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Optimal feedback control for a thermoviscoelastic model of the motion of water polymer solutions. (Russian, English) Zbl 1438.76005

Mat. Tr. 21, No. 2, 181-203 (2018); translation in Sib. Adv. Math. 29, No. 2, 137-152 (2019).
Summary: We study an optimal feedback control problem for an initial boundary value problem of a thermoviscoelastic model describing the motion of weakly concentrated water polymer solutions in the presence of dependence of the viscosity on the temperature. We prove the existence of an optimal solution minimizing to a given bounded lower semicontinuous quality functional. For proving the existence of an optimal solution, we use the topological approximation method for studying problems in hydrodynamics.

MSC:

76A10 Viscoelastic fluids
35Q35 PDEs in connection with fluid mechanics
49K21 Optimality conditions for problems involving relations other than differential equations
93C20 Control/observation systems governed by partial differential equations
93B52 Feedback control
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[1] S. N. Antontsev, A. V. Kazhikhov, and V. N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids (Nauka, Novosibirsk, 1983; North-Holland, Amsterdam, 1990). · Zbl 0568.76001
[2] J. P. Aubin and A. Cellina, Differential Inclusions. Set-Valued Maps and Viability Theory (Springer-Verlag; Berlin etc., 1984). · Zbl 0538.34007 · doi:10.1007/978-3-642-69512-4
[3] Yu. G. Borisovich, B. D. Gel’man, A. D. Myshkis, and V. V. Obukhovskiĭ, Introduction to the Theory of Multi-Valued Mappings and Differential Inclusions (Librokom, Moscow, 2011) [in Russian]. · Zbl 1231.54001
[4] H. Choi, R. Temam, P. Moin, and J. Kim, “Feedback control for unsteady flow and its application to the stochastic Burgers equation,” J. Fluid Mech. 253, 509 (1993). · Zbl 0810.76012 · doi:10.1017/S0022112093001880
[5] A. F. Filippov, “On certain questions in the theory of optimal control,” Vestn. Mosk. Univ., Ser. Mat. Mekh. Astron. Fiz. Khim. 14 (2), 25 (1959) [J. Soc. Ind. Appl.Math., Ser. A, Control 1, 76 (1962)]. · Zbl 0139.05102
[6] A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, (Nauchnaya Kniga, Novosibirsk, 1999; AMS, Providence, RI, 2000). · Zbl 0938.93003 · doi:10.1090/mmono/187
[7] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow (Nauka, Moscow, 1970; Gordon and Breach Science Publishers, New York-London-Paris, 1969). · Zbl 0184.52603
[8] A. P. Oskolkov, “Some quasilinear systems occurring in the study of the motion of viscous fluids,” Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 52, 128 (1975) [J. Sov.Math. 9, 765 (1978)]. · Zbl 0358.76025
[9] J. Simon, “Compact sets in the space <Emphasis Type=”Italic“>Lp <Emphasis Type=”Italic“>(0, T; B),” Ann. Mat. Pura Appl. 146, 65 (1987). · Zbl 0629.46031 · doi:10.1007/BF01762360
[10] R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis (AMS, Providence, RI, 2001; Mir, Moscow, 1981). · Zbl 0981.35001
[11] I. I. Vorovich and V. I. Yudovich, “Steady flow of a viscous incompressible fluid,” Mat. Sb. (N.S.) 53(95) (4) 393 (1961). · Zbl 0096.41301
[12] A. V. Zvyagin, “An optimal control problem with feedback for a mathematical model of the motion of weakly concentrated water polymer solutions,” Sib. Mat. Zh. 54, 807 (2013) [Sib. Math. J. 54, 640 (2013)]. · Zbl 1275.49009 · doi:10.1134/S003744661304006X
[13] A. V. Zvyagin, “Solvability for equations of motion of weak aqueous polymer solutions with objective derivative,” Nonlinear Anal., TheoryMethods Appl. Ser. A 90, 70 (2013). · Zbl 1404.76016 · doi:10.1016/j.na.2013.05.022
[14] A. V. Zvyagin, “Solvability of the stationary mathematical model of a non-Newtonian fluid motion with the objective derivative,” Fixed Point Theory 15, 623 (2014). · Zbl 1298.76022
[15] A. V. Zvyagin, “Study of Solvability of a Thermoviscoelastic Model Describing the Motion of Weakly ConcentratedWater Solutions of Polymers,” Sib. Mat. Zh. 59, 1066 (2018) [Sib.Math. J. 59, 843 (2018)]. · Zbl 1411.35233
[16] A. V. Zvyagin and V. P. Orlov, “Solvability of the thermoviscoelasticity problem for linearly elastically retarded Voigt fluid,” Mat. Zametki 97, 681 (2015) [Math. Notes 97, 694 (2015)]. · Zbl 1326.35297 · doi:10.4213/mzm10508
[17] V. G. Zvyagin, “Topological approximation approach to study of mathematical problems of hydrodynamics,” CMFD, 46, 92 (2012) [J.Math. Sci., 201, 830 (2014)]. · Zbl 1310.35124
[18] V. Zvyagin, V. Obukhovskii, and A. Zvyagin, “On inclusions with multivalued operators and their applications to some optimization problems,” J. Fixed Point Theory Appl. 16, 27 (2014). · Zbl 1316.49025 · doi:10.1007/s11784-015-0219-2
[19] V. G. Zvyagin and V. P. Orlov, “On certain mathematical models in continuum thermomechanics,” J. Fixed Point Theory Appl. 15, 3 (2014). · Zbl 1315.80004 · doi:10.1007/s11784-014-0179-y
[20] V. G. Zvyagin and V. P. Orlov, “Solvability of a parabolic problem with non-smooth data,” J. Math. Anal. Appl. 453, 589 (2017). · Zbl 1404.35225 · doi:10.1016/j.jmaa.2017.04.028
[21] V. G. Zvyagin and M. V. Turbin, Mathematical Problems of the Hydrodynamics of Viscoelastic Media, (URSS, Moscow, 2012) [in Russian].
[22] V. G. Zvyagin and M. V. Turbin, “Optimal feedback control in the mathematical model of low concentrated aqueous polymer solutions,” J. Optim. Theory Appl. 148, 146 (2011). · Zbl 1211.49045 · doi:10.1007/s10957-010-9749-3
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