Refined Asymptotics for Singularly Perturbed Reachable Sets
We study, in the spirit of A. L. Dontchev, J. I. Slavov, Systems & Control Letters, Vol. 11, Issue 5, November 1988, reachable sets for singularly perturbed linear control systems. The fast component of the phase vector is assumed to be governed by a strictly stable linear system. It is shown in loc.cit. that the reachable sets converge as the small parameter $\varepsilon $ tends to $0$, and the rate of convergence is $O(\varepsilon ^\alpha )$, where
$0<\alpha <1$ is arbitrary. In fact, the said rate of convergence is $\varepsilon \log 1/\varepsilon $. Under an extra assumption pertaining to singularities of the boundaries of sets of admissible controls, we find the coefficient of
$\varepsilon \log 1/\varepsilon $ in the asymptotics of the support function of the reachable set.