Stability of manoeuvre regulation using arbitrary transverse foliations

In previous work (Hauser and Hindman, 1995, and Hindman and Hauser, 1996), we have shown that a trajectory tracking control law could be converted to a manoeuvre regulation control law by using a manoeuvre projection derived from a valid Lyapunov function. The given sufficient condition provided a safe way to construct a manoeuvre regulation control law (providing essentially orbital stability). In this paper, we explore the converse of this problem. The specific question of study is: given a trajectory tracking control law and a manoeuvre projection, can we find a compatible Lyapunov function? We show that this question is equivalent to an (infinite dimensional) convex feasibility problem. Using this result, we develop and implement an algorithm for constructing such a Lyapunov function (if feasible). An example is given that illustrates the nature of the problem, providing key new insights.