Global Stabilization of BBM-Burgers' Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis
For researchers in control of PDEs and numerical analysis, this work offers the first superconvergence results for boundary feedback control laws, though the problem is specialized.
This paper achieves global stabilization of BBM-Burgers' type equations using nonlinear Neumann boundary feedback control, and provides optimal error estimates and superconvergence results for finite element approximations. Numerical experiments confirm the theoretical findings.
In this article, global stabilization results for the Benjamin-Bona-Mahony-Burgers' (BBM-B) type equations are obtained using nonlinear Neumann boundary feedback control laws. Based on the $C^0$-conforming finite element method, global stabilization results for the semidiscrete solution are also discussed. Optimal error estimates in $L^\infty(L^2)$, $L^\infty(H^1)$ and $L^\infty(L^\infty)$-norms for the state variable are derived, which preserve exponential stabilization property. Moreover, for the first time in the literature, superconvergence results for the boundary feedback control laws are established. Finally, several numerical experiments are conducted to confirm our theoretical findings.