h$ and $\dot V$. One must pay attention to the Zeno behavior (cf. [@Lygeros2010]) where a system can make an infinite number of jumps in a finite amount of time. With these considerations in minds, one can use a numerical procedure to integrate eq.\[eq:roc\] and eq.\[eq:acceleration\].

Trajectory Generation
---------------------

In order to generate the trajectory, one must specify eq.\[eq:roc\] and eq.\[eq:acceleration\] as functions of time. With respect to BADA, let $Thr(h(t),t) = Thr(h(t))$, $D(h(t), V(t),t)=D(h(t),V(t))$ and $T(h(t),t) = T(h(t))$, that is the evolution of the thrust, the temperature and the drag are time-invariant. Moreover, the aircraft dynamic functions shall be specified with respect to the flight envelope constraints. In this study, we choose a nominal thrust function. In eq.\[eq:roc\], $Thr$ is replaced by the maximum thrust $Thr_{max}$ and the whole equation is multiplied by a reduced climb power coefficient $C_{red}$, which is supposed to give realistic profiles (cf. [@Bada2012] p.24). To simulate the system, a common fourth-order Runge-Kutta method is used. Let $f_1(t,h,V,q) = \dot h(V,q)$ and $f_2(t,V,h, \dot h) = \dot V(h,\dot h)$. Then, one obtains the following integration scheme: $$\begin{aligned}
  dh_1 &=& f_1(t_n,h_n,V_n,q(V_n,h_n)) \nonumber \\
  dv_1 &=& f_2(t_n,V_n,h_n, \dot h_n) \nonumber \\
  dh_2 &=& f_1( t_n + \frac{\Delta 