This paper studies boundary controllability of the Korteweg-de Vries equation posed on a finite interval, in which, because of the third-order character of the equation, three boundary conditions are required to secure the well-posedness of the system. We consider the cases where one, two, or all three of those boundary data are employed as boundary control inputs. The system is first linearized around the origin and the corresponding linear system is proved to be exactly boundary controllable if using two or three boundary control inputs. In the case where only one control input is allowed to be used, the linearized system is known to be only null controllable if the single control input acts on the left end of the spatial domain. By contrast, if the single control input acts on the right end of the spatial domain, the linearized system is shown to be exactly controllable if and only if the length of the spatial domain does not belong to a set of critical values. Moreover, the nonlinear system is shown to be locally exactly boundary controllable via the contraction mapping principle if the associated linearized system is exactly controllable. © 2013 Society for Industrial and Applied Mathematics.
Cerpa, E., Rivas, I., & Zhang, B. Y. (2013). Boundary controllability of the Korteweg-de Vries equation on a bounded domain. SIAM Journal on Control and Optimization, 2976-3010. https://doi.org/10.1137/120891721