STABILITY OF CONDITIONALLY INVARIANT-SETS AND CONTROLLED UNCERTAIN DYNAMIC-SYSTEMS ON TIME SCALES

A basic feedback control problem is that of obtaining some desired sta bility property from a system which contains uncertainties due to unkn own inputs into the system. Despite such imperfect knowledge in the se lected mathematical model, we often seek to devise controllers that wi ll steer the system in a certain required fashion. Various classes of controllers whose design is based on the method of Lyapunov are known for both discrete [4], [10], [15], and continuous [3-9], [11] models d escribed by difference and differential equations, respectively. Recen tly, a theory for what is known as dynamic systems on time scales has been built which incorporates both continuous and discrete times, name ly, time as an arbitrary closed sets of reals, and allows us to handle both systems simultaneously [1], [2], [12], [13]. This theory permits one to get some insight into and better understanding of the subtle d ifferences between discrete and continuous systems. We shall, in this paper, utilize the framework of the theory of dynamic systems on time scales to investigate the stability properties of conditionally invari ant sets which are then applied to discuss controlled systems with unc ertain elements. For the notion of conditionally invariant set and its stability properties, see [14]. Our results offer a new approach to t he problem in question.