Feedback control for fluid mixing via advection. (English) Zbl 1527.35286
Summary: This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain \(\Omega \subset \mathbb{R}^d\), with \(d = 2\) or \(d = 3\). The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space \(( H^1 ( \Omega ) )^\prime\) of \(H^1(\Omega)\) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q93 | PDEs in connection with control and optimization |
| 35Q49 | Transport equations |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 76F25 | Turbulent transport, mixing |
| 93B52 | Feedback control |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
Keywords:
fluid mixing; transport equation; Stokes equations; nonlinear feedback control; instantaneous control; asymptotic behaviorReferences:
| [1] | Aamo, O. M.; Krstić, M.; Bewley, T. R., Control of mixing by boundary feedback in 2d channel flow, Automatica, 39, 1597-1606 (2003) · Zbl 1055.93039 |
| [2] | Alberti, G., Exponential self-similar mixing by incompressible flows, J. Am. Math. Soc., 32, 445-490 (2019) · Zbl 1456.35158 |
| [3] | Alberti, G.; Crippa, G.; Mazzucato, A., Exponential self-similar mixing by incompressible flows, J. Am. Math. Soc., 32, 445-490 (2019) · Zbl 1456.35158 |
| [4] | Azmi, B.; Kunisch, K., On the stabilizability of the Burgers equation by receding horizon control, SIAM J. Control Optim., 54, 1378-1405 (2016) · Zbl 1352.49003 |
| [5] | Balogh, A.; Aamo, O. M.; Krstic, M., Optimal mixing enhancement in 3-d pipe flow, IEEE Trans. Control Syst. Technol., 13, 27-41 (2004) |
| [6] | Barbu, V.; Precupanu, T., Convexity and Optimization in Banach Spaces (2012), Springer Science & Business Media · Zbl 1244.49001 |
| [7] | Bewley, T. R.; Moin, P.; Temam, R., Dns-based predictive control of turbulence: an optimal benchmark for feedback algorithms, J. Fluid Mech., 447, 179-225 (2001) · Zbl 1036.76027 |
| [8] | Biswas, A.; Foias, C.; Larios, A., On the attractor for the semi-dissipative Boussinesq equations, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 34, 381-405 (2017) · Zbl 1361.35138 |
| [9] | Bressan, A., A lemma and a conjecture on the cost of rearrangements, Rend. Semin. Mat. Univ. Padova, 110, 97-102 (2003) · Zbl 1114.05002 |
| [10] | Brezis, H.; Gallouet, T., Nonlinear Schrödinger evolution equations, Nonlinear Anal., Theory Methods Appl., 4, 677-681 (1980) · Zbl 0451.35023 |
| [11] | Chang, Y.; Collis, S. S., Active control of turbulent channel flows based on large eddy simulation, (ASME Paper No. FEDSM-99, vol. 6929 (1999)), 1-8 |
| [12] | Choi, H., Suboptimal control of turbulent flow using control theory, (Int Symp. on Math. Modelling of Turbulent Flows. Int Symp. on Math. Modelling of Turbulent Flows, Tokyo, Japan (1995)) |
| [13] | Choi, H.; Hinze, M.; Kunisch, K., Instantaneous control of backward-facing step flows, Appl. Numer. Math., 31, 133-158 (1999) · Zbl 0939.76027 |
| [14] | Choi, H.; Temam, R.; Moin, P.; Kim, J., Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech., 253, 509-543 (1993) · Zbl 0810.76012 |
| [15] | Cockburn, B.; Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16, 173-261 (2001) · Zbl 1065.76135 |
| [16] | Crippa, G.; Lucà, R.; Schulze, C., Polynomial mixing under a certain stationary Euler flow, Phys. D, Nonlinear Phenom., 394, 44-55 (2019) · Zbl 1451.76035 |
| [17] | Danckwerts, P. V., The definition and measurement of some characteristics of mixtures, Appl. Sci. Res., Sect. A, 3, 279-296 (1952) |
