Limit behavior of attainable sets of singular perturbed linear autonomous control systems

We consider a singularly-perturbed linear control system with a small parameter epsilon as a coefficient of the derivatives of the fast components of the state vector, over a finite time interval t epsilon [0, T], and investigate the asymptotic behaviour of its attainable sets K(epsilon, t) as epsilon --> 0. It has been proved [1] that if the system is stable with respect to the fast variables, then K(epsilon, t) converges. For systems without slow variables the convergence has been proved [2] for the shapes of the attainable sets rather than for the attainable sets themselves (by the spape of a set we mean the entity of all its images under non-singular linear transformations). In the general case considered here, it is possible to indicate a matrix scaling function R(epsilon, t) such that the product of this function and the attainable set K(epsilon, t) tend to a limit as epsilon --> 0, describing in this way the asymptotic properties of the attainable sets themselves. In the language of shapes (applicable only to systems such that their attainable sets are bodies), this means that the shapes of the attainable sets K(epsilon, t) converge.