Stability of Linear Dynamic Systems on Time Scales

Continuous and discrete dynamical systems have a number of significant differences mainly due to the topological fact that in one case the time scale T R, real numbers, and the corresponding trajectories are connected while in other case T Z, integers, they are not. The correct way of dealing with this duality is to provide separate proofs. All investigations on the two time scales show that much of the analysis is analogous but, at the same time, usually additional assumptions are needed in the discrete case in order to overcome the topological deficiency of lacking connectedness. Thus, we need to establish a theory that allows us to handle systematically both time scales simultaneously. To create the desired theory requires to setup a certain structure of Twhich is to play the role of the time scale generalizing R and Z. Furthermore, an operation on the space of functions from T to the state space has to be defined generalizing the differential and difference operations. This work was initiated by Hilger 1 in the name of “calculus on measure chains or time scales.” In this paper, we examine the various types of stability-stability, uniform stability, asymptotic stability, strong stability, restrictive stability, and so forth, for the solutions of linear dynamic systems on time scales and give two examples.