Barriers in Nonlinear Control Systems with Mixed Constraints

In this paper, we propose an extension to mixed multidimensional constraints of the problem of state and input constrained control introduced in DeDona and L\'evine, SIAM J. Control and Optimiz., Vol. 51, 2013, where the admissible set, namely the subset of the state space where the state and input constraints can be satisfied for all times, was studied, with focus on its boundary. The latter may be divided in two parts, one of them being called barrier, a semipermeable surface. We extend this notion of barrier to the mixed case and prove that it can be constructed via a minimum-like principle involving the Karush-K\"uhn-Tucker multipliers associated to the constraints and a generalised gradient condition at its endpoints.