  We propose the new notion of Visco-Energetic solutions to rate-independent
systems $(X,\mathcal E,\mathsf d)$ driven by a time dependent energy $\mathcal
E$ and a dissipation quasi-distance $\mathsf d$ in a general metric-topological
space $X$. As for the classic Energetic approach, solutions can be obtained by
solving a modified time Incremental Minimization Scheme, where at each step the
dissipation (quasi-)distance $\mathsf d$ is incremented by a viscous correction
$\delta$ (e.g.~proportional to the square of the distance $\mathsf d$), which
penalizes far distance jumps by inducing a localized version of the stability
condition.
  We prove a general convergence result and a typical characterization by
Stability and Energy Balance in a setting comparable to the standard energetic
one, thus capable to cover a wide range of applications. The new refined Energy
Balance condition compensates the localized stability and provides a careful
description of the jump behavior: at every jump the solution follows an optimal
transition, which resembles in a suitable variational sense the discrete scheme
that has been implemented for the whole construction.
