## 5 Numerical Experiments  
Let us consider the problem  
\[ \begin{array}{ll}\dot{x}_{1}=u_{1}+\varepsilon(x_{3}+u_{2}),\\ \dot{x}_{2}=x_{1}+x_{3}u_{1},\\ \dot{x}_{3}=u_{2},\end{array} \qquad x(0)= \begin{bmatrix}5\\ -2\\ 3\end{bmatrix} = \begin{bmatrix}x_{0,1}\\ x_{0,2}\\ x_{0,3}\end{bmatrix} ,\quad x(2)= \begin{bmatrix}0\\ 0\\ 0\end{bmatrix} . \tag{16}\]  
This problem has no practical meaning, but used to demonstrate the effectiveness of the proposed approach. The corresponding unperturbed dynamical system  
\[ \begin{array}{ll}\dot{x}_{1}=u_{1},\\ \dot{x}_{2}=x_{1}+x_{3}u_{1},\\ \dot{x}_{3}=u_{2},\end{array}  \tag{17}\]  
is flat with the flat output  
\[y_{1}=x_{1},\quad y_{2}=x_{2}. \tag{18}\]  
Indeed, all state and control variables can be expressed in terms of flat output as follows:  
\[x_{1}=y_{1},\quad x_{2}=y_{2},\quad u_{1}=\dot{y}_{1},\quad x_{3}=\frac{\dot{ y}_{2}-y_{1}}{\dot{y}_{1}},\quad u_{2}=\frac{(\dot{y}_{2}-\dot{y}_{1})\dot{y}_{1 }-(y_{2}-y_{1})\dot{y}_{1}}{\dot{y}_{1}^{2}}. \tag{19}\]  
It can be shown that the original system in (16) is not flat. For (16) we have  
\[F_{0}= \begin{bmatrix}u_{1}\\ x_{1}+x_{3}u_{1}\\ u_{2}\end{bmatrix} ,\quad F_{1}= \begin{bmatrix}x_{3}+u_{2}\\ 0\\ 0\end{bmatrix} ,\quad F_{0,x}(t,x_{0}(t),u_{0}(t))= \begin{bmatrix}0&0&0\\ 1&0&u_{0,1}\\ 0&0&0\end{bmatrix} ,\] \[F_{0,u}(t,x_{0}(t),u_{0}(t))= \begin{bmatrix}1&0\\ x_{0,3}&0\\ 0&1\end{bmatrix} .\]The matrix \(\Phi\) for the homogeneous system corresponding to (12), the kernel matrix \(K\), and the vector \(b\) have the following representations  
\[\Phi(t)= \begin{bmatrix}1&0&0\\ t&1&\int\limits_{0}^{t}u_{0,1}(s)ds\\ 0&0&1\end{bmatrix} ,\quad K(t,x_{0}(t),u_{0}(t))= \begin{bmatrix}1&0\\ -\tau+x_{0,3}(t)&-\left(\int\limits_{0}^{t}u_{0,1}(s)ds\\ 0&1\end{bmatrix} \right),\] \[b(t,x_{0}(t),u_{0}(t))=-\int\limits_{0}^{T} \begin{bmatrix}x_{0,3} (\tau)+u_{0,2}(\tau)\\ -\tau x_{0,3}(\tau)-\tau u_{0,2}(\tau)\\ 0\end{bmatrix} d\tau.\]  
Following the algorithm of the covering method [9; 10], we introduce the additional state variable  
\[\xi=u_{1}=\dot{y}_{1}.\]  
The extended system  
\[ \begin{array}{l}\dot{x}_{1}=\xi,\\ \dot{x}_{2}=x_{1}+x_{3}\xi,\\ \dot{x}_{3}=\frac{1}{\xi}(\dot{y}_{2}-\xi-x_{3}\dot{y}_{1}),\\ \dot{\xi}=\dot{y}_{1}\end{array}  \tag{20}\]  
is equivalent to the normal Brunovsky form of the second order  
\[\dot{y}_{1}=v_{1},\quad\dot{y}_{2}=v_{2}, \tag{21}\]  
where \(v_{1}\), \(v_{2}\) stands for the new control input and \(\xi\) represents the dynamic feedback. We extend the boundary problem (16) with the initial conditions on \(\xi\) and rewrite the problem in terms of flat output as follows  
\[ \begin{array}{l}y_{0,1}^{0}=y_{1}(0)=x_{0,1}^{0},\quad y_{0,2}^{0}=y_{2}(0)= x_{0,2}^{0},\quad\dot{y}_{0,1}^{0}=\dot{y}_{1}(0)=\xi_{0},\\ y_{0,1}^{\prime}=y_{1}(2)=x_{0,1}^{\prime},\quad y_{0,2}^{\prime}=y_{2}(2)=x_ {0,2}^{0},\quad\dot{y}_{0,1}^{\prime}=\dot{y}_{1}(2)=\xi_{f},\\ \dot{y}_{0,2}^{\prime}=\dot{y}_{2}(0)=x_{0,1}^{0}+x_{0,3}^{0}\xi_{0},\\ \dot{y}_{0,2}^{\prime}=\dot{y}_{2}(2)=x_{0,1}^{\prime}+x_{0,3}^{\prime}\xi_{f}.\end{array}  \tag{22}\]  
We suggest that we find the solution in the space of third-order polynomials  
\[y_{i}^{(4)}=0,\quad i=\{1,2\}. \tag{23}\]  
Let us consider the function  
\[p_{i}=y_{i}-\frac{1}{2}(t_{f}-t)^{2}\dot{y}_{i}-\frac{1}{3}(t_{f}-t)^{3} \dddot{y}_{i},\quad i=1,2. \tag{24}\]  
The time derivatives of these functions are  
\[ \begin{array}{l}\dot{p}_{i}=\dot{y}_{i}+(t_{f}-t)^{2}\dot{y}_{i}+\frac{1}{2 }(t_{f}-t)^{3}\dddot{y}_{i},\\ \dot{p}_{i}=0,\quad i=1,2.\end{array}