Vibrosolutions to differential equations in distributions with discontinuous regular functions on the right-hand side

We study $n$-dimensional differential equations in distributions of the form
$$ \dot x(t)=f(x,u,t)+ b(x,u,t)\dot u(t), $$
where $f(x,u,t)$ and $b(x,u,t)$ are piecewise continuous functions and $u(t)$ is an $m$-dimensional function of bounded variation with nondecreasing components. The notion of vibrosolution is introduced for equations of this type, and necessary and sufficient conditions for the existence of vibrosolutions are derived. The transition to an equivalent equation with measure is carried out, thus making it possible to explicitly calculate the jumps of the vibrosolutions at the points of discontinuity of $u(t)$.