Let $(\bar x(t),\bar u(t)),\, t\in [0,1]$ solution of $$ \left \{ \begin{array}{l} \dot x_1 (t) = u(t)\, f(x(t)) \\ \dot x_2 (t) = u(t)\\ x_1(0) = 0, x_1(1)=1 \\ x_2(0) = x_2(1) \end{array} \right. $$ where $x=(x_1,x_2) \in \mathbb{R}^2$ is the state, $f$ is a given vector field and $u$ is the control function.
For $\alpha \in \mathbb{R}$, we define $v(t) = \bar u(t+\alpha)$ (well defined because $\bar u$ is periodic).
Do we have that $(\bar x(t), v(t)),\, t\in[0,1] $ solution of the previous control system?