[Paper Review] Internal controllability of non-localized solution for the Kodomtsev-Petviashvili II equation

Chenmin Sun, Ivonne Rivas|arXiv (Cornell University)|Nov 26, 2017

Advanced Mathematical Physics Problems14 references3 citations

TL;DR

This paper establishes internal exact controllability of the KP-II equation on the 2D torus from vertical strips using the Hilbert Uniqueness Method and semiclassical/microlocal analysis. It proves that for any initial and final data in $L^2_0(\mathbb{T}^2)$, a control localized in a vertical strip can steer the solution from the initial to the target state within finite time, while showing that controllability fails from horizontal strips due to wave packet propagation constraints.

ABSTRACT

The internal control problem for the Kadomstev-Petviashvili II equation, known as KP-II, is the object of study in this paper. The controllability in $L^2(T)$ from vertical strip is proved using the Hilbert Unique Method through the techniques of semiclassical and microlocal analysis. Additionally, a negative result for the controllability in $L^2(T)$ from horizontal strip is also showed.

Motivation & Objective

• To establish exact internal controllability of the KP-II equation in $L^2(\mathbb{T}^2)$ from a vertical strip.
• To analyze the limitations of controllability when the control region is a horizontal strip.
• To apply the Hilbert Uniqueness Method combined with semiclassical and microlocal techniques to prove controllability results.
• To extend the controllability results from the linearized to the nonlinear KP-II equation using perturbative arguments.
• To demonstrate the failure of controllability from horizontal strips by constructing a sequence of solutions with vanishing control energy.

Proposed method

• Uses the Hilbert Uniqueness Method (HUM) to reduce the controllability problem to a spectral inequality involving the dual system.
• Applies semiclassical analysis to study the propagation of wave packets in the frequency domain, focusing on high-frequency oscillations.
• Employs microlocal techniques to analyze the behavior of solutions near the characteristic variety and control region.
• Constructs a sequence of solutions $u_n$ to the linearized KP-II equation with initial $L^2$-norm bounded away from zero but control energy over the vertical strip tending to zero.
• Uses the Poisson summation formula and integration by parts to estimate the $L^2$-norm of solutions in the control region, showing decay as $\epsilon_n^{1/2}$.
• Adapts a perturbative argument based on the Cauchy theory of KP-II to extend local controllability to the nonlinear case.

Experimental results

Research questions

• RQ1 Can the KP-II equation be exactly controlled in $L^2(\mathbb{T}^2)$ from a vertical strip using internal controls?
• RQ2 Why does controllability fail when the control is restricted to a horizontal strip?
• RQ3 What role does the direction of wave packet propagation play in the controllability of dispersive PDEs like KP-II?
• RQ4 How do semiclassical and microlocal methods help in proving controllability for nonlinear dispersive equations?
• RQ5 Can the local controllability result for the nonlinear KP-II equation be extended to small data in $L^2(\mathbb{T}^2)$?

Key findings

• The KP-II equation is exactly controllable in $L^2(\mathbb{T}^2)$ from a vertical strip for any $T>0$ and any initial and final data in $L^2_0(\mathbb{T}^2)$.
• The control operator $\mathcal{G}$, defined via a vertical window function $g(x)$, ensures that the control input preserves the horizontal mean value and acts locally in $x$.
• A sequence of solutions $u_n$ is constructed such that $\|u_n(0)\|_{L^2} \sim 1$ but $\int_0^T \int_\omega |\mathcal{G}u_n(t,x,y)|^2 \, dxdydt \to 0$ as $n \to \infty$, proving the failure of the spectral inequality for horizontal control.
• The failure of controllability from horizontal strips is due to wave packets with high frequency and large group velocity escaping the control region.
• For the nonlinear KP-II equation, exact controllability holds locally in time for small initial and target data in $L^2(\mathbb{T}^2)$, with a radius $R>0$ depending on $T$.
• The results extend to $H^s_0(\mathbb{T}^2)$ for $s \geq 0$, though the paper focuses on $L^2$ due to the conservation of $L^2$-norm along the flow.