Uniform stabilization of the fourth order Schrödinger equation

The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

“…If s ∈ 9 2 , 17 2 , then 2s+3 8 > 3 2 , and therefore h 0 (0), h 0 (0) are both well-defined, but z x (0, 0) = 0, and hence we must have h 0 (0) = 0 in addition to h 0 (0) = 0. More generally, if s ∈ 1 2 + 4(j − 1), 1 2 + 4j for some j ≥ 1, then using also the main equation, we deduce that ∂ k t h 0 (0) = 0 is a necessary condition for all 0 ≤ k ≤ j − 1. By similar arguments, we deduce that if s ∈ 3 2 + 4(l − 1), 3 2 + 4l for some l ≥ 1, then ∂ k t h 1 (0) = 0 is a necessary condition for all 0 ≤ k ≤ l − 1.…”

“…Step 5 -Strichartz estimates. We treat the low regularity case s < 1 2 for the nonlinear model by proving Strichartz estimates by using the oscillatory integral theory.…”

“…For instance, [48], [49], and [50] studied the wellposedness and exact controllability of the linear biharmonic Schrödinger equation on a bounded domain Ω ⊂ R n . Most recently, [1] studied the stabilization of the linear biharmonic Schrödinger equation on a bounded domain with a locally supported internal damping.…”

Stabilization and Control for the Biharmonic Schrödinger Equation

“…x u + λ |u| 2 u = 0 on T× (0, T ) . Here b > 1 2 and we assume that u ∈ C ∞ (ω × (0, T )), where ω ⊂ T nonempty set. Then, u ∈ C ∞ (T× (0, T )) .…”

“…Lastly, to get a general outline of the control theory already done for the system (1.1), two interesting problems were studied recently by Aksas and Rebiai [1] and Peng [15]: Stochastic control problem and uniform stabilization, in a smooth bounded domain Ω of R n and on the interval I = (0, 1) of R, respectively. The first work, by introducing suitable dissipative boundary conditions, the authors proved that the solution decays exponentially in L 2 (Ω) when the damping term is effective on a neighborhood of a part of the boundary.…”