  Given a training set in the form of a paired $(\mathcal{X},\mathcal{Y})$, we
say that the control system $\dot x = f(x,u)$ has learned the paired set via
the control $u^*$ if the system steers each point of $\mathcal{X}$ to its
corresponding target in $\mathcal{Y}$. If the training set is expanded, most
existing methods for finding a new control $u^*$ require starting from scratch,
resulting in a quadratic increase in complexity with the number of points. To
overcome this limitation, we introduce the concept of $\textit{ tuning without
forgetting}$. We develop $\textit{an iterative algorithm}$ to tune the control
$u^*$ when the training set expands, whereby points already in the paired set
are still matched, and new training samples are learned. At each update of our
method, the control $u^*$ is projected onto the kernel of the end-point mapping
generated by the controlled dynamics at the learned samples. It ensures keeping
the end-points for the previously learned samples constant while iteratively
learning additional samples.
