Comparison of path-complete Lyapunov functions via template-dependent lifts

Necessary and Sufficient Conditions for Template-Dependent Ordering of Path-Complete Lyapunov Methods

“…Following [9], we relate the validity of a lift L to the closure property of a template, and to the ordering relation in Definition 4 that it induces between a graph G and its lifted graph L(G).…”

“…The validity of the min lift (in the sense of the relation ( 4) in Definition 4) has already been proved regarding the class of templates closed under pointwise minimum in [9]. Indeed, we have proved that for any switched system F , if there exists a solution V := {V s | s ∈ S } in such a template V admissible for F and a pathcomplete graph G, then the graph G min also admits a solution in V for F , denoted by W .…”

“…Theorem 2 (Theorem 3 in [9]). Consider a family of binary operations {⋆ n } n ∈N such that ∀n ∈ N, ⋆ n corresponds to the pointwise minimum (resp.…”

“…We now focus on (9). Consider ẽ1 = (P 1 , Q 1 , i) ∈ E min , by the same argument as in (11) and [8][Lemma 1], we have to prove that…”

“…As expected, the validity of these lifts in the sense of (4) in Definition 4 holds for any template closed under sum. Indeed in [9], we proved that, for any T ∈ N and for any switched system F , if there exists a solution V in a template V closed under sum admissible for F and a path-complete graph G, then the candidate path-complete Lyapunov function W defined as…”