  We study a certain class of weak solutions to rate-independent systems, which
is constructed by using the local minimality in a small neighborhood of order
$\varepsilon$ and then taking the limit $\varepsilon \to 0$. We show that the
resulting solution satisfies both the weak local stability and the new
energy-dissipation balance, similarly to the BV solutions constructed by
vanishing viscosity introduced recently by Mielke, Rossi and Savar\'e.
