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.