  Trajectory Optimization (TO) and Reinforcement Learning (RL) are powerful and
complementary tools to solve optimal control problems. On the one hand, TO can
efficiently compute locally-optimal solutions, but it tends to get stuck in
local minima if the problem is not convex. On the other hand, RL is typically
less sensitive to non-convexity, but it requires a much higher computational
effort. Recently, we have proposed CACTO (Continuous Actor-Critic with
Trajectory Optimization), an algorithm that uses TO to guide the exploration of
an actor-critic RL algorithm. In turns, the policy encoded by the actor is used
to warm-start TO, closing the loop between TO and RL. In this work, we present
an extension of CACTO exploiting the idea of Sobolev learning. To make the
training of the critic network faster and more data efficient, we enrich it
with the gradient of the Value function, computed via a backward pass of the
differential dynamic programming algorithm. Our results show that the new
algorithm is more efficient than the original CACTO, reducing the number of TO
episodes by a factor ranging from 3 to 10, and consequently the computation
time. Moreover, we show that CACTO-SL helps TO to find better minima and to
produce more consistent results.
