Controlling trajectories globally via spatiotemporal finite-time optimal control. (English) Zbl 1458.49003
Summary: The problems of (i) maximizing or minimizing Lagrangian mixing in fluids via the introduction of a spatiotemporally varying control velocity and (ii) globally controlling the finite-time location of trajectories beginning at all initial conditions in a chaotic system are considered. A particular form of solution to these is designed which uses a new methodology for computing a spatiotemporally dependent optimal control. An \(L^2\)-error norm for trajectory locations over a finite-time horizon is combined with a penalty energy norm for the control velocity in defining the global cost function. A computational algorithm for cost minimization is developed, and theoretical results on global error and cost presented. Numerical simulations (using velocities which are specified, and obtained as data from computational fluid dynamics simulations) are used to demonstrate the efficacy and validity of the approach in determining the required spatiotemporally defined control velocity.
MSC:
| 49J15 | Existence theories for optimal control problems involving ordinary differential equations |
| 34H10 | Chaos control for problems involving ordinary differential equations |
Keywords:
global optimal control; Lagrangian trajectories; trajectory control; chaos control; flow controlSoftware:
gridfitReferences:
| [1] | M. Aghababa and H. Aghababa, Finite-time stabilization of uncertain non-autonomous chaotic gyroscopes with nonlinear inputs, Appl. Math. Mech., 33 (2012), pp. 155-164. · Zbl 1233.93081 |
| [2] | M. P. Aghababa and H. P. Aghababa, Adaptive finite-time synchronization of non-autonomous chaotic systems with uncertainty, J. Comput. Nonlinear Dyn., 8 (2013), 031006. · Zbl 1281.34062 |
| [3] | V. Arnol’d, Sur la topologie des écoulements stationnaires des fluides parfaits, C. R. Acad. Sci. Paris, 261 (1965), pp. 17-20. · Zbl 0145.22203 |
| [4] | S. Balasuriya, Optimal frequency for microfluidic mixing across a fluid interface, Phys. Rev. Lett., 105 (2010), 064501. |
| [5] | S. Balasuriya, Unsteadily manipulating internal flow barriers, J. Fluid Mech., 818 (2017), pp. 382-406. · Zbl 1383.76101 |
| [6] | S. Balasuriya, Stochastic sensitivity: A computable Lagrangian measure of uncertainty for unsteady flows, SIAM Rev., to appear. |
| [7] | S. Balasuriya and M. Finn, Energy constrained transport maximization across a fluid interface, Phys. Rev. Lett., 108 (2012), 244503. |
| [8] | S. Balasuriya, N. T. Ouellette, and I. I. Rypina, Generalized Lagrangian coherent structures, Phys. D, 372 (2018), pp. 31-51. · Zbl 1391.76002 |
| [9] | S. Balasuriya and K. Padberg-Gehle, Accurate control of hyperbolic trajectories in any dimension, Phys. Rev. E, 90 (2014), 032903. · Zbl 1341.93043 |
| [10] | V. Y. Belozyorov, Universal approach to the problem of emergence of chaos in autonomous dynamical systems, Nonlinear Dynam., 95 (2019), pp. 579-595. · Zbl 1439.34044 |
| [11] | T. Bewley, P. Moin, and R. Temam, DNS-based predictive control of turbulence: An optimal benchmark for feedback algorithms, J. Fluid Mech., 447 (2001), pp. 179-225. · Zbl 1036.76027 |
| [12] | S. Boccaletti, J. Kurths, G. Osipov, D. Valladares, and C. Zhou, The synchronization of chaotic systems, Phys. Rep., 366 (2002), pp. 1-101. · Zbl 0995.37022 |
| [13] | O. Bokanowski, A. Briani, and H. Zidani, Minimum time control problems for non-autonomous differential equations, Systems Control Lett., 58 (2009), pp. 742-746. · Zbl 1181.49034 |
| [14] | A. E. Botha, I. Rahmonov, and Y. Shukrinov, Spontaneous and controlled chaos synchronization in intrinsic Josephson junctions, IEEE Trans. Appl. Superconductivity, 28 (2018), 1800806. |
| [15] | T. Botmart, P. Niamsup, and X. Liu, Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control, Commun. Nonlinear Sci., 17 (2012), pp. 1894-1907. · Zbl 1239.93043 |
