Hamilton–Jacobi–Bellman Equation for Control Systems With Friction

This article proposes a new framework for modeling control systems, in which a dynamic friction occurs. The model consists of a controlled differential inclusion with a discontinuous 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.