Approximate controllability and irreducibility of the 2D Zakharov–Kuznetsov–Burgers equation

The present paper is a continuation of [Chen and Rosier, Discrete Continuous Dyn. Syst.- B 25 (2020) 3889–3916] and [Chen, Appl. Math. Optim. 87 (2023) 12]. In this paper, we consider the 2D Zakharov–Kuznetsov–Burgers equation posed on the periodic domain T2. We show that the 2D Zakharov–Kuznetsov–Burgers equation is globally approximately controllable by a finite dimensional external force, the proof is based on the Agrachev–Sarychev approach in geometric control theory. As an application of the approximate controllability result, we establish the irreducibility of the transition semigroup associated with stochastic Zakharov–Kuznetsov–Burgers equation driven by a degenerate noise.