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Erschienen in:
1995 | OriginalPaper | Buchkapitel
Stabilization of the Korteweg-de Vries Equation on a Periodic Domain
verfasst von : David L. Russell, Bing-Yu Zhang
Erschienen in: Control and Optimal Design of Distributed Parameter Systems
Verlag: Springer New York
Enthalten in: Professional Book Archive
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We study solutions of the Korteweg-de Vries (KdV) equations $$ {u_t} + u{u_x} + {u_{xxx}} = f $$ and $$ {u_t} + u{u_x} + {u_{xxx}} = 0 $$ for t ≥ 0 and 0 ≤ x ≤ 1 where the subscripts denote partial derivatives. hi the first case, periodic boundary conditions are imposed at 0 and 1, and the distributed control f is assumed to be generated by a linear feedback control law conserving the “volume” or “mass” ∫01u(x, t)dx which monotonically reduces the “energy” ∫01u(x, t)2dx. For the second equation a feedback boundary control is applied having the same properties. In both cases we obtain uniform exponential decay of the solutions to a constant state.