Global Stabilization of The Generalized Burgers-Korteweg-de Vries Equation by Boundary Control

Global Stabilization of The Generalized Burgers-Korteweg-de Vries Equation by Boundary Control

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Abstract

The generalized Korteweg-de Vries (gKdV) equation is a nonlinear partial differential equation describing the propagation of nonlinear dispersive waves. For higher-order non- linearities, the equation has unstable solitary wave solutions. To take account for dissipative mechanisms, a Burgers-type term is often added which yields the generalized Burgers-Korteweg- de Vries (gBKdV) equation. In this paper, stabilizing boundary control laws are constructed for the gBKdV equation using the Lyapunov method. The control laws achieve global exponential stability in the L2 -sense. Adaptive control laws are also constructed for the case of an unknown dissipation coefficient. Other control laws where the dissipation coefficient e = 0 and the dispersion coefficient δ = 0 are also treated and stability is proved. Finally, an observer for the control laws where measurements inside the domain is needed is derived in the last section. Again closed loop exponential stability is shown.
Content

The generalized Korteweg-de Vries (gKdV) equation is a nonlinear partial differential equation describing the propagation of nonlinear dispersive waves. For higher-order non- linearities, the equation has unstable solitary wave solutions. To take account for dissipative mechanisms, a Burgers-type term is often added which yields the generalized Burgers-Korteweg- de Vries (gBKdV) equation. In this paper, stabilizing boundary control laws are constructed for the gBKdV equation using the Lyapunov method. The control laws achieve global exponential stability in the L2 -sense. Adaptive control laws are also constructed for the case of an unknown dissipation coefficient. Other control laws where the dissipation coefficient e = 0 and the dispersion coefficient δ = 0 are also treated and stability is proved. Finally, an observer for the control laws where measurements inside the domain is needed is derived in the last section. Again closed loop exponential stability is shown.

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