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In earlier works, global inverse optimal controllers were proposed for input affine nonlinear systems with given control Lyapunov functions. However, these controllers do not provide information about the convergence rates. If the systems have local homogeneous approximations, we can employ homogeneous controllers, which locally asymptotically stabilize the origin and specify the convergence rates. However, homogeneous controllers generally do not attain global stability for nonhomogeneous systems. In this paper, we design global inverse optimal controllers with guaranteed local convergence rates by utilizing local homogeneity of input affine nonlinear systems. If we do not consider inverse optimality, we can liberally adjust the sector margins. We also clarify that local convergence rates and sector margins are invariant under coordinate transformations.