Consider the scalar nonlinear system $$\frac{d}{dt}x=\underbrace{\sin x+u}_{f(x,u)}$$
with equilibrium point $(x^{\ast},u^{\ast})=(0,0)$
We have $\frac{\partial f}{\partial x}=\cos x$ and $\frac{\partial f}{\partial u}=1$.
Thus $\frac{\partial f}{\partial x}(x^{\ast},u^{\ast})=\cos(0)=1$ and $\frac{\partial f}{\partial u}(x^{\ast},u^{\ast})=1$
So linearisation around the equilibrium yields $$\frac{d}{dt}\Delta_{x}=A\Delta_{x}+B\Delta_{u}$$ with $A=\frac{\partial f}{\partial x}(x^{\ast},u^{\ast})$ and $B=\frac{\partial f}{\partial u}(x^{\ast},u^{\ast})$. Hence $$\frac{d}{dt}\Delta_{x}=\Delta_{x}+\Delta_{u}$$
I want to determine if $(A,B)$, i.e. $(1,1)$ is controllable, but can this be determined for singular matrices such as these? For instance, I don't believe the controllability matrices, $\mathfrak{C}=\begin{bmatrix}B & AB & A^{2}B & \cdots & A^{n-1}B\end{bmatrix}$ can be computed.
In this case $n=1$, so controllability matrix is $1 \times 1$ matrix which is equal to $B$. Since $B$ is nonsingular, i.e. $\neq 0$ when scalar, the system is controllable.