Comparative analysis of the asymptotic dynamics of reachable sets to linear systems

Explicit asymptotic formulas for reachable sets of linear dynamic systems with two types of control bounds, geometric constraints and constraints on the total impulse of the control action, are compared. A detailed proof of asymptotic formulas for the case of geometric constraints was first given. It was shown that, despite the similarity of the problem statement and methods to study the asymptotics of reachable sets for the considered types of constraints, the results obtained are essentially different. For example, for the impulsive case in the space of shapes of reachable sets, a multidimensional attractor may arise, while, in the case of geometric constraints, the corresponding attractor is reduced to a single point.