In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on t = T. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in L2 have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

中文翻译：

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内部孤立波强相互作用模型系统的局部零点可控性

在本文中，我们证明了两个 Korteweg-de Vries 方程的非线性耦合系统的局部零点可控性，该方程位于有界区间上，源项呈指数衰减吨 = 吨. 该系统由 Gear 和 Grimshaw 引入，用于模拟二维长内部重力波在分层流体中传播的相互作用。我们通过支持域的内部开放子集并仅作用于一个方程的控制来解决可控性问题。证明主要包括证明线性化系统的可控性，这是通过对伴随系统进行卡尔曼估计来完成的。在做 Carleman 时，我们改进了处理具有源项的色散方程和抛物方程的解这一事实的技术大号2具有有限的规律性。应用局部反演定理得到非线性系统的结果。