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

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 $(x_{eq},u_{eq})$ 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.

--Elisa