A metric approach to the converse Lyapunov theorem for continuous multivalued dynamics

We construct a smooth Lyapunov pair for a continuous differential inclusion possessing a compact attractor. Our approach is based on the introduction of an intrinsic length functional, together with the corresponding length distance, defined starting from the polar cone of the multivalued vector field. This method allows a more geometrical insight into the subject and leads to a proof of the result which is, in our opinion, quite simple compared with the others available in the literature. © 2007 IOP Publishing Ltd. and London Mathematical Society.