth global solutions of initial boundary-value problems for the equations of Oldroyd fluids and of their ε-approximations // Journal of Mathematical Sciences. – 1998. – Vol. 89. – P. 1750–1763. – DOI: 10.1007/BF02355375.
- Doubova A., Fernandez-Cara E. On the control of viscoelastic Jeffreys fluids // Systems & Control Letters. – 2012. – Vol. 61. – P. 573–579. – DOI: 10.1016/j.sysconle.2012.02.003.
- Hatzikiriakos S. G. Wall slip of molten polymers // Progress in Polymer Science. – 2012. – Vol. 37. – P. 624–643. – DOI: 10.1016/j.progpolymsci.2011.09.004.
- Slip effects on creeping flow of slightly non-Newtonian fluid in a uniformly porous slit / H. Ullah, A. M. Siddiqui, H. Sun, T. Haroon // Journal of the Brazilian Society of Mechanical Sciences and Engineering. – 2019. – Vol. 41. – Article ID 412. – DOI: 10.1007/s40430-019-1917-2.
- Burmasheva N. V., Prosviryakov E. Yu. An exact solution for describing the unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Temperature field investigation // Diagnostics, Resource and Mechanics of materials and structures. – 2020. – Iss. 1. – P. 6–23. – DOI: 10.17804/2410-9908.2020.1.006-023. – URL: https://dream-journal.org/DREAM_Issue_2_2020_Burmasheva_N.V._et_al._061_075.pdf
- Burmasheva N. V., Prosviryakov E. Yu. Exact solution for describing a unidirectional Marangoni flow of a viscous incompressible fluid with the Navier boundary condition. Pressure field investigation // Diagnostics, Resource and Mechanics of materials and structures. – 2020. – Iss. 2. – P. 61–75. – DOI: 10.17804/2410-9908.2020.2.061-075. – URL: https://dream-journal.org/DREAM_Issue_2_2020_Burmasheva_N.V._et_al._061_075.pdf
- Kinderlehrer D., Stampacchia G. An Introduction to Variational Inequalities and Their Applications. – New York : Academic Press, 1980.
Библиографическая ссылка на статью
Artemov M. A., Baranovskii E. S. An Extremum Problem for a Linear Integro-Differential System Describing Creeping Flows of a Viscoelastic Fluid // Diagnostics, Resource and Mechanics of materials and structures. -
2021. - Iss. 2. - P. 52-63. - DOI: 10.17804/2410-9908.2021.2.052-063. -
URL: http://dream-journal.org/issues/2021-2/2021-2_318.html (accessed: 19.04.2024).
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