Franz Rußwurm, Jean Lévine, Stefan Streif
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 robust admissible set must be tangent to the state constraint set and separate the interior of the robust admissible set and its complement, a property that we call the ultimate locally separating hyperplane condition. Moreover, we prove that the barrier must satisfy a saddle-point principle on a Hamiltonian, based on Pontryagin's maximum principle, whose final condition is precisely the ultimate locally separating condition, thus providing a set of differential equations made of the system and its adjoint for a direct construction of the barrier. Lastly, we illustrate our results by calculating the robust admissible set for an adaptive cruise control example.