parameter $s$ not too large, the effect of the control term is still effective and the barrier is restored close to $$\begin{aligned}
p(x,t)&=&\frac{v_1+v_2}{2}+\frac{2\varepsilon_1\cos(k_1x-\omega_1t+\varphi_1)}
{v_1-v_2} \nonumber
    \\[0.5\baselineskip]
&&-\frac{2\varepsilon_2\cos(k_2x-\omega_2t+\varphi_2)}{v_1-v_2}.\end{aligned}$$

Numerical results {#loc:simulation}
-----------------

In this section we perform a numerical investigation of the effect of the exact and approximate control terms on the electron beam dynamics. We introduce the parameter $r$ given by the ratio of the two wave amplitudes $r=\varepsilon_1/\varepsilon_2$. In order to reproduce as close as possible the experimental set-up described in the next section (see also [@Macor1]), we consider the following values of amplitudes, wave numbers, frequencies and phases of the two electrostatic waves: $(\varepsilon_1,k_1,\omega_1,\varphi_1)=(\varepsilon r,1,0,0)$ and $(\varepsilon_2,k_2,\omega_2,\varphi_2)=(\varepsilon,k,k,0)$. Thus Hamiltonian (\[Ham1\]) can be written as $$H(p,x,t)=\frac{p^2}{2}+\varepsilon r\cos x+\varepsilon \cos
[k(x-t)], \label{Ham:sim}$$ i.e. $v_1=0$ and $v_2=1$. We perform simulations with $r=0.082$ and $k=5/3$. The amplitudes of the waves are determined by $r$ and $\varepsilon$ that are related to the Chirikov parameter by the following equation: $$s=2\sqrt{\varepsilon}(\sqrt{r}+1).$$ The value of $\varepsilon$ will be given by $s$. In the following we consider two values of $s$, that is $s=0.85$ ($\varepsilon\sim0.11$) and $s=1.27$ ($\varepsilon\sim0.24$). In this case the expression of the exact control term given by Eq. (\[eqn:fTWT\]) becomes $$f(x,t)= -2\varepsilon^2(r\cos x-\cos k(x-t))^2,
    