The goal of this article is to study the feedback control for non-stationary three-dimensional Navier–Stokes–Voigt equations. Based on the existence, uniqueness, and boundedness result of the weak solutions to the equations, we obtain the existence of solutions to the feedback control system. An existence result for an optimal control problem is also given. We illustrate our main result with an evolutionary hemivariational inequality.
OskolkovAP.The uniqueness and solvability in the large of boundary value problems for the equations of motion of aqueous solutions of polymers. Nauchn Semin LOMI1973; 38: 98–136.
2.
CaoYLunasinEMTitiES.Global well-posedness of the three-dimensional viscous and inviscid simplified Bardina turbulence models. Commun Math Sci2006; 4: 823–848.
3.
AbergelFTemamR.On some control problems in fluid mechanics. Theoret Comput Fluid Dynam1990; 1: 303–325.
4.
AnhCTNguyetTM.Optimal control of the instationary three dimensional Navier–Stokes–Voigt equations. Numer Funct Anal Optimiz2016; 37(4): 415–439.
5.
AnhCTTrangPT.Pull-back attractors for three-dimensional Navier–Stokes–Voigt equations in some unbounded domains. Proc Royal Soc Edinburgh Sect A2013; 143: 223–251.
6.
CelebiAOKalantarovVKPolatM.Global attractors for 2D Navier–Stokes-Voight equations in an unbounded domain. Appl Anal2009; 88: 381–392.
7.
FattoriniHOSritharanS.Necessary and sufficient for optimal controls in viscous flow problems. Proc Royal Soc Edinburgh1994; 124: 211–251.
8.
FursikovAVGunzburgerMDHouLS.Optimal boundary control for the evolutionary Navier–Stokes system: The three-dimensional case. SIAM J Control Optim2005; 43: 2191–2232.
9.
García-LuengoJMarín-RubioPRealJ.Pullback attractors for three-dimensional non-autonomous Navier–Stokes–Voigt equations. Nonlinearity2012; 25: 905–930.
10.
GunzburgerMDManservisiS.The velocity tracking problem for Navier–Stokes flows with bounded distributed controls. SIAM J Control Optim1999; 37: 1913–1945.
11.
HinzeM.Optimal and Instantaneous Control of the Instationary Navier–Stokes Equations. Berlin: TU Berlin, 2002.
12.
HinzeMKunischK.Second order methods for optimal control of time-dependent fluid flow. SIAM J Control Optim2001; 40: 925–946.
13.
KalantarovVKTitiES.Global attractor and determining modes for the 3D Navier–Stokes–Voight equations. Chin Ann Math Ser B2009; 30: 697–714.
14.
KalantarovVKTitiES.Gevrey regularity for the attractor of the 3D Navier–Stokes-Voight equations. J Nonlinear Sci2009; 19: 133–152.
15.
TröltzschFWachsmuthD.Second-order sufficient optimality conditions for the optimal control of Navier–Stokes equations. ESAIM Control Optim Calc Var2006; 12: 93–119.
16.
WangG.Optimal controls of 3-dimensional Navier–Stokes equations with state constraintsSIAM J Control Optim2002; 41: 583–606.
17.
YueGZhongCK.Attractors for autonomous and nonautonomous 3D Navier–Stokes–Voight equations. Discrete Cont Dynam Syst Ser B2011; 16: 985–1002.
18.
LiXJYongJM.Optimal Control Theory for infinite Dimensional Systems. Boston, MA: Birkhäuser, 1995.
19.
LiuZHMigorskiSZengB.Optimal feedback control and controllability for hyperbolic evolution inclusions of Clarke’s subdifferential type. Comput Math Appl2017; 74: 3183–3194.
20.
ZengB.Feedback control systems governed by evolution equations. Optimization2019; 68: 1223–1243.
21.
ZengBLiuZH.Existence results for impulsive feedback control systems. Nonlin Anal Hybrid Syst2019; 33: 1–16.
22.
MigórskiSOchalASofoneaM.Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems. New York: Springer, 2013.
23.
NaniewiczZPanagiotopoulosPD.Mathematical Theory of Hemivariational Inequalities and Applications. New York: Marcel Dekker, Inc., 1995.
24.
PanagiotopoulosPD.Hemivariational Inequalities, Applications in Mechanics and Engineering. Berlin: Springer, 1993.
25.
RobinsonJC.Infinite-Dimensional Dynamical Systems. Cambridge: Cambridge University Press,2001.
26.
TemamR.Navier–Stokes Equations: Theory and Numerical Analysis. 2nd ed.Amsterdam: North-Holland, 1979.
27.
ContantinPFoiasC.Navier–Stokes Equations (Chicago Lectures in Mathematics). Chicago, IL: University of Chicago Press, 1988.
28.
ClarkeFH.Optimization and Nonsmooth Analysis. New York: John Wiley & Sons, Inc., 1983.
