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The viability of a switched system on a bounded polyhedron set, which is expressed by some linear inequalities, is investigated. Based on nonsmooth analysis and the properties of the tangent cone, a necessary and sufficient condition for viability is proposed. It is shown that the viability of a system is equivalent to the consistency of some systems of linear inequalities. Specifically, a viability condition for a switched system on a bounded polyhedron is presented. According to this condition, determining the viability of a bounded polyhedron can be transformed into verifying certain conditions at vertices of each facet. The method of determining viability, which transforms verifying the condition from infinite points to finite ones, can be implemented easily in practice. An algorithm to determine the viability for the switched system is constructed by using convex analysis. In addition, the approach can be extended to the switched system in which a control input is present. Finally, an example is listed to illustrate the effectiveness of the results.

The computation of the viability kernel provides the guarantee for the security evolution of the systems. In this paper, we focus on the computation of the viability kernel for discrete-time and continuous-time switched systems. A connection between the backward reachable set and the viability kernel for switched systems is established. The methods of computing the viability kernel for switched systems are constructed by using this connection. First, a method of computing the viability kernel for discrete-time switched systems is proposed. Then, taking into account the special structure of switched linear systems, a simple algorithm that is easy to implement is developed. Moreover, the methods of dealing with the discrete systems are extended to the continuous systems, and the algorithms of computing the viability kernel for continuous-time switched systems and switched linear systems are proposed. Finally, examples are listed to illustrate the effectiveness of the main results.