| [16] | A. E. Bryson, Applied Optimal Control: Optimization, Estimation and Control, Routledge, London, 2018. |
| [17] | Y. Chen, X. Wu, and Z. Gui, Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control, Appl. Math. Model., 34 (2010), pp. 4161-4170. · Zbl 1201.93045 |
| [18] | I. Couchman, E. Kerrigan, and J. Vassilicos, Optimization-based feedback control of mixing in a Stokes fluid flow, in Proceedings of the European Control Conference, IEEE, 2009, pp. 1227-1232. |
| [19] | J. de Jong, R. Lammertink, and M. Wessling, Membranes and microfluidics: A review, Lab Chip, 6 (2006), pp. 1125-1139. |
| [20] | J. D’Errico, Surface Fitting Using Gridfit, MATLAB Central File Exchange, https://au.mathworks.com/matlabcentral/fileexchange/8998-surface-fitting-using-gridfit, 2016. |
| [21] | T. Dombre, U. Frisch, J. Greene, M. Hénon, A. Mehr, and A. Soward, Chaotic streamlines in the ABC flows, J. Fluid Mech., 167 (1986), pp. 353-391. · Zbl 0622.76027 |
| [22] | O. E, C. Grebogi, and J. Yorke, Controlling chaos, Phys. Rev. Lett., 64 (1990), pp. 1196-1199. · Zbl 0964.37501 |
| [23] | M. El-Dessoky, M. Yassen, and E. Aly, Bifurcation analysis and chaos control in Shimizu-Morioka chaotic system with delayed feedback, Appl. Math. Comp., 243 (2014), pp. 283-297. · Zbl 1335.37013 |
| [24] | G. Froyland and N. Santitissadeekorn, Optimal mixing enhancement, SIAM J. Appl. Math., 77 (2017), pp. 1444-1470. · Zbl 1373.37179 |
| [25] | T. Glad and L. Ljung, Control Theory, CRC Press, Boca Raton, FL, 2014. |
| [26] | E. F. D. Goufo, M. Mbehou, and M. M. K. Pene, A peculiar application of Atangana-Baleanu fractional derivative in neuroscience: Chaotic burst dynamics, Chaos Solitons Fractals, 115 (2018), pp. 170-176. · Zbl 1416.92036 |
| [27] | R. Hajiloo, H. Salarieh, and A. Alasty, Chaos control in delayed phase space constructed by the Takens embedding theory, Commun, Nonlinear Sci. Numer. Simul., 54 (2018), pp. 453-465. · Zbl 1510.37121 |
| [28] | G. Haller, Lagrangian coherent structures, Annu. Rev. Fluid Mech., 47 (2015), pp. 137-162. |
| [29] | C. Hernandez, Y. Bernard, and A. Razek, A global assessment of piezoelectric actuated micro-pumps, Eur. Phys. J. Appl. Phys., 51 (2010), 20101. |
| [30] | K. Hicke, X. Porte, and I. Fischer, Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback, Phys. Rev. E, 88 (2013), 052904. |
| [31] | M. Hinze and K. Kunisch, Second-order methods for optimal control of time-dependent fluid flow, SIAM J. Control Optim., 40 (2001), pp. 925-946. · Zbl 1012.49026 |
| [32] | C.-M. Ho and Y.-C. Tai, Micro-electro-mechanical systems (mems) and fluid flows, Annu. Rev. Fluid Mech., 30 (1998), pp. 579-612. |
| [33] | A.-C. Huang and Y.-C. Chen, Adaptive multiple-surface sliding control for non-autonomous systems with mismatched uncertainties, Automatica, 40 (2004), pp. 1939-1945. · Zbl 1070.93012 |
| [34] | K. Iwamoto, Y. Suzuki, and N. Kasagi, Reynolds number effect on wall turbulence: Towards effective feedback control, Int. J. Heat Fluid Flow, 23 (2002), pp. 678-689. |
| [35] | J.-D. Jansen, O. H. Bosgra, and P. M. Van den Hof, Model-based control of multiphase flow in subsurface oil reservoirs, J. Process Contr., 18 (2008), pp. 846-855. |
| [36] | J. Kabziński, Synchronization of an uncertain Duffing oscillator with higher order chaotic systems, Int. J. Applied Math. Comput. Sci., 28 (2018), pp. 625-634. · Zbl 1416.93103 |
| [37] | H. B. Keller, Numerical Methods for Two-Point Boundary-Value Problems, Dover, New York, 2018. · Zbl 1409.65001 |
| [38] | J. Kim, Control of turbulent boundary layers, Phys. Fluids, 15 (2003), pp. 1093-1105. · Zbl 1186.76283 |
| [39] | D. Lawden, Analytical Methods of Optimization, Scottish Academic Press, Edinburgh, 1975. · Zbl 0379.49001 |
