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

Abstract

This paper concerns control BVPs, driven by ODEs x′(t) = u(t), using controls u0(·) & u1(·) in L1((a, 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 \overline{x}(·) ∈ W1,1([a, 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 x1(·) ≤ \overline{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.