Control of a unit mass
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 
on the mass at time intervals 
,
.
Denoting by 
the position at the final instant 
, we can express via Newton’s law the relationship between the force vector 
and position/velocity vector 
as 
, where 
.
Now assume that we would like to find the smallest-norm (in the Euclidean sense) force that puts the mass at 
at the final time. This is the problem of finding the minimum-norm solution to the equation 
. The solution is 
.
