  The task of inducing, via continuous static state-feedback control, an
asymptotically stable heteroclinic orbit in a nonlinear control system is
considered in this paper. The main motivation comes from the problem of
ensuring convergence to a so-called point-to-point maneuver in an underactuated
mechanical system. Namely, to a smooth curve in its state--control space, which
is consistent with the system dynamics and connects two (linearly) stabilizable
equilibrium points. The proposed method uses a particular parameterization,
together with a state projection onto the maneuver as to combine two
linearization techniques for this purpose: the Jacobian linearization at the
equilibria on the boundaries and a transverse linearization along the orbit.
This allows for the computation of stabilizing control gains offline by solving
a semidefinite programming problem. The resulting nonlinear controller, which
simultaneously asymptotically stabilizes both the orbit and the final
equilibrium, is time-invariant, locally Lipschitz continuous, requires no
switching, and has a familiar feedforward plus feedback--like structure. The
method is also complemented by synchronization function--based arguments for
planning such maneuvers for mechanical systems with one degree of
underactuation. Numerical simulations of the non-prehensile manipulation task
of a ball rolling between two points upon the "butterfly" robot demonstrates
the efficacy of the synthesis.
