  Optimal state-feedback controllers, capable of changing between different
objective functions, are advantageous to systems in which unexpected situations
may arise. However, synthesising such controllers, even for a single objective,
is a demanding process. In this paper, we present a novel and straightforward
approach to synthesising these policies through a combination of trajectory
optimisation, homotopy continuation, and imitation learning. We use numerical
continuation to efficiently generate optimal demonstrations across several
objectives and boundary conditions, and use these to train our policies.
Additionally, we demonstrate the ability of our policies to effectively learn
families of optimal state-feedback controllers, which can be used to change
objective functions online. We illustrate this approach across two trajectory
optimisation problems, an inverted pendulum swingup and a spacecraft orbit
transfer, and show that the synthesised policies, when evaluated in simulation,
produce trajectories that are near-optimal. These results indicate the benefit
of trajectory optimisation and homotopy continuation to the synthesis of
controllers in dynamic-objective contexts.
