On the asymptotics of eigenvalues of the controllability gramian of a linear system with a small parameter

In this paper, the problem on asymptotics of eigenvalues of the controllability Gramian for a linear system with a small parameter is considered. This problem arises in the study of the convexity of reachable sets for control systems with integral constraints on the control on a small time interval. To verify the asymptotics of the eigenvalues, the controllability Gramian is expanded in a series in powers of a small parameter. We show that coefficients of this expansion can be calculated recursively. Also, we present examples demonstrating applications of the approach proposed to second-order systems.