Unique Continuation for Fifth-Order KP Equation and its application to control problems

TL;DR

This paper establishes unique continuation, observability, stabilization, and local controllability results for the fifth-order KP equation on a torus, using advanced harmonic analysis and control theory techniques.

Contribution

It extends unique continuation and control methods to the fifth-order KP equation, introducing a novel combination of techniques inspired by Benjamin--Ono analysis.

Findings

Derived a $5/2$--derivative gain in space--time norms.

Proved exponential stabilization for small initial data.

Established local exact controllability with $L^2$ controls.

Abstract

We develop a framework for the fifth-order Kadomtsev--Petviashvili equation on $\mathbb{T}_x \times \mathbb{R}_y$ within a mean-zero KP-adapted Sobolev scale. A localized high-order feedback acting on the periodic variable yields a $5/2$--derivative gain in suitable space--time norms, leading to propagation of regularity and a unique continuation property for the linear dynamics. As a consequence, we derive an observability inequality for the adjoint system and establish exponential stabilization of the nonlinear closed-loop equation: for small initial data in $X_{s,0}$, $s>2$, solutions are global and decay exponentially in $X_s$. Combining observability with the Hilbert Uniqueness Method and a fixed-point argument, we obtain local exact controllability near the origin, with $L^2$ controls supported in the feedback region and cost linear in the data size. The analysis relies on a novel combination of unique continuation, frequency grouping, and the one-sided Fourier vanishing mechanism introduced for the Benjamin--Ono equation by Linares and Rosier in \textit{Trans. Amer. Math. Soc.} (2015)~\cite{LR}, here extended to the fifth-order Kadomtsev--Petviashvili equation.