| [18] | Doering, C. R.; Wu, J.; Zhao, K.; Zheng, X., Long time behavior of the two-dimensional Boussinesq equations without buoyancy diffusion, Phys. D, Nonlinear Phenom., 376, 144-159 (2018) · Zbl 1398.35164 |
| [19] | Elgindi, T. M.; Zlatoš, A., Universal mixers in all dimensions, Adv. Math., 356, Article 106807 pp. (2019) · Zbl 1428.35357 |
| [20] | Foias, C.; Manley, O.; Rosa, R.; Temam, R., Navier-Stokes Equations and Turbulence, vol. 83 (2001), Cambridge University Press · Zbl 0994.35002 |
| [21] | Garcia, C. E.; Prett, D. M.; Morari, M., Model predictive control: theory and practice—a survey, Automatica, 25, 335-348 (1989) · Zbl 0685.93029 |
| [22] | Grüne, L., Analysis and design of unconstrained nonlinear mpc schemes for finite and infinite dimensional systems, SIAM J. Control Optim., 48, 1206-1228 (2009) · Zbl 1194.49045 |
| [23] | Gubanov, O.; Cortelezzi, L., Towards the design of an optimal mixer, J. Fluid Mech., 651, 27-53 (2010) · Zbl 1189.76168 |
| [24] | Hinze, M., Optimal and instantaneous control of the instationary Navier-stokes equations (2000) |
| [25] | Hinze, M.; Kunisch, K., Control strategies for fluid flows-optimal versus suboptimal control, TU, Fachbereich, 3 (1998) · Zbl 1068.76512 |
| [26] | Hinze, M.; Pinnau, R.; Ulbrich, M.; Ulbrich, S., Optimization with PDE Constraints, vol. 23 (2008), Springer Science & Business Media · Zbl 1167.49001 |
| [27] | Hinze, M.; Volkwein, S., Analysis of instantaneous control for the Burgers equation, Nonlinear Anal., 50, 1-26 (2002) · Zbl 1022.49001 |
| [28] | Hu, W., An approximating control design for optimal mixing by Stokes ows, Appl. Math. Optim., 1-28 (2018) |
| [29] | Hu, W., Boundary control for optimal mixing by Stokes flows, Appl. Math. Optim., 78, 201-217 (2018) · Zbl 1398.35253 |
| [30] | Hu, W.; Wang, Y.; Wu, J.; Xiao, B.; Yuan, J., Partially dissipative 2d Boussinesq equations with Navier type boundary conditions, Phys. D, Nonlinear Phenom., 376, 39-48 (2018) · Zbl 1398.35007 |
| [31] | Hu, W.; Wu, J., Boundary control for optimal mixing via Navier-Stokes flows, SIAM J. Control Optim., 56, 2768-2801 (2018) · Zbl 1472.49005 |
| [32] | Zheng, X.; Hu, W.; Wu, J., Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows (2023), arXiv preprint · Zbl 1539.76056 |
| [33] | Hu, W.; Kukavica, I.; Ziane, M., On the regularity for the Boussinesq equations in a bounded domain, J. Math. Phys., 54, Article 081507 pp. (2013) · Zbl 1293.35246 |
| [34] | Hu, W.; Wu, J., An approximating approach for boundary control of optimal mixing via Navier-Stokes flows, J. Differ. Equ., 267, 5809-5850 (2019) · Zbl 1428.35642 |
| [35] | Iyer, G.; Kiselev, A.; Xu, X., Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows, Nonlinearity, 27, 973 (2014) · Zbl 1293.35248 |
| [36] | Ju, N., Global regularity and long-time behavior of the solutions to the 2d Boussinesq equations without diffusivity in a bounded domain, J. Math. Fluid Mech., 19, 105-121 (2017) · Zbl 1457.76054 |
| [37] | Kukavica, I.; Wang, W., Long time behavior of solutions to the 2d Boussinesq equations with zero diffusivity, J. Dyn. Differ. Equ., 32, 2061-2077 (2020) · Zbl 1452.35039 |
| [38] | Kunisch, K.; Pfeiffer, L., The effect of the terminal penalty in receding horizon control for a class of stabilization problems, ESAIM Control Optim. Calc. Var., 26, 58 (2020) · Zbl 1448.49007 |
