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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.

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 [

In the study of hybrid systems, the concept of viability is more prevalent. The notion of viability was first introduced by Aubin [

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 [

In this paper, we mainly consider the viability condition of a hybrid differential inclusion on a region with sub-differentiable boundary. Based on nonsmooth analysis theory, a method for checking the validity of the viability is given for such case as mapping of the set valued at the right hand of the differential inclusion is a polyhedron, the boundary function of the region is sub-differentiable, and its sub-differential is a convex hull with finite point set.

The paper is organized as follows. Section

Consider the general form of nonlinear control system

Let

Let

Let

Let

Let

A hybrid differential inclusion is a collection

A run of a hybrid differential inclusion

discrete evolution: for all

continuous evolution: if

We use

Let

The closed set

To discuss the problem in

Consider the following hybrid system

Consider the following region

For hybrid time set

On the other hand, we assume that the discrete transition does not occur infinite times within the limited time. The set

Let

In nonsmooth optimization, two frequently used constraint qualifications:

constraint qualification

constraint qualification

If the set

According to [

Assume that constraint qualification 1 or 2 is satisfied; then

Consider a hybrid system

Before we state Theorem

According to [

For the above hybrid system

discrete transition (or jump) must take place:

continuous section: for each fixed point

Under the above assumptions, it is sufficient to show that Theorem

In Lemma

The Lemma

Hybrid differential inclusion can describe a hybrid system in a wide range of significance.

Consider the following hybrid differential system

Consider the following region

For the above hybrid differential inclusion

discrete transition (or jump) must take place:

continuous section: Optimal value of the following linear programming problem

where

Under the above assumptions, it is sufficient to show that Theorem

In Lemma

In Lemma

Let hybrid differential inclusion be

discrete transition (or jump) must take place:

uncertainty section:

continuous section:

Under the above assumptions, it is sufficient to show that Theorem

In Lemma

In Lemma

We provide here an example that better illustrates the class of hybrid systems where our theoretical framework is relevant.

Consider the differential inclusion

We can easily conclude that

Viability discrimination of the point

Obviously, the point

Viability discrimination of the point

The authors declare that there is no conflict of interests regarding the publication of this paper.