On the Boundary of the Robust Admissible Set in State and Input Constrained Nonlinear Systems

In this paper, we consider nonlinear control systems subject to bounded disturbances and to both state and input constraints. We introduce the definition of robust admissible set - the set of all initial states from which the state and input constraints can be satisfied for all times against all admissible disturbances. We focus on its boundary that can be decomposed into the usable part on the state constraint boundary and the barrier, interior to the state constraints. We show that, at the intersection of these two components, the boundary of the admissible set must be tangent to the state constraints and separate the interior of the robust admissible set and its complement. Moreover, we prove that the barrier must satisfy a saddle-point principle on a Hamiltonian, in the spirit of Pontryagin's maximum principle, thus providing a direct computational tool. Lastly, we illustrate our results by calculating the robust admissible set for an adaptive cruise control example.