| [40] | D. Lee, W. Yoo, and S. Won, An integral control for synchronization of a class of unknown non-autonomous chaotic systems, Phys. Lett. A, 374 (2010), pp. 4231-4237. · Zbl 1238.34119 |
| [41] | Z. Lin, J.-L. Thiffeaut, and C. Doering, Optimal stirring strategies for passive scalar mixing, J. Fluid Mech., 675 (2011), pp. 465-476. · Zbl 1241.76361 |
| [42] | M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), pp. 287-289. · Zbl 1383.92036 |
| [43] | T. Medjo, R. Temam, and M. Ziane, Optimal and robust control of fluid flows: Some theoretical and computational aspects, Appl. Mech. Rev., 61 (2008), 010802. · Zbl 1145.76341 |
| [44] | S. Miah, M. M. Fallah, and D. Spinello, Non-autonomous coverage control with diffusive evolving density, IEEE Trans. Automat. Control, 62 (2017), pp. 5262-5268. · Zbl 1390.93062 |
| [45] | C. Miles and C. Doering, A shell model for optimal mixing, J. Nonlinear Sci., 28 (2018), pp. 2153-2186. · Zbl 1448.76147 |
| [46] | N. Mishchuk, T. Heldal, T. Volden, J. Auerswald, and H. Knapp, Microfluidic pump based on the phenomenon of electroosmosis of the second kind, Microfluid. Nanofluid., 11 (2011), pp. 675-684. |
| [47] | J. Nicholson, regularizeNd, MATLAB Central File Exchange, https://www.mathworks.com/matlabcentral/fileexchange/61436-regularizend, 2020. |
| [48] | S. Ober-Blöbaum and K. Padberg-Gehle, Multiobjective optimal control of fluid mixing, Proc. Appl. Math. Mech., 15 (2015), pp. 639-640. |
| [49] | Y. Orlov, Finite time stability and robust control synthesis of uncertain switched systems, SIAM J. Control Optim., 43 (2004), pp. 1253-1271. · Zbl 1085.93021 |
| [50] | L. Pecora and I. Carrol, Synchronization in chaotic systems, Phys. Rev. Lett., 64 (1990), 8212. · Zbl 0938.37019 |
| [51] | K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A, 170 (1992), pp. 421-428. |
| [52] | M. Rafikov and J. Balthazar, On an optimal control design for Rössler system, Phys. Lett. A, 333 (2004), pp. 241-245. · Zbl 1123.49300 |
| [53] | S. S. Ravindran, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Int. J. Numer. Methods Fluids, 34 (2000), pp. 425-448. · Zbl 1005.76020 |
| [54] | K. L. Schlueter-Kuck and J. O. Dabiri, Coherent structure colouring: Identification of coherent structures from sparse data using graph theory, J. Fluid Mech., 811 (2017), pp. 468-486. · Zbl 1383.76372 |
| [55] | T. Shinbrot, C. Grebogi, J. A. Yorke, and E. Ott, Using small perturbations to control chaos, Nature, 363 (1993), pp. 411-417. |
| [56] | S. Strogatz, Nonlinear Dynamics and Chaos, Perseus Books, New York, 1994. |
| [57] | S. Varghese, M. Speetjens, and R. Trieling, Lagrangian transport and chaotic advection in two-dimensional anisotropic systems, Transp. Porous Media, 119 (2017), pp. 225-246. |
| [58] | C.-H. Wang and G.-B. Lee, Automatic bio-sampling chips integrated with micro-pumps and micro-valves for disease detection, Biosensors Bioelectronics, 21 (2005), pp. 419-425. |
| [59] | J. Wang, Y. Li, S. Zhong, and X. Hou, Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer-Meinhardt system, Chaos Solitons Fractals, 118 (2019), pp. 1-17. · Zbl 1442.35214 |
| [60] | P. Woias, Micropumps-summarizing the first two decades, Proc. SPIE, 4560 (2001), pp. 39-52. |
| [61] | Y. Yang and X. Wu, Global finite-time synchronization of a class of the non-autonomous chaotic systems, Nonlinear Dynam., 70 (2012), pp. 197-208. · Zbl 1267.93150 |
| [62] | Q. You, Q. Wen, J. Fang, M. Guo, Q. Zhang, and C. Dai, Experimental study on lateral flooding for enhanced oil recovery in bottom-water reservoir with high water cut, J. Petrol. Sci. Eng., 174 (2019), pp. 747-756. |
| [63] | J. Zhou, W. Zhou, T. Chu, Y.-x. Chang, and M.-j. Huang, Bifurcation, intermittent chaos and multi-stability in a two-stage Cournot game with R&D spillover and product differentiation, Appl. Math. Comput., 341 (2019), pp. 358-378. · Zbl 1429.91203 |
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