Consider the control system $$ \dot q = u_1 X(q) + u_2 Y(q) = F(q) U, \qquad q\in \mathbb{R}^3 $$ where $U=(u,v)$ is a control, and $X,Y$ are vector fields such that $X(2)=Y(3)=1$ and $X(3)=Y(2)=0$.
Let $g$ be the (Riemannian) metric $$ g = a(q) u_1^2+2b(q)u_1u_2 +c(q)u_2^2 = U^TSU $$ for some (smooth) functions $a,b,c$.
I would like to find a feedback $U = \beta(q)V$ such that the metric $\tilde g$ associated with the new controls $V$ is Euclidean, that is $$ \tilde g = v_1^2+v_2^2 $$
The problem is that if I can't find $\beta$ on all the domain so I was thinking to make an expansion in space for $\beta$ and find its coefficients to rectify the metric.
I would like some references about such calculations.
The system that I have to solve is $$ \beta^T S \beta = Id_2 $$
Shall I write $\beta_{11} = \sum_{i,j,k} \alpha_{i,j,k} q_1^i q_2^j q_3^k$ where $q=(q_1,q_2,q_3)$ and $\beta=(\beta_{ij})$ ? and so on ? and then find $\alpha_{i,j,k}$ ?
Your $S(q)$ is a symmetric matrix. According to Sylvester's law of inertia, a symmetric matrix can always be transformed by a congruence transformation (=P^TSP) to a diagonal form with only $1$, $-1$, and $0$'s along the diagonal and the number of respective elements is an invariant of $S$. The conclusion is that you can transform your $S(q)$ to an identity matrix if $S(q)$ has positive eigenvalues for all $q$.