On One Control Problem for Zakharov–Kuznetsov Equation

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 ∫0Tu(t,x,y)ω(x,y)dxdy=φ(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. © Springer Nature Switzerland AG 2019.