notice that for $\varepsilon=0$, the Hamiltonian $H$ has an invariant torus with frequency vector ${\bm\omega}$ at ${\bf A}={\bf 0}$ for any $Q$ not necessarily small. The controlled Hamiltonian we construct is $$\label{eqn:gene} H_c({\bf A},{\bm\theta})={\bm \omega}\cdot {\bf
A}+ W({\bf A},{\bm\theta})+ f({\bm \theta}).$$ The control term $f$ we construct only depends on the angle variables and is given by $$\label{eqn:exf} f({\bm\theta})=W({\bf 0},{\bm\theta})-
W\left(-\Gamma \partial_{\bm\theta} W({\bf
0},{\bm\theta}),{\bm\theta}\right),$$ where $\partial_{\bm\theta}$ is the derivative operator with respect to $\bm\theta$, and $\Gamma$ is a linear operator defined as a pseudo-inverse of ${\bm\omega}\cdot
\partial_{\bm\theta}$, i.e. acting on $W=\sum_{{\bf k}}W_{{\bf
k}} {\mathrm e}^{i{\bf k}\cdot{\bm\theta}}$ as $$\Gamma W=\sum_{{\bm\omega}\cdot{\bf k}\not= 0} \frac{W_{{\bf
k}}}{i{\bm\omega}\cdot{\bf k}} {\mathrm e}^{i{\bf
k}\cdot{\bm\theta}}.$$ Note that $f$ is of order $\varepsilon^2$. For any function $W$, Hamiltonian (\[eqn:gene\]) has an invariant torus with frequency vector close to ${\bm\omega}$. The equation of the torus which is restored by the addition of $f$ is $$\label{eqn:eto} {\bf A}=-\Gamma \partial_{\bm\theta} W({\bf
 0},{\bm\theta}),$$ which is of order $\varepsilon$ for $W$ given by Eq. (\[eqn:e4V\])

![TWT dispersion relation (circles) with the helix mode at 30[MHz]{} (square) and the beam mode at the same frequency but with propagation velocity choose equal to about $2.5\times
10^6$[m/s]{} (triangle); the control wave corresponds to the beating of these 