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 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.

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