Internal null controllability of the generalized Hirota-Satsuma system

Abstract

The generalized Hirota-Satsuma system consists of three coupled nonlinear Korteweg-de Vries (KdV) equations. By using two distributed controls it is proven in this paper that the local null controllability property holds when the system is posed on a bounded interval. First, the system is linearized around the origin obtaining two decoupled subsystems of third order dispersive equations. This linear system is controlled with two inputs, which is optimal. This is done with a duality approach and some appropriate Carleman estimates. Then, by means of an inverse function theorem, the local null controllability of the nonlinear system is proven.

Local null controllability of a model system for strong interaction between internal solitary waves

“…• We can slightly improve the result presented in [16] and show that the general Hirota-Satsuma system can be controlled with two forces supported in any two control domains (in [16] the authors need that one of the control domains touches the boundary). The proof is very similar to the one of Theorem 1.1, as we only have to do some minor adjustments.…”

“…More recently, by using the approach used in the present paper, the internal null controllability for the Hirota-Satsuma system was addressed in [16]. However, due to the complexity of the system (1.1), which has coupling constituted by third order terms, the control problem presents new difficulties we have to deal with.…”

“…However, due to the complexity of the system (1.1), which has coupling constituted by third order terms, the control problem presents new difficulties we have to deal with. Then, we develop techniques applicable to this more complicated situation, which allow to slightly improve the result present in [16] and, in addition, can also be applied to other dispersive and parabolic systems.…”

“…It has also been studied, for example, the controllability in cascade-like systems [29,15], the null controllability in the context of linear thermoelasticity [31], the controllability to trajectories in phase-field models [3], the existence of insensitizing controls for the heat equation [20], and the controllability in reaction-diffusion systems [2]. However, as for KdV-like systems, it is a very recent research topic which has been studied only in [10] and [16].…”