#### Problem

Consider the control system

(1)on $N$, where $\tau (t)$ and $\varsigma (t)$ are real-valued control inputs. Is this system locally accessible near $f(0, \frac{\pi}{6})$? Small-time locally controllable near $f(\frac{\pi}{2}, 0)$?

#### Solution

The first thing to note is that this system is drift-less, i.e. if you do not give it any control inputs it will not move. Because of this fact, the questions of whether or not the system is locally accessible or small-time locally controllable(STLC) are equivalent. To test for local accessibility (and STLC for the drift-less case), check:

(2)For this problem, $n=3$, there is no term without a control input (no $g_0$), and there are two terms ($g_1$ and $g_0$) with control inputs control inputs ($\tau (t)$ and $\varsigma (t)$). For this problem:

(3)The Jacobi-Lie bracket $g_1$ and $g_2$ is:

(4)So, in order to check for local accessibility (and STLC) we need:

(5)For this problem:

(6)Plugging these functions into eq(5), we are left with:

(7)In order to check for local accessibility near $f\left(0,\frac{\pi}{6}\right)$, simply plug the values into equation 7 and check if the determinant of the matrix consisting of the three vectors is non-zero.

(8)Since the determinant is non-zero, the three vectors are linearly independent and span $\mathbb R^3$. Therefore, the system is locally accessible (and small-time locally controllable since drift-less) near $f\left(0,\frac{\pi}{6}\right)$.

To check whether or not the system is small-time locally controllable near $f\left(\frac{\pi}{2},0\right)$, again plug in the values into equation 7 and check the determinant.

(9)Again, the determinant is non-zero and therefore the three vectors span $\mathbb R^3$ , so the system is small-time locally controllable (also locally accessible since drift-less) near $f\left(\frac{\pi}{2},0\right)$.

Scott agrees!