Asymptotic behavior of attainable sets of linear periodic control systems

In this paper we try to study in a more general setup a phenomenon, found in [1], where the asymptotic behavior of attainable sets for linear autonomous control systems was investigated. The main result of [1] 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 result of [1] shows that the shapes of attainable sets for linear autonomous control systems always possesses a limit as t --> infinity in a natural metric of the infinite-dimensional space of forms. At present the range of this phenomenon is not clearcut. 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) over bar(t), t > 0 of forms of attainable sets approaches, in general, not a point, but a closed curve (a limit cycle). More precisely, there exist limits of forms of attainable sets which corresponds to time instants with the same residue module the period of the system. These limits are just the points of the said closed limit curve. Such a behavior of the forms of attainable sets is due to presence of exponentially stable solutions to the system with zero control. If such solutions are absent, then the limit curve for a periodic system consists of a single point in the space of forms, just like the cage of an autonomous system. Thus, in our infinite-dimensional problem there arises a finite-dimensional (in fact, one dimensional) attractor. The fact that in the periodic setup the attractor arising is one dimensional, seems to be accidental. We have Some examples of (nonperiodic) linear control systems such that the dimension of the corresponding attractor exceeds any prescribed limit.