Abstract
We study solutions of the Korteweg-de Vries (KdV) equations
and
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” ∫ 10 u(x, t)dx which monotonically reduces the “energy” ∫ 10 u(x, t)2 dx. 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.
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Russell, D.L., Zhang, BY. (1995). Stabilization of the Korteweg-de Vries Equation on a Periodic Domain. In: Lagnese, J.E., Russell, D.L., White, L.W. (eds) Control and Optimal Design of Distributed Parameter Systems. The IMA Volumes in Mathematics and its Applications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8460-1_9
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