Free Access
Volume 47, Number 5, September-October 2013
Page(s) 1465 - 1492
Published online 30 July 2013
  1. V.S. Afraimovich, M.K. Muezzinoglu and M.I. Rabinovich, Metastability and Transients in Brain Dynamics: Problems and Rigorous Results, in Long-range Interactions, Stochasticity and Fractional Dynamics; Nonlinear Physical Science, edited by Albert C.J. Luo and Valentin Afraimovich. Springer-Verlag (2010) 133–175. [Google Scholar]
  2. E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations. Math. Comput. Model. 35 (2002) 1165–1195. [CrossRef] [Google Scholar]
  3. E. Allen, J.A. Burns and D.S. Gilliam, On the use of numerical methods for analysis and control of nonlinear convective systems, in Proc. of 47th IEEE Conference on Decision and Control (2008) 197–202. [Google Scholar]
  4. J.A. Atwell and B.B. King, Stabilized Finite Element Methods and Feedback Control for Burgers’ Equation, in Proc. of the 2000 American Control Conference (2000) 2745–2749. [Google Scholar]
  5. D.H. Bailey and J.M. Borwein, Exploratory Experimentation and Computation, Notices AMS 58 (2011) 1410–1419. [Google Scholar]
  6. A. Balogh, D.S. Gilliam and V.I. Shubov, Stationary solutions for a boundary controlled Burgers’ equation. Math. Comput. Model. 33 (2001) 21–37. [CrossRef] [Google Scholar]
  7. M. Beck and C.E. Wayne, Using Global Invariant Manifolds to Understand Metastability in the Burgers Equation With Small Viscosity. SIAM Review 53 (2011) 129–153 [Published originally SIAM J. Appl. Dyn. Syst. 8 (2009) 1043–1065]. [CrossRef] [Google Scholar]
  8. T.R. Bewley, P. Moin and R. Temam, Control of Turbulent Flows, in Systems Modelling and Optimization, Chapman and Hall CRC, Boca Raton, FL (1999) 3–11. [Google Scholar]
  9. J.T. Borggaard and J.A. Burns, A PDE Sensitivity Equation Method for Optimal Aerodynamic Design. J. Comput. Phys. 136 (1997) 366–384. [CrossRef] [Google Scholar]
  10. J. Burns, A. Balogh, D. Gilliam and V. Shubov, Numerical stationary solutions for a viscous Burgers’ equation. J. Math. Syst. Estim. Control 8 (1998) 1–16. [Google Scholar]
  11. J.A. Burns and S. Kang, A control problem for Burgers’ equation with bounded input/output. Nonlinear Dyn. 2 (1991) 235–262. [CrossRef] [Google Scholar]
  12. J.A. Burns and S. Kang, A Stabilization problem for Burgers’ equation with unbounded control and observation, in Estimation and Control of Distributed Parameter Systems. Int. Ser. Numer. Math. vol. 100, edited by W. Desch, F. Rappel, K. Kunisch. Springer-Verlag (1991) 51–72. [Google Scholar]
  13. J.A. Burns and H. Marrekchi, Optimal fixed-finite-dimensional compensator for Burgers’ Equation with unbounded input/output operators. ICASE Report No. 93-19. Institute for Comput. Appl. Sci. Engrg., Hampton, VA. (1993). [Google Scholar]
  14. J.A. Burns and J.R. Singler, On the Long Time Behavior of Approximating Dynamical Systems, in Distributed Parameter Control, edited by F. Kappel, K. Kunisch and W. Schappacher. Springer-Verlag (2001) 73–86. [Google Scholar]
  15. C.I. Byrnes and D.S. Gilliam, Boundary control and stabilization for a viscous Burgers’ equation. Computation and Control, Progress in Systems Control Theory, vol. 15. Birkhäuser Boston, Boston, MA (1993) 105–120. [Google Scholar]
  16. C.I. Byrnes, D.S. Gilliam and V.I. Shubov, Convergence of trajectories for a controlled viscous Burgers’ equation, Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena. Int. Ser. Numer. Math., vol. 118, edited by W. Desch, F. Rappel, K. Kunisch. Birkhäuser, Basel (1994) 61–77. [Google Scholar]
