On One Control Problem for Zakharov–Kuznetsov Equation

Abstract

An initial-boundary value problem for Zakharov–Kuznetsov equation u _t + bu _x + u _xxx + u _xyy + uu _x = f ₀(t)g(t, x, y) posed on a rectangle (0, R) × (0, L) for t ∈ (0, T) under certain initial and boundary conditions is considered. Here, the function f ₀ is unknown and is referred as a control, and the function g is given. The problem is to find a pair (f ₀, u), satisfying the additional condition $\int _0^T u(t,x,y)\omega (x,y)\,dxdy = \varphi (t)$, where the functions ω, φ are given and u is the solution to the corresponding initial-boundary value problem. It is shown that under certain smallness assumptions on input data such a pair exists and is unique. For the corresponding linearized equation, a similar result is obtained without any smallness assumptions.