Topological Methods in Nonlinear Analysis

Optimal feedback control in the problem of the motion of a viscoelastic fluid

Valeri Obukhovskiĭ, Pietro Zecca, and Victor G. Zvyagin

Full-text: Open access

Abstract

We study an optimization problem for the feedback control system emerging as a regularized model for the motion of a viscoelastic fluid subject to the Jeffris-Oldroyd rheological relation. The approach includes systems governed by the classical Navier-Stokes equation as a particular case. Using the topological degree theory for condensing multimaps we prove the solvability of the approximating problem and demonstrate the convergence of approximate solutions to a solution of a regularized one. At last we show the existence of a solution minimizing a given convex, lower semicontinuous functional.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 23, Number 2 (2004), 323-337.

Dates
First available in Project Euclid: 31 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1464731412

Mathematical Reviews number (MathSciNet)
MR2078195

Zentralblatt MATH identifier
1259.49006

Citation

Obukhovskiĭ, Valeri; Zecca, Pietro; Zvyagin, Victor G. Optimal feedback control in the problem of the motion of a viscoelastic fluid. Topol. Methods Nonlinear Anal. 23 (2004), no. 2, 323--337. https://projecteuclid.org/euclid.tmna/1464731412


Export citation

References

  • V. Barbu, Optimal control of Navier–Stokes equations with periodic inputs, Nonlinear Anal., 31(1998), 15–31 \ref\key 2
  • V. T. Dmitrienko and V. G. Zvyagin, The topological degree method for equations of the Navier–Stokes type , Abstr. Appl. Anal., 2(1997), 1–45 \ref\key 3
  • I. Ekeland and R. Témam, Convex Analysis and Variational Problems, Classics Appl. Math., 28, SIAM, Philadelfia(1999) \ref\key 4
  • A. V. Fursikov, Some control problems and results related to the unique solvability of the mixed boundary value problem for the Navier–Stokes and Euler three-dimensional systems , Dokl. Akad. Nauk SSSR, 252(1980), 1066–1070, (Russian) \ref\key 5 ––––, Control problems and theorems concerning unique solvability of a mixed boundary value problem for the Navier–Stokes and Euler three-dimensional equations , Math. Sb. (N.S.), 115 (1981), 281–306, 320, (Russian) \ref\key 6 ––––, Optimal Control of Distributed Systems. Theory and Applications , Transl. of Math. Monographs, 187, Amer. Math. Soc., Providence, RI, (2000) \ref\key 7
  • F. Gozzi, S. S. Sritharam and A. Swikech, Viscosity solutions of dynamic-programming equations for the optimal control of the two-dimensional Navier–Stokes equations, Arch. Rational Mech. Anal., 163(2002), 295–237 \ref\key 8
  • M. Kamenskiĭ, V. Obukhovskiĭ and P. Zecca, Condensing multivalued maps and semilinear differential inclusions in Banach spaces , de Gruyter Series in Nonlinear Analysis and Applications, 7 , Walter de Gruyter & Co., Berlin–New York (2001) \ref\key 9
  • J.-L. Lions, Controle Optimal de Systemes Gouvernes par des Equations aux Derivees Partielles, Dunod, Gauthier–Villars, Paris (1968) \ref\key 10
  • R. Témam, Navier-Stokes equations. Theory and numerical analysis , Stud. Math. Appl., 2 , North Holland Publishing Co., Amsterdam–New York–Oxford (1977) \ref\key 11
  • V. G. Zvyagin and V. T. Dmitrienko, On weak solutions of an initial-boundary value problem for the equation of motion of a viscoelastic fluid , Dokl. Akad. Nauk, 380(2001), 308–311, (Russian) \ref\key 12 ––––, On weak solutions of a regularized model of a viscoelastic fluid , Differrentsial'nye Uravneniya, 38 (2002), 1633–1645, (Russian)