Feedback Control Problem for Modified Kelvin-Voigt Model

Abstract

This paper deals with the optimal feedback control problem for the modified Kelvin-Voigt model. The considered model describes the motion of weakly concentrated aqueous polymer solutions. In our case, the control function (the external force) depends on the velocity of the fluid. In such a way, the control is not selected from a finite set of available controls, but belongs to the image of some multi-valued map. The solution for the control problem of fluid motion is a pair: the velocity of the fluid and the control (the density of external forces). Since there can be many such pairs, the concept of optimal solution naturally arises, which gives a minimum to specified cost functional. For the considered optimal feedback control problem the existence theorem on weak solution is proved.

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Funding

The research of the first author (Theorem 1 on the existence of a weak solution for the feedback control problem) was supported by the Russian Foundation for Basic Research, project no. 20-01-00051. The research of the second author (Theorem 2 on the existence of a solution minimizing the given cost functional) was supported by the Russian Science Foundation, project no. 19-11-00146.

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Correspondence to Mikhail Turbin.

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Ustiuzhaninova, A., Turbin, M. Feedback Control Problem for Modified Kelvin-Voigt Model. J Dyn Control Syst (2021). https://doi.org/10.1007/s10883-021-09539-0

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Keywords

  • Feedback control
  • Control problem
  • Modified Kelvin-Voigt model
  • Weak solution
  • Multi-valued map
  • Existence theorem

Mathematics Subject Classification (2010)

  • 76D55
  • 49J20
  • 76A05
  • 49J53