Asymptotic behavior of attainable sets of linear periodic control systems

We study in a more general setup a phenomenon concerning the asymptotic behavior of attainable sets for linear autonomous control systems. Oseevich´s main result (1991) consists in discovering a simple behavior in a long run of shapes of attainable sets, unlike that of attainable sets itself. Here, shape stands for the entity of all images of a set under nonsingular linear transformations. More precisely, the shapes of attainable sets for linear autonomous control systems always possess a limit as t→∞ in a natural metric of the infinite-dimensional space of forms. At present the range of this phenomenon is not clear-cut. In a search for its limits we consider here the asymptotic behavior of attainable sets for linear periodic control systems. Our main result establishes both a similarity and a distinction between the case under consideration and the autonomous case. We show that the curve t→D¯(t), t>0 of forms of attainable sets approaches, in general, not a point, but a limit cycle