Optimal Controls 3

Non-holonomic integrator is given by the following dynamics;

\[\begin{align} \dot x_1 &= u_1 \\ \dot x_2 &= u_2 \\ \dot x_3 &= x_1 u_2 - x_2 u_1 \end{align}\]

Consider the objective of driving the system from \((0, 0, 0)^T\) to \((0, 0, a)^T\) by minimizing the following objective (minimum energy).

\[J({\bf u}) = \int_{t=0}^{t=1} {\bf u}^T(t) {\bf u}(t) dt\]

Turns out that for such a state transfer, optimal inputs are sinusoidal with fundamental frequency \(2\pi\).