Boundary controllability of a nonlinear coupled system of two Korteweg–de Vries equations with critical size restrictions on the spatial domain
详细信息 查看全文
- 作者：Roberto A. Capistrano-Filho…
- 关键词：Gear–Grimshaw system ; Exact boundary controllability ; Neumann boundary conditions ; Dirichlet boundary conditions ; Critical set
- 刊名：Mathematics of Control, Signals, and Systems
- 出版年：2017
- 出版时间：March 2017
- 年：2017
- 卷：29
- 期：1
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Systems Theory, Control; Control, Robotics, Mechatronics; Communications Engineering, Networks;
- 出版者：Springer London
- ISSN：1435-568X
- 卷排序：29
文摘
This article is dedicated to improve the controllability results obtained by Cerpa and Pazoto (Commun Contemp Math 13:183–189, 2011) and by Micu et al. (Commun Contemp Math 11(5):779–827, 2009) for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval. Initially, Micu et al. (2009) proved that the nonlinear system is exactly controllable by using four boundary controls without any restriction on the length L of the interval. Later on, in Cerpa and Pazoto (2011), two boundary controls were considered to prove that the same system is exactly controllable for small values of the length L and large time of control T. Here, we use the ideas contained in Capistrano-Filho et al. (Z Angew Math Phys 67(5):67–109, 2016) to prove that, with another configuration of four controls, it is possible to prove the existence of the so-called critical length phenomenon for the linear system, i.e., whether the system is controllable depends on the length of the spatial domain. In addition, when we consider only one control input, the boundary controllability still holds for suitable values of the length L and time of control T. In both cases, the control spaces are sharp due a technical lemma which reveals a hidden regularity for the solution of the adjoint system.