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The stabilization problem of the upright position of the sherical pendulum is treated in detail. This problem is reduced to the stabilization of the stable manifold Omega_st of the upright position of the unforced spherical pendulum. It is shown that for any smooth feedback control derived by the speed-gradient algorithm with the objective to stabilize Omega_st the closed loop system has a limit cycle Gamma, which does not belong to the desired attractor Omega_st. It is shown that Gamma is hyperbolic

Stabilization of Stable Manifold of Upright Position of the Spherical Pendulum