Geometric Characterization of Validity of the Lyapunov Convexity Theorem in the Plane for Two Controls under a Pointwise State Constraint

This paper concerns control BVPs, driven by ODEs x′t=ut, using controls u0· &u1· in L1a,b,R2. We ask these two controls to satisfy a very simple restriction: at points where their first coordinates coincide, also their second coordinates must coincide; which allows one to write (u1−u0)·=v·1,f· for some f·. Given a relaxed non bang-bang solution x¯·∈W1,1a,b,R2, a question relevant to applications was first posed three decades ago by A. Cellina: does there exist a bang-bang solution x^· having lower first-coordinate x^1·≤x¯1·? Being the answer always yes in dimension d=1, hence without f·, as proved by Amar and Cellina, for d=2 the problem is to find out which functions f· “are good”, namely “allow such 1-lower bang-bang solution x^· to exist”. The aim of this paper is to characterize “goodness of f·” geometrically, under “good data”. We do it so well that a simple computational app in a smartphone allows one to easily determine whether an explicitly given f· is good. For example: non-monotonic functions tend to be good; while, on the contrary, strictly monotonic functions are never good.