How do we investigate stability of a system that has inputs?

From MurrayWiki

Stability of a linear system $\dot{x} = Ax + Bu, y = Cx + Du$ is a property that only depends on the eigenvalues of the matrix A.

For a nonlinear system $\dot{x}=f(x,u)$ things are more complicated, and the equilibria usually vary as a function of the chosen input. Stability of the linearized system around the equilibria $(x e q, u e q)$ will depend on both equilibrium state and input, but it's always the $\frac{df}{dx}(x_{eq},u_{eq})$ matrix that one has to look at.