Estimates for Trajectories Confined to a Cone in Rn

We develop estimates on the distance of a trajectory, associated with a differential inclusion $\dot{x}\in F$ in $\mathbb{R}^n$, to a given set of feasible $F$-trajectories, where “feasible” means “with values confined to a given cone.” When the cone is a half space, it is known that a (linear) $K\,h(x)$ estimate of the $\mathbf{W}^{1,1}$ distance is valid, where $K$ is a constant independent of the initial choice of trajectory and $h(x)$ is a measure of the constraint violation. A recent counterexample has unexpectedly demonstrated that linear estimates of the distance are no longer valid when the state constraint set is the intersection of two half spaces. This paper addresses fundamental questions concerning the approximation of general $F$-trajectories by $F$-trajectories confined to a cone which is the intersection of two half spaces. In this context, we establish the validity of a (superlinear) $K\,h(x)\,|\ln h(x)|$ estimate of the distance. We demonstrate by means of an example that the structure of this estimate is optimal. We show furthermore that a linear estimate can be recovered in the case when the velocity set $F$ is strictly convex.