Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-18T21:41:37.429Z Has data issue: false hasContentIssue false
  • English
  • Français

Distributed Control of the Stochastic Burgers Equation with Random Input Data

Part of: Existence theories Incompressible viscous fluids

Published online by Cambridge University Press:  27 January 2016

Hyung-Chun Lee  and
Yun Nam
Hyung-Chun Lee*
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 16499
Yun Nam
Affiliation:
Department of Mathematics, Ajou University, Suwon, Korea 16499
*
*Corresponding author. Email addresses:hclee@ajou.ac.kr (H.-C. Lee), namy0727@ajou.ac.kr (Y. Nam)
Get access

Abstract

We discuss a control problem involving a stochastic Burgers equation with a random diffusion coefficient. Numerical schemes are developed, involving the finite element method for the spatial discretisation and the sparse grid stochastic collocation method in the random parameter space. We also use these schemes to compute closed-loop suboptimal state feedback control. Several numerical experiments are performed, to demonstrate the efficiency and plausibility of our approximation methods for the stochastic Burgers equation and the related control problem.

Keywords

Finite element methodfeedback controlstochastic Burgers equationsparse grid collocation

MSC classification

Secondary: 49J20: Optimal control problems involving partial differential equations 76D05: Navier-Stokes equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 6 , Issue 1 , February 2016 , pp. 89 - 108
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Atwell, J.A., Borggaard, J.T. and King, B.B., Reduced order controller for Burgers’ equation with a nonlinear observer, Int. J. Appl. Math. Comp. Sci. 11, 1311–1330 (2001).Google Scholar
[2]Bungartz, H.J. and Griebel, M., Sparse grids, Acta Num. 13, 1–123 (2004).Google Scholar
[3]Burns, J.A. and Kang, S., A control problem for Burgers’ equation with bounded input/output, Nonlinear Dynamics 2, 235–262 (1991).Google Scholar
[4]Balogh, A. and Krstic, M, Burgers equation with nonlinear boundary feedback: H 1 stability, well-posedness and simulation, Machematical Problems in Engineering 6, 189–200 (2000).Google Scholar
[5]Barthelmann, V., Novak, E. and Ritter, K., High dimensional polynomial interpolation on sparse grids, Adv. Comp. Math. 12, 273–288 (2000).Google Scholar
[6]Babuska, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal. 45, 1005–1034 (2007).CrossRefGoogle Scholar
[7]Babuska, I., Tempone, R. and Zouraris, G.E., Galerkin Finite Element Approximations of Stochastic Elliptic Partial Differential Equations, SIAM J. Num. Anal. 42, 800–825 (2005).Google Scholar
[8]Chen, C.T., Linear System Theory and Design, Holt, Rinehart and Winston, New York (1984).Google Scholar
[9]Choi, H., Temam, R., Moin, P. and Kim, J., Feedback control for unsteady flow and its application to the stochastic Burgers equation, J. Fluid Mech. 253, 509–543 (1993).Google Scholar
[10]Chen, P., Quarteroni, A. and Rozza, G., A weighted reduced basis method for elliptic partial differential equations with random input data, SIAM J. Num. Anal. 51, 3163–3185 (2013).CrossRefGoogle Scholar
[11]Durrett, R., Probability: Theory and Examples, 2nd ed., Duxbury Press, Belmont CA (1964).Google Scholar
[12]Da Prato, G. and Debussche, A, Control of the stochastic Burgers model of turbulence, SIAM J. Control Optim. 37, 1123–1149 (1999).Google Scholar
[13]El-Beltagy, M., Wafa, M. and Galal, O., Upwind finite-volume solution of the stochastic Burgers equation, Appl. Math. 3, 1818–1825 (2012).Google Scholar
[14]El-Beltagy, M. and Wafa, M., Stochastic 2D incompressible Navier-Stokes solver using the vorticity stream function formulation, J. Appl, Math. 2013, 1–14 (2013).Google Scholar
[15]Fletcher, C.A.J., Burgers Equation: A model for all reasons, Numerical Solutions of Partial Differential Equations, Noye, J. (Ed.), North-Holland (1982).Google Scholar
[16]Gunzburger, M.D. and Labovsky, A., Effects of approximate deconvolution models on the solution of the stochastic Navier-Stokes equations, J. Comp. Math. 29 131–140 (2011).Google Scholar
[17]Krstic, M., On global stabilization of Burgers equation by boundary control, Systems Control Letts. 37, 123–141 (1999).CrossRefGoogle Scholar
[18]Kim, N. and Lee, H.-C., Sparse grid collocation method for an optimal control problem involving a stochastic partial differential equation with random inputs, East Asian J. Appl. Math. 4, 166–188 (2014).Google Scholar
[19]Kunisch, K. and Volkwein, S., Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102, 345–371 (1999).Google Scholar
[20]Loève, M., Probability Theory, 4th ed., Springer, New York (1977).Google Scholar
[21]Lee, H.-C. and Piao, G.-R., Boundary feedback control of the Burgers equations by a reduced-order approach using centroidal Voronoi tessellations, J. Sci. Comp. 43 369–387 (2010).Google Scholar
[22]Novak, E. and Ritter, K., High dimensional integration of smooth functions over cubes, Num. Math. 75, 79–97 (1996).Google Scholar
[23]Novak, E., Ritter, K., Schmitt, R. and Steinbauer, A., On an interpolatory method for high dimensional integration, J. Comp. Appl. Math. 112, 215–228 (1999).CrossRefGoogle Scholar
[24]Nobile, F., Tempone, R. and Webster, C.G., A sparse grid stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal. 46, 2309–2345 (2008).CrossRefGoogle Scholar
[25]Nobile, F., Tempone, R. and Webster, C.G., An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal. 46, 2411–2442 (2008).Google Scholar
[26]Piao, G.-R. and Lee, H.-C., Distributed feedback control of the Benjamin-Bona-Mahony-Burgers equation by a reduced-order model, East Asian J. Appl. Math. 5, 61–74 (2015).CrossRefGoogle Scholar
[27]Piao, G.-R., Lee, H.-C. and Lee, J.-Y., Distributed feedback control of the Burgers equation by a reduced-order approach using weighted centroidal Voronoi tessellation, J. KSIAM 13, 293–305 (2009).Google Scholar
[28]Smolyak, S.A., Quadrature and interpolation formulas for tensor products of certain classes of functions, Dokl. Akad. Nauk SSSR 4, 240–243 (1963).Google Scholar
[29]Wasilkowski, G.W., Woźniakowski, H., Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity 11 1–56 (1995).Google Scholar