I'm reading a very informative paper. But I met some formulations hard to accept.
In the stability proof section (Sec. V, Theorem 5.2), they define a Lyapunov function as $V(s) = \frac{1}{2}m\|s\|^2$ and obtain the derivative using the relation $m\dot{s}=\epsilon(\zeta,u-K_v s)$.
However, the following formulation seems hard to accept since they suddenly have discrete terms as:
$\dot{V}=s^\top(-K_v s + \hat{f}_a(\zeta_k,u_k)-\hat{f}_a(\zeta_k,u_{k-1})+\epsilon(\zeta_k,u_k))\cdots(A)$.
If I understand the context correctly, the derivative should look like,
$\dot{V}=s^\top(-K_v s + f_a(\zeta,u)-\hat{f}_a(\zeta,u))\cdots(B)$.
How is it possible to bridge the gap between (A) and (B)? How can one have discrete terms in a continuous Lyapunov function without any notation?
*FYI, I asked them this question directly 2 month ago (plus the reminder 2 weeks ago) and I do not have the reply yet.