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Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation. (English) Zbl 1376.93119

Summary: We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.

MSC:

93E25 Computational methods in stochastic control (MSC2010)
35R60 PDEs with randomness, stochastic partial differential equations
93B52 Feedback control
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[1] Cuerno, A.; Makse, H. A.; Tomassone, S.; Harrington, S. T.; Stanley, H. E., Stochastic erosion for surface erosion via ion sputtering: Dynamical evolution from ripple morphology to rough morphology, Phys. Rev. Lett., 75, 4464-4467 (1995)
[2] Cuerno, R.; Barabasi, A.-L., Dynamic scaling of ion-sputtemaroon surfaces, Phys. Rev. Lett., 74, 23, 4746-4749 (1995)
[3] Buceta, J.; Pastor, J.; Rubio, M. A.; de la Rubia, F. J., The stochastic Kuramoto-Sivashinsky equation: a model for compact electrodeposition growth, Phys. Lett. A, 235, 464-468 (1997)
[4] Buceta, J.; Pastor, J.; Rubio, M. A.; de la Rubia, F. J., Small scale properties of the stochastic stabilized Kuramoto-Sivashinsky equation, Physica D, 113, 166-171 (1998) · Zbl 0946.60095
[5] Hu, G.; Orkoulas, G.; Christofides, P. D., Stochastic modeling and simultaneous regulation of surface roughness and porosity in thin film deposition, Ind. Eng. Chem. Res., 48, 6690-6700 (2009)
[6] Hu, G.; Lou, Y.; Christofides, P. D., Dynamic output feedback covariance control of stochastic dissipative partial differential equations, Chem. Eng. Sci., 63, 4531-4542 (2008)
[7] Alava, M.; Dubé, M.; Rost, M., Imbibition in disordemaroon media, Adv. Phys., 53, 2, 83-175 (2004)
[8] Soriano, J.; Mercier, A.; Planet, R.; Hernández-Machado, A.; Rodríguez, M. A.; Ortín, J., Anomalous roughening of viscous fluid fronts in spontaneous imbibition, Phys. Rev. Lett., 95, Article 104501 pp. (2005)
[9] Pradas, M.; Hernández-Machado, A., Intrinsic versus superrough anomalous scaling in spontaneous imbibition, Phys. Rev. E, 74, Article 041608 pp. (2006)
[10] Bouchbinder, Eran; Procaccia, Itamar; Santucci, Stéphane; Vanel, Loïc, Fracture surfaces as multiscaling graphs, Phys. Rev. Lett., 96, Article 055509 pp. (2006)
[11] Kalliadasis, S.; Ruyer-Quil, C.; Scheid, B.; Velarde, M. G., Falling Liquid Films, Applied Mathematical Sciences, Vol. 176 (2012), Springer · Zbl 1231.76001
[12] Diez, J. A.; González, A. G., Metallic-thin-film instability with spatially correlated thermal noise, Phys. Rev. E, 93, Article 013120 pp. (2016)
[13] Nesic, S.; Cuerno, R.; Moro, E.; Kondic, L., Fully nonlinear dynamics of stochastic thin-film dewetting, Phys. Rev. E, 92, 061002(R) (2015)
[14] Grün, G.; Mecke, K.; Rauscher, M., Thin-film flow influenced by thermal noise, J. Stat. Phys., 122, 6, 1261-1294 (2006) · Zbl 1107.82039
[15] Blömker, D.; Gugg, C.; Raible, M., Thin-film growth models: roughness and correlation functions, European J. Appl. Math., 13, 385-402 (2002) · Zbl 1020.82014
[16] Barabasi, A.-L.; Stanley, H. E., Fractal Concepts in Surface Growth (1995), Cambridge University Press · Zbl 0838.58023
[17] Gomes, S. N.; Papageorgiou, D. T.; Pavliotis, G. A., Stabilising nontrivial solutions of the generalised Kuramoto-Sivashinsky equation using feedback and optimal control, IMA J. Appl. Math., 82, 1, 158-194 (2017) · Zbl 1406.35276
[18] Gomes, S. N.; Pradas, M.; Kalliadasis, S.; Papageorgiou, D. T.; Pavliotis, G. A., Controlling spatiotemporal chaos in active dissipative-dispersive nonlinear systems, Phys. Rev. E, 92, Article 022912 pp. (2015) · Zbl 1376.93119
[19] Hu, G.; Orkoulas, G.; Christofides, P. D., Modeling and control of film porosity in thin film deposition, Chem. Eng. Sci., 64, 3668-3682 (2009)
[20] Hu, G.; Orkoulas, G.; Christofides, P. D., Regulation of film thickness, surface roughness and porosity in thin film growth using deposition rate, Chem. Eng. Sci., 64, 3903-3913 (2009)
[21] Lou, Y.; Christofides, P. D., Feedback control of surface roughness in sputtering processes using the stochastic Kuramoto-Sivashinsky equation, Comput. Chem. Eng., 29, 741-759 (2005)
[22] Lou, Y.; Christofides, P. D., Nonlinear feedback control of surface roughness using a stochastic PDE: Design and application to a sputtering process, Ind. Eng. Chem. Res., 45, 7177-7189 (2006)
