%0 Journal Article
%T Viability Discrimination of a Class of Control Systems on a Nonsmooth Region
%A Na Zhao
%A Jianfeng Lv
%A Jinlin Yang
%A Xinzhi Liu
%J Discrete Dynamics in Nature and Society
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/127185
%X The viability problem is an important field of study in control theory; the corresponding research has profound significance in both theory and practice. In this paper, we consider the viability for both an affine nonlinear hybrid system and a hybrid differential inclusion on a region with subdifferentiable boundary. Based on the nonsmooth analysis theory, we obtain a method to verify the viability condition at a point, when the boundary function of the region is subdifferentiable and its subdifferential is convex hull of many finite points. 1. Introduction Hybrid systems have been used to describe complex dynamic systems that involve both continuous and discrete systems. Such hybrid systems can be extensively used in robotics, automated highway systems, air traffic management systems, manufacturing, communication networks, and computer synchronization, and so forth. There has been significant research activity in the area of hybrid systems in the past decade involving researchers from several areas [1¨C8]. In recent years, the viability of systems is an important research topic; it has been widely used in both reach-ability and designing security domain. In the study of hybrid systems, the concept of viability is more prevalent. The notion of viability was first introduced by Aubin [9]. Viability property provides a very nice theoretical framework for a hybrid controller design problem. Many researchers have considered the problem of viability for the analysis and control of hybrid systems [10¨C14]. The nonsampling viability problem was examined in the pioneering work of Aubin and coworkers [10] in which impulse differential inclusions are used to describe hybrid behavior. As an important part of hybrid system, studies in the viability theory include two topics. One is to verify viability condition for a given set. Another one is to design a viable solution within a viable set. Viability conditions for a linear control system have been studied widely in recent years; see [15, 16]. A necessary and sufficient viability condition for a differential inclusion was given in [8, 17], but it is a hard work to check that condition in most applications directly. In the literature [10], the authors give the necessary and sufficient condition of the viability, but it is still very difficult to judge quantitatively. Gao in [18] discusses the viability discrimination for an affine nonlinear control system on a smooth region; it gives some results on continuous system. There is certain limitation in the application of the literature [18]. The limitation is that the
%U http://www.hindawi.com/journals/ddns/2014/127185/