Boundary controllability of the Korteweg-de Vries equation: The Neumann case

2023/10/08 by Roberto de A. Capistrano–Filho, João S. Silva, Capistrano-Filho, R. de A. +1
Engineering · Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2310.04977

openalex publication_date 2023/10/08 · openalex created_date 2023/10/12 · openalex updated_date 2026/07/29

Abstract

This article gives a necessary first step to understanding the critical set phenomenon for the Korteweg-de Vries (KdV) equation posed on interval [0,L] considering the Neumann boundary conditions with only one control input. We showed that the KdV equation is controllable in the critical case, i.e., when the spatial domain L belongs to the set R_c, where c≠-1 and R_c:=\(2π)/(√(3(c+1)))√(m²+ml+m²); m,l∈ ℕ^*\∪\(mπ)/(√(c+1)); m∈ ℕ^*\, the KdV equation is exactly controllable in L²(0,L). The result is achieved using the return method together with a fixed point argument.

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