| [39] | Lai, S.; Wu, J.; Zhong, Y., Stability and large-time behavior of the 2d Boussinesq equations with partial dissipation, J. Differ. Equ., 271, 764-796 (2021) · Zbl 1454.35288 |
| [40] | Lee, C.; Kim, J.; Choi, H., Suboptimal control of turbulent channel flow for drag reduction, J. Fluid Mech., 358, 245-258 (1998) · Zbl 0907.76039 |
| [41] | Liu, M.; Muzzio, F.; Peskin, R., Quantification of mixing in aperiodic chaotic flows, Chaos Solitons Fractals, 4, 869-893 (1994) · Zbl 0823.76020 |
| [42] | Lunasin, E.; Lin, Z.; Novikov, A.; Mazzucato, A.; Doering, C. R., Optimal mixing and optimal stirring for fixed energy, fixed power, or fixed palenstrophy flows, J. Math. Phys., 53, Article 115611 pp. (2012) · Zbl 1302.76076 |
| [43] | Mathew, G.; Mezić, I.; Petzold, L., A multiscale measure for mixing, Phys. D, Nonlinear Phenom., 211, 23-46 (2005) · Zbl 1098.37067 |
| [44] | Min, C.; Choi, H., Suboptimal feedback control of vortex shedding at low Reynolds numbers, J. Fluid Mech., 401, 123-156 (1999) · Zbl 0968.76017 |
| [45] | V. Nevistic, J. Primbs, Finite receding horizon control: a general framework for stability and performance analysis, preprint, 1997. · Zbl 0990.93024 |
| [46] | Ottino, J. M., The Kinematics of Mixing: Stretching, Chaos, and Transport, vol. 3 (1989), Cambridge University Press · Zbl 0721.76015 |
| [47] | Rawlings, J. B.; Muske, K. R., The stability of constrained receding horizon control, IEEE Trans. Autom. Control, 38, 1512-1516 (1993) · Zbl 0790.93019 |
| [48] | Seis, C., Maximal mixing by incompressible fluid flows, Nonlinearity, 26, 3279 (2013) · Zbl 1396.76042 |
| [49] | Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis, Studies in Mathematics and Its Applications, vol. 2 (1977), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0383.35057 |
| [50] | Temam, R., Navier-Stokes Equations: Theory and Numerical Analysis, vol. 343 (2001), American Mathematical Soc. · Zbl 0981.35001 |
| [51] | Thiffeault, J.-L., Using multiscale norms to quantify mixing and transport, Nonlinearity, 25, R1 (2012) · Zbl 1325.37007 |
| [52] | Thiffeault, J.-L.; Lin, Z.; Doering, C. R., Optimal stirring strategies for passive scalar mixing, J. Fluid Mech., 675, 465-476 (2011) · Zbl 1241.76361 |
| [53] | Tröltzsch, F., Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, vol. 112 (2010), American Mathematical Soc. · Zbl 1195.49001 |
| [54] | Unger, A.; Tröltzsch, F., Fast solution of optimal control problems in the selective cooling of steel, ZAMM-J. Appl. Math. Mech./Z. Angew. Math. Mech.: Appl. Math. Mech., 81, 447-456 (2001) · Zbl 0993.49024 |
| [55] | Waitz, I. A.; Qiu, Y. J.; Manning, T. A.; Fung, A. K.S.; Elliot, J. K.; Kerwin, J. M.; Krasnodebski, J. K.; O’Sullivan, M. N.; Tew, D. E.; Greitzer, E. M.; Marble, F. E.; Tan, C. S.; Tillman, T. G., Enhanced mixing with streamwise vorticity, Prog. Aerosp. Sci., 33, 323-351 (1997) |
| [56] | Warhaft, Z., Passive scalars in turbulent flows, Annu. Rev. Fluid Mech., 32, 203-240 (2000) · Zbl 0988.76042 |
| [57] | Wiggins, S.; Ottino, J. M., Foundations of chaotic mixing, Philos. Trans. R. Soc., Math. Phys. Eng. Sci., 362, 937-970 (2004) · Zbl 1069.76502 |
| [58] | Zlatoš, A.; Yao, Y., Mixing and un-mixing by incompressible flows, J. Eur. Math. Soc., 19, 1911-1948 (2017) · Zbl 1369.35071 |
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