Explicit asymptotic formulas for the reachable sets of linear dynamical systems with constraints on the total impulse of control action under different assumptions concerning the spectrum of the system’s matrix are obtained. It is shown that for large time, the reachable sets can be approximately represented in the form of the product of the scaling matrix and the normalized reachable set, where the matrix is an elementary function of time, and the normalized reachable set depends on time quasi-periodically. Analysis of asymptotic formulas has shown that, generally speaking, there exists a continuum of limit shapes, the aggregate of which produces a multidimensional attractor.

Asymptotics of reachable sets of linear dynamical systems with impulsive control

We study, in the spirit of [1], 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 e tends to 0, and the rate of convergence is O(eα), where 0<α<1 is arbitrary In fact, the said rate of convergence is elog1/e Under an extra smoothness assumption we find the coefficient of elog1/e in the asymptotics of the support function of the reachable set.

Asymptotics for singularly perturbed reachable sets

The controllability of linear systems with large and small time constants (singularly perturbed systems) is established. The time-optimal control of such systems is shown to be separable into two time scales related to the slow and fast modes of the system. An approximate design method of the time-optimal control is proposed, which is based on the separability of the fast and slow controls.

A.I. Ovseyevich

Papers

The asymptotic behaviour of attainable sets of singularly-perturbed linear autonomous control systems☆☆☆