This chapter undertakes an examination of the Lipschitz regularity of the value function within the context of infinite horizon control problems, with a particular emphasis on those incorporating discount factors, time-dependent data, and state constraints. To ensure feasibility and to derive local estimates within the ensemble of admissible trajectories, we impose controllability assumptions. We establish the Lipschitz continuity of the value function and demonstrate its asymptotic convergence to zero at infinity within the feasible set—contingent upon a discounted Lagrangian.