In this talk we will propose a new framework to model control systems in which a dynamic friction occurs. In particular, such a framework is motivated by the study of the movement of a robotic root tip in the soil. The model consists in a controlled differential inclusion with a dissipative, upper semi-continuous right hand side, which still preserves existence and uniqueness of the solution for each given input function u(t). Under general hypotheses, we are able to derive the Hamilton-Jacobi-Bellman equation for the related free time optimal control problem and to characterize the value function as the unique, locally Lipschitz continuous viscosity solution.