Narrow Results

This paper proves the local exact boundary controllability property of a nonlinear system of two coupled Korteweg–de Vries equations which models the interactions of weakly nonlinear gravity waves (see [10]). Following the method in [24], which combines the analysis of the linearized system and the Banach's fixed point theorem, the controllability problem is reduced to prove a nonstandard unique continuation property of the eigenfunctions of the corresponding differential operator.

This note concerns the paper "On the controllability of a coupled system of two Korteweg–de Vries equations" by Micu et al. [2]. They study a nonlinear coupled system of two Korteweg–de Vries equations and prove that the system is controllable by using four boundary controls. Here, we prove that in some cases it is possible to get the controllablity of the system by using only two controls. This can be done depending on both the spatial domain and the control time.

In this paper, the superconvergence results of nonconforming finite element method are derived for a class of mixture boundary control problems governed by Stokes equations. We derive superconvergence results for both the control and the state approximations. Base on superconvergence results, we obtain asymptotically exact a posteriori error estimates. Moreover, numerical examples demonstrating our theoretical results are provided in this paper.