Abstract
In this article, we discuss global stabilization results for the Burgers’ equation using nonlinear Neumann boundary feedback control law. As a result of the nonlinear feedback control, a typical nonlinear problem is derived. Then, based on C 0-conforming finite element method, global stabilization results for the semidiscrete solution are analyzed. Further, introducing an auxiliary projection, optimal error estimates in \(L^{\infty }(L^{2})\), \(L^{\infty }(H^{1})\) and \(L^{\infty }(L^{\infty })\)-norms for the state variable are obtained. Moreover, superconvergence results are established for the first time for the feedback control laws, which preserve exponential stabilization property. Finally, some numerical experiments are conducted to confirm our theoretical findings.
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Allen, E.J., Burns, J.A., Gilliam, D.S., Hill, J., Shubov, V.: The impact of finite precision arithmetic and sensitivity on the numerical solution of partial differential equations. Math. Comput. Model. 35, 1165–1195 (2002)
Allen, E.J., Burns, J.A., Gilliam, D.S.: Numerical approximations of the dynamical system generated by Burgers equation with Neumann—Dirichlet boundary conditions. ESAIM: Math. Modell. Numer. Anal. 47, 1465–1492 (2013)
Balogh, A., Krstic, M.: Burgers’ equation with nonlinear boundary feedback: H 1 stability well-posedness and simulation. Math. Problems Engg. 6, 189–200 (2000)
Burns, J.A., Kang, S.: A stabilization problem for Burgers’ equation with unbounded control and observation. In: Proceedings of an International Conference on Control and Estimation of Distributed Parameter Systems. Vorau (1990)
Burns, J.A., Kang, S.: A control problem for Burgers’ equation with bounded input/output. Nonlinear Dyn. 2, 235–262 (1991)
Burns, J.A., Marrekchi, H.: Optimal fixed-finite-dimensional compensator for Burgers’ equation with unbounded input/output operators. Comput. Control III(2), 83–104 (1993)
Byrnes, C.I., Gilliam, D.S., Shubov, V.I.: On the global dynamics of a controlled viscous Burgers’ equation. J. Dyn. Control Syst. 4, 457–519 (1998)
Doss, L.J.T., Pani, A.K., Padhy, S.: Galerkin method for a Stefan-type problem in one space dimension. Numer. Methods Partial Diff. Equa. 13, 393–416 (1997)
Ito, K., Kang, S.: A dissipative feedback control for systems arising in fluid dynamics. SIAM J. Control Optim. 32, 831–854 (1994)
Ito, K., Yan, Y.: Viscous scalar conservation laws with nonlinear flux feedback and global attractors. J. Math. Anal. Appl. 227, 271–299 (1998)
Krstic, M.: On global stabilization of Burgers’ equation by boundary control. Syst. Control Lett. 37, 123–141 (1999)
Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasilinear equations of parabolic type. In: Translations of AMS, vol. 23 (1968)
Liu, W.J., Krstic, M.: Adaptive control of Burgers equation with unknown viscosity. Int. J. Adapt. Control Signal Process. 15, 745–766 (2001)
Ly, H.V., Mease, K.D., Titi, E.S.: Distributed and boundary control of the viscous Burgers’ equation. Numer. Funct. Anal. Optim. 18, 143–188 (1997)
Marrekchi, H.: Dynamic compensators for a nonlinear conservation law, PhD. Thesis Department of Mathematics. Virginia Polytechnic Institute and State University (1993)
Pani, A.K.: A finite element method for a diffusion equation with constrained energy and nonlinear boundary conditions. J. Austral. Math. Soc. Ser. B 35, 87–102 (1993)
Smaoui, N.: Nonlinear boundary control of the generalized Burgers equation. Nonlinear Dyn. 37, 75–86 (2004)
Smaoui, N.: Boundary and distributed control of the viscous Burgers equation. J. Comput. Appl. Math. 182, 91–104 (2005)
Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer-Verlag, Berlin (1997)
Acknowledgements
The second author acknowledges the support provided by the National Programme on Differential Equations: Theory, Computation and Applications (NPDE-TCA) vide the DST project No.SR/S4/MS:639/90. The first author acknowledges the financial support from UGC, Govt. India. Both authors thank the referees for their valuable suggestions.
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Communicated by: Enrique Zuazua
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Kundu, S., Pani, A.K. Finite element approximation to global stabilization of the Burgers’ equation by Neumann boundary feedback control law. Adv Comput Math 44, 541–570 (2018). https://doi.org/10.1007/s10444-017-9553-9
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DOI: https://doi.org/10.1007/s10444-017-9553-9
Keywords
- Burgers’ equation
- Boundary feedback control
- Stabilization
- Finite element method
- Optimal error estimate
- Superconvergence results and numerical experiments