On BV-extension of asymptotically constrained control-affine systems and complementarity problem for measure differential equations

The goal of the paper is to design a constructive impulsive trajectory extension for a class of control-affine dynamical systems subject to a asymptotic mixed constraint of complementarity type. An inspiration for the addressed models comes from the framework of Lagrangian mechanical systems with impactively blockable degrees of freedom. The constraint formalizes the requirement that "control actions steer the system's state from one prescribed configuration \begin{document}$\mathcal{Z}_-$\end{document} to another one \begin{document}$\mathcal{Z}_+$\end{document} ". This issue is also closely connected with the problem of continuous trajectory approximation of hybrid systems with control switches.