Studies On The Viability For A Class Of Hybrid Systems

The viability problem is an important area of research in the control theory, the results of their research have profound significance in both theory and practice. This paper firstly introduces the viability of a hybrid differential inclusion on a region with sub-differentiable boundary. Based on non-smooth analysis theory, the viability criterion is verified under the condition that the set-valued mapping in the right side of differential inclusion is a poly-tope and the boundary function of the region is sub-differentiable and its sub-differential is a convex hull of many finite points. This verification can be implemented by determining the consistency of a group of linear inequalities, or equivalently, by solving a linear programming problem. And the viability of epigraph of a sub-differentiable function is discussed. Examples are presented to illustrate the results. Secondly, the viability criterions of certain and uncertain hybrid system are discussed. We define three operators, then give some basic properties. For a fixed region, we obtain a method that determines whether it has been a viable set. If it is not, an approximation algorithm is given to solve the viable kernel in the region. Finally, specific examples are provided to illustrate the algorithm.