<p>Here we consider a time-optimal control problem for systems with backlash and inelastic impact. Such systems are described by second order differential equations coupled with restrictions modeling the inelastic shocks. A main feature of such systems is the lack of uniqueness of solution to the Cauchy problem. Here, we introduce approximating systems where the forces during the impact are taken into account. The approximating systems are crucial for two reasons. Firstly, they allow us to define a set of solutions as limits of the solutions to the approximating systems. This set may be smaller than the set of the solutions usually considered in the literature. Secondly, such approximations are tailored to derive necessary condition to the time-optimal control of interest. To the best of our knowledge, this is the first attempt to derive necessary conditions of optimality for optimal control problems involving systems with backlash. The case of systems with backlash and elastic impact is covered in a forthcoming and companion paper.</p>