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

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 $5\mathrm{/}2$–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 $\mathit{\text{Trans.}}\mathit{\text{Amer.}}\mathit{\text{Math.}}\mathit{\text{Soc.}}$ (2015)~\cite{LR}, here extended to the fifth-order Kadomtsev–Petviashvili equation.