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

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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.