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.
Highlights
In the eighties, Hirota and Satsuma introduced in [15] the set of two coupled Korteweg-de Vries (KdV) equations, ut − 1 4 uxxx = 3uux 6vvx, vt + 1 2 vxxx −3uvx, (1.1)describing the interaction of two long waves with different dispersion relations
We can see that the first equation in (1.2) is of KdV type with a negative dispersive term whereas the two others have positive dispersive term. Considering these facts, we propose to study equations (1.2) on a spatial domain [0, L] with the usual boundary conditions for KdV equations, as for instance in [19], and the initial conditions u(t, 0)
More related to this paper we can cite [5] where the authors study the internal control of a KdV equation on a bounded domain with the same kind of boundary conditions than here
Summary
Hirota and Satsuma introduced in [15] the set of two coupled Korteweg-de Vries (KdV) equations, ut. More related to this paper we can cite [5] where the authors study the internal control of a KdV equation on a bounded domain with the same kind of boundary conditions than here. They use duality arguments and a Carleman estimate to prove an observability inequality. Concerning the internal control of dispersive systems, the closest works are [18] where Ingham theorems are used to prove some observability inequalities for Boussinesq systems and [3] where a Carleman estimates approach is used to get the null controllability of a linear system coupling a KdV equation with a Schrodinger equation. We end this paper with some comments and related open problems
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