Approximability and Variational Description of the Zeno Behavior in Affine Switched Systems

This paper deals with a new theoretic description of a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno dynamics. We consider solutions to the closed loop realizations of control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real-world processes do not present the corresponding behavior, mathematical models of some interconnected engineering systems may be Zeno due to the formal abstraction. In this paper we give a variational interpretation of the Zeno-like dynamics. A generic trajectory associated with this dynamics is finally specified as a result of a particular optimization procedure applied to the original model. The variational description of the Zeno effect provides an analytic basis for the constructive approximations techniques. We also study an intersection of a class of switched systems and the conventional sliding mode control processes.