Consider the problem of transferring a unit mass at rest sliding on a plane from a point to another at a unit distance. We can exert a constant force of magnitude $x_i$ on the mass at time intervals $i-1<t \leq i $,$i=1,\ldots,10$ .

Denoting by $y_1$ the position at the final instant $T=10$, we can express via Newton’s law the relationship between the force vector $x$ and position/velocity vector $y$ as $y=Ax$, where $A \inR^{2 \times 10}$.

Now assume that we would like to find the smallest-norm (in the Euclidean sense) force that puts the mass at $y=(1,0)$ at the final time. This is the problem of finding the minimum-norm solution to the equation $Ax=y$. The solution is $x_{LN}=A^T(AA^T)^{-1}y$.