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Register a new userGlobal Stabilization of 2D Forced Viscous Burgers Equation Around Nonconstant Steady State Solution by Nonlinear Neumann Boundary Feedback Control:Theory and Finite Element Analysis
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Sudeep Kundu
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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Global stabilization of viscous Burgers equation around constant steady state solution has been discussed in the literature. The main objective of this paper is to show global stabilization results for the 2D forced viscous Burgers equation around a nonconstant steady state solution using nonlinear Neumann boundary feedback control law, under some smallness condition on that steady state solution. On discretizing in space using $C^0$ piecewise linear elements keeping time variable continuous, a semidiscrete scheme is obtained. Moreover, global stabilization results for the semidiscrete solution and optimal error estimates for the state variable in $L^infty(L^2)$ and $L^infty(H^1)$-norms are derived. Further, optimal convergence result is established for the boundary feedback control law. All our results in this paper preserve exponential stabilization property. Finally, some numerical experiments are documented to confirm our theoretical findings.
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In this article, global stabilization results for the two dimensional (2D) viscous Burgers equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $C^0$-conforming finite element method in spatial direction, optimal error estimates in $L^infty(L^2)$ and in $L^infty(H^1)$- norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.
The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the State-dependent Riccati Equation (SDRE) is presented for H2 and Hinf control problems. Depending on the nonlinearity and the dimension of the resulting problem, offline, online, and hybrid offline-online alternatives to the SDRE synthesis are proposed. The hybrid offline-online SDRE method reduces to the sequential solution of Lyapunov equations, effectively enabling the computation of suboptimal feedback controls for two-dimensional PDEs. Numerical tests for the Sine-Gordon, degenerate Zeldovich, and viscous Burgers PDEs are presented, providing a thorough experimental assessment of the proposed methodology.
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.
We study the problem of global exponential stabilization of original Burgers equations and the Burgers equation with nonlocal nonlinearities by controllers depending on finitely many parameters. It is shown that solutions of the controlled equations are steering a concrete solution of the non-controlled system as $trightarrow infty$ with an exponential rate.
The paper is concerned with the steady-state Burgers equation of fractional dissipation on the real line. We first prove the global existence of viscosity weak solutions to the fractal Burgers equation driven by the external force. Then the existence and uniqueness of solution with finite $H^{frac{alpha}{2}}$ energy to the steady-state equation are established by estimating the decay of fractal Burgers solutions. Furthermore, we show that the unique steady-state solution is nonlinearly stable, which means any viscosity weak solution of fractal Burgers equation, starting close to the steady-state solution, will return to the steady state as $trightarrowinfty$.
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