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

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Stability of a linear system is a property that only depends on the eigenvalues of the matrix A.

For a nonlinear system things are more complicated, and the equilibria usually vary as a function of the chosen input. Stability of the linearized system around the equilibria will depend on both equilibrium state and input, but it's always the matrix that one has to look at.

--Elisa