  17. C.I. Byrnes, D. Gilliam, V. Shubov and Z. Xu, Steady state response to Burgers’ equation with varying viscosity, in Progress in Systems and Control: Computation and Control IV, edited by K. L.Bowers and J. Lund. Birkhäuser, Basel (1995) 75–98. [Google Scholar]
  18. C.I. Byrnes, D.S. Gilliam and V.I. Shubov, High gain limits of trajectories and attractors for a boundary controlled viscous Burgers’ equation. J. Math. Syst. Estim. Control 6 (1996) 40. [Google Scholar]
  19. C.I. Byrnes, A. Balogh, D.S. Gilliam and V.I. Shubov, Numerical stationary solutions for a viscous Burgers’ equation. J. Math. Syst. Estim. Control 8 (1998) 16 (electronic). [Google Scholar]
  20. C.I. Byrnes, D.S. Gilliam and V.I. Shubov, On the Global Dynamics of a Controlled Viscous Burgers’ Equation. J. Dyn. Control Syst. 4 (1998) 457–519. [CrossRef] [Google Scholar]
  21. C.I. Byrnes, D.S. Gilliam and V.I. Shubov, Boundary Control, Stabilization and Zero-Pole Dynamics for a Nonlinear Distributed Parameter System. Int. J. Robust Nonlinear Control 9 (1999) 737–768. [CrossRef] [Google Scholar]
  22. C. Cao and E. Titi, Asymptotic Behavior of Viscous Burgers’ Equations with Neumann Boundary Conditions, Third Palestinian Mathematics Conference, Bethlehem University, West Bank. Mathematics and Mathematics Education, edited by S. Elaydi, E. S. Titi, M. Saleh, S. K. Jain and R. Abu Saris. World Scientific (2002) 1–19. [Google Scholar]
  23. M.H. Carpenter, J. Nordström and D. Gottlieb, Revisiting and extending interface penalties for multi-domain summation-by-parts operators. J. Sci. Comput. 45 (2010) 118–150. [CrossRef] [Google Scholar]
  24. J. Carr and J.L. Pego, Metastable patterns in solutions of ut = ϵ2uxxf(u). Comm. Pure Appl. Math. 42 (1989) 523–576. [CrossRef] [MathSciNet] [Google Scholar]
  25. J. Carr, D.B. Duncan and C.H. Walshaw, Numerical approximation of a metastable system. IMA J. Numer. Anal. 15 (1995) 505–521. [CrossRef] [Google Scholar]
  26. C.A.J. Fletcher, Burgers’ equation: A model for all reasons, in Numerical Solutions of J. Partial Differ. Eqns., edited by J. Noye. North-Holland Publ. Co. Amsterdam (1982) 139–225. [Google Scholar]
  27. A.V. Fursikov and R. Rannacher, Optimal Neumann Control for the 2D Steady-State Navier-Stokes equations, in New Directions in Math. Fluid Mech. The Alexander. V. Kazhikhov Memorial Volume. Advances in Mathematical Fluid Mechanics, Birkhauser, Berlin (2009) 193–222. [Google Scholar]
  28. G. Fusco, G. and J. K. Hale, Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Differ. Eqns. 1 (1989) 75–94. [CrossRef] [Google Scholar]
  29. T. Gallay and C.E. Wayne, Invariant manifolds and the long-time asymptotics of the navier-stokes and vorticity equations on R2. Arch. Rational Mech. Anal. 163 (2002) 209–258. [CrossRef] [MathSciNet] [Google Scholar]
  30. T. Gallay and C.E. Wayne, Global stability of vortex solutions of the two-dimensional Navier-Stokes equation. Commun. Math. Phys. 255 (2005) 97–129. [CrossRef] [Google Scholar]
  31. M. Garbey and H.G. Kaper, Asymptotic-Numerical Study of Supersensitivity for Generalized Burgers’ Equation. SIAM J. Sci. Comput. 22 (2000) 368–385. [CrossRef] [Google Scholar]
  32. S. Gottlieb, D. Gottlieb and C.-W. Shu, Recovering High-Order Accuracy in WENO Computations of Steady-State Hyperbolic Systems. J. Sci. Comput. 28 (2006) 307–318. [CrossRef] [Google Scholar]
