Observability of dispersive equations from line segments on the torus

Abstract

We investigate the observability of a general class of linear dispersive equations on the torus $\mathbb{T}$. We take one line segment or two line segments in space-time region as the observable set. We give the characteristic on the slopes of the line segments to guarantee the qualitative observability and quantitative observability respectively. The one line segment case, is simple, follows directly from the Ingham's inequality. However, the two line segments case is difficult, the statement of results and the proof rely heavily on the language of graph theory. We also apply our results to (higher order) Schrödinger equations and the linear KdV equation.

Figures (9)

• Figure 1: Observability from a positive measure set in $[0,2\pi]^2$.
• Figure 2: Observability from one line segments holds if the slope is not an integer (see left). Observability from two line segments fails if the slopes are integers (see middle). Observability from $N\ge 3$ line segments fails if the slopes are integers (see right and note that they have the common starting point).
• Figure 3: Graphs of polynomial function $y=p(x)-vx$: (1) if $\mathrm{deg}\,p$ is odd, then for $|k|$ large enough, $n_k(v)=1$ (see left); (2) if $\mathrm{deg}\,p$ is even, then for $|k|$ large enough, $n_k(v)\le 2$ (see right).
• Figure 4: Two consequtive vertices in the subsequence.
• Figure 5: Examples of the red-portion of $k$ in $G(v_1,v_2)$

Theorems & Definitions (31)

• Definition 1.1
• Definition 1.2
• Remark 1.3
• Proposition 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Definition 2.1
• Theorem 2.2: Ingham
• Theorem 2.3: Kahane-Haraux