[23] Lou, Y.; Hu, G.; Christofides, P. D., Model pmaroonictive control of nonlinear stochastic partial differential equations with application to a sputtering process, Process Syst. Eng., 54, 8, 2065-2081 (2008)
[24] Lou, Y.; Hu, G.; Christofides, P. D., Model predictive control of nonlinear stochastic PDEs: Application to a sputtering process. American Control Conference 2009, ACC ’09 (2009)
[25] Zhang, X.; Hu, G.; Orkoulas, G.; Christofides, P. D., Pmaroonictive control of surface mean slope and roughness in a thin film deposition process, Chem. Eng. Sci., 65, 4720-4731 (2010)
[26] Harrison, M. P.; Bradley, R. M., Producing virtually defect-free nanoscale ripples by ion bombardment of rocked solid surfaces, Phys. Rev. E, 93, 040802(R) (2016)
[27] Pradas, M.; Tseluiko, D.; Kalliadasis, S.; Papageorgiou, D. T.; Pavliotis, G. A., Noise induced state transitions, intermittency, and universality in the noisy Kuramoto-Sivashinsky equation, Phys. Rev. Lett., 106, Article 060602 pp. (2011)
[28] Pradas, M.; Pavliotis, G. A.; Kalliadasis, S.; Papageorgiou, D. T.; Tseluiko, D., Additive noise effects in active nonlinear spatially extended systems, Eur. J. Appl. Math., 23, 563-591 (2012) · Zbl 1279.60081
[29] Thompson, A. B.; Gomes, S. N.; Pavliotis, G. A.; Papageorgiou, D. T., Stabilising falling liquid film flows using feedback control, Phys. Fluids, 28, Article 012107 pp. (2016)
[30] Da Prato, G.; Zabczyk, J., Stochastic Equations in Infinite Dimensions (2014), Cambridge University Press · Zbl 1317.60077
[31] Duan, J.; Ervin, V. J., On the stochastic Kuramoto-Sivashinsky equation, Nonlinear Anal.-Theor., 44, 205-216 (2001) · Zbl 0976.60059
[32] Ferrario, B., Invariant measures for a stochastic Kuramoto-Sivashinsky equation, Stoch. Anal. Appl., 26, 2, 379-407 (2008) · Zbl 1145.60035
[33] Lauritsen, K. B.; Cuerno, R.; Makse, H. A., Noisy Kuramoto-Sivashinsky equation for an erosion model, Phys. Rev. E, 54, 4, 3577-3580 (1996)
[34] Rost, M.; Krug, J., Anisotropic Kuramoto-Sivashinsky equation for surface growth and erosion, Phys. Rev. Lett., 75, 21, 3894-3897 (1995)
[35] Kardar, M.; Parisi, G.; Zhang, Y.-C., Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56, 889-892 (1986) · Zbl 1101.82329
[36] Hairer, M., Solving the KPZ equation, Ann. of Math., 178, 2, 559-664 (2013) · Zbl 1281.60060
[37] Yakhot, V., Large-scale properties of unstable systems governed by the Kuramoto-Sivashinksi equation, Phys. Rev. A, 24, 642-644 (1981)
[38] Procaccia, I.; Jensen, M. H.; L’vov, V. S.; Sneppen, K.; Zeitak, R., Surface roughening and the long-wavelength properties of the Kuramoto-Sivashinsky equation, Phys. Rev. A, 46, 3220-3224 (1992)
[39] Elezgaray, J.; Berkooz, G.; Holmes, P., Large-scale statistics of the Kuramoto-Sivashinsky equation: A wavelet-based approach, Phys. Rev. E, 54, 224-230 (1996)
[40] Krug, J., Origins of scale invariance in growth processes, Adv. Phys., 46, 2, 139-282 (1997)
[41] Nicoli, M.; Cuerno, R.; Castro, M., Unstable nonlocal interface dynamics, Phys. Rev. Lett., 102, Article 256102 pp. (2009)
[42] Corwin, I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices: Theory Appl., 1, 1130001 (2012) · Zbl 1247.82040
[43] Robinson, J. C., Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors (2001), Cambridge University Press · Zbl 0980.35001
[44] Akrivis, G.; Papageorgiou, D. T.; Smyrlis, Y.-S., Linearly implicit methods for a semilinear parabolic system arising in two-phase flows, IMA J. Numer. Anal., 31, 299-321 (2011) · Zbl 1428.35339
[45] Jimenez, J. C., Simplified formulas for the mean and variance of linear stochastic differential equations, Appl. Math. Lett., 49, 12-19 (2015) · Zbl 1381.60093
[46] Gomes, S. N.; Tate, S. J., On the solution of a Lyapunov type matrix equation arising in the control of stochastic partial differential equations, IMA J. Appl. Math. (2016), (submited for publication)
[47] Lelievre, T.; Nier, F.; Pavliotis, G. A., Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, J. Stat. Phys., 152, 2, 237-274 (2013) · Zbl 1276.82042
[48] Duncan, A. B.; Lelievre, T.; Pavliotis, G. A., Variance maroonuction using nonreversible langevin samplers, J. Stat. Phys. (2016) · Zbl 1343.82036
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