  33. M. Gunzburger, L. Hou and T. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with distributed and Neumann controls. Math. Comput. 57 (1991) 123–151. [CrossRef] [Google Scholar]
  34. M.D. Gunzburger, H.C. Lee and J. Lee, Error estimates of stochastic optimal Neumann boundary control problems. SIAM J. Numer. Anal. 49 (2011) 1532–1552. [CrossRef] [Google Scholar]
  35. J.S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time Dependent Problems, Cambridge Monographs on Applied and Computational Mathematics, vol. 21. Cambridge University Press (2006). [Google Scholar]
  36. IEEE Computer Society, IEEE Standard for Binary Floating-Point Arithmetic, IEEE Std 754-1985 (1985). [Google Scholar]
  37. A. Kanevsky, M.H. Carpenter, D. Gottlieb and J. S. Hesthaven, Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225 (2007) 1753–1781. [CrossRef] [Google Scholar]
  38. R. Kannan and Z.J. Wang, A high order spectral volume solution to the Burgers’ equation using the Hopf–Cole transformation. Int. J. Numer. Meth. Fluids (2011). Available on DOI: 10.1002/fld.2612. [Google Scholar]
  39. O.A. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of the AMS, vol. 23 (1968). [Google Scholar]
  40. J.G.L. Laforgue and R.E. O’Malley, Supersensitive Boundary Value Problems, Asymptotic and Numerical Methods for Partial Differential Equations with Critical Parameters, edited by H.G. Kaper and M. Garbey. Kluwer Publishers (1993) 215–224. [Google Scholar]
  41. H.V. Ly, K.D. Mease and E.S. Titi, Distributed and boundary control of the viscous Burgers’ equation. Numer. Funct. Anal. Optim. 18 (1997) 143–188. [CrossRef] [MathSciNet] [Google Scholar]
  42. H. Marrekchi, Dynamic Compensators for a Nonlinear Conservation Law, Ph.D. Thesis, Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061 (1993). [Google Scholar]
  43. V.Q. Nguyen, A Numerical Study of Burgers’ Equation With Robin Boundary Conditions, M.S. Thesis. Department of Mathematics, Polytechnic Institute and State University, Blacksburg, VA, 24061 (2001). [Google Scholar]
  44. P. Pettersson, J. Nordström and G. Laccarino, Boundary procedures for the time-dependent Burgers’ equation under uncertainty. Acta Math. Sci. 30 (2010) 539–550. [Google Scholar]
  45. J.T. Pinto, Slow motion manifolds far from the attractor in multistable reaction-diffusion equations. J. Differ. Eqns. 174 (2001) 101–132. [CrossRef] [Google Scholar]
  46. S.M. Pugh, Finite element approximations of Burgers’ Equation, M.S. Thesis. Departmant of Mathematics, Polytechnic Institute and State University, Blacksburg, VA, 24061 (1995). [Google Scholar]
  47. G.R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143. Springer-Verlag (2002). [Google Scholar]
  48. Z.-H. Teng, Exact boundary conditions for the initial value problem of convex conservation laws. J. Comput. Phys. 229 (2010) 3792–3801. [CrossRef] [Google Scholar]
  49. M.J. Ward and L.G. Reyna, Internal layers, small eigenvalues, and the sensitivity of metastable motion. SIAM J. Appl. Math. 55 (1995) 425–445. [CrossRef] [Google Scholar]
  50. T.I. Zelenjak, Stabilization of solutions of boundary value problems for a second-order parabolic equation with one space variable, Differentsial’nye Uravneniya 4 (1968) 34D45. [Google Scholar]
  51. T.I. Zelenyak, M.M. Lavrentiev Jr. and M.P. Vishnevskii, Qualitative Theory of Parabolic Equations, Part 1, VSP, Utrecht, The Netherlands (1997). [Google Scholar]

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