Determining the Viability for Differential Inclusions

Yan Gao

School of Management, University of Shanghai for Science and Technology,

Shanghai 200093, China, email: gaoyan@usst.edu.cn

Abstract: Consider the following differential inclusion:

, (1)

where is mapping. Both linear and nonlinear control systems are special cases of the differential inclusion (1). Let . The set is said to be variable if for any initial point , there exists a solution of (1) such that for all . The closed set is viable for the differential inclusion (1) if and only if

(2)

where is tangent cone of . This paper is devoted to verifying the viability condition for a kind of differential and a kind of affine nonlinear control system on a region which is expressed by inequality constraints．Based on nonsmooth analysis, a method of determining the viability condition at a point is given. In these methods, determining the viability is transformed into determining the consistency of a system of convex inequalities. Then, a project method is used to solving convex inequalities.

Key words: differential inclusion, viability; nonsmooth analysis, nonlinear control.