Viability Discrimination of a Class of Control Systems on a Nonsmooth Region

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.


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][2][3][4][5][6][7][8].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][11][12][13][14].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 region must be smooth; in fact most of the region's boundaries are nonsmooth.Ahmed considers the viability criteria for a hybrid differential inclusions on smooth region in [19].Gao in [20] gives viability criteria for differential inclusions on a nonsmooth region.
In this paper, we mainly consider the viability condition of a hybrid differential inclusion on a region with subdifferentiable 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 2 states the main assumption, definitions and describes the hybrid dynamics.Section 3 overcomes these limitations in the literature [18]; we deal with the viability criteria for a hybrid system on a region with sub-differentiable boundary.Section 4 considers the viability of a hybrid differential inclusion.Section 5 shows an example.

Preliminaries
Consider the general form of nonlinear control system where  ∈ R  denotes the state variable,  ∈  denotes the control variable,  ⊂ R  , and (, ) is a Lipschitz function which is from R + to R  .
Definition 1 (see [8]).Let  ⊂ R  be a subset of R  , for any initial states  0 ∈ , if there exists one solution () of the system (1), such that () ∈  for all  ≥ 0; then we call the subset  viable under the system (1); the solution () is called viable solution.
Definition 2 (see [8]).Let  ⊆ R  be a nonempty subset of R  ; the tangent cone of the set  at  ∈  is given by the formula where   () is distance from the point  ∈ R  to the set .
for all  1 ,  2 ∈ , then  is said to be Lipschitz, where  > 0 is a Lipschitz constant.
Definition 6 (see [10], hybrid differential inclusion).A hybrid differential inclusion is a collection  = (, , , ), consisting of a finite dimensional vector space , a set valued map  :  → 2  , regarded as a differential inclusion ẋ () ∈ (), a set valued map  :  → 2  , regarded as a reset map, and a set  ⊆ , regarded as a forced transition set.
Proposition 9 (see [8]).The closed set  ⊂ R  is said to be viable under the system (1), if and only if for any  ∈ , the following formula is satisfied: For any interior point  in the set , the tangent cone   () = R  , so the above formula is satisfied.Hence, if we want to judge the above formula, we should only consider the boundary point.

The Viability of a Hybrid System
To discuss the problem in R  , we assume  = R  in the following paper.Consider the following hybrid system  = (, , , ), and where  : R  → R  and  : R  → R + are both Lipschitz functions. ⊂ R  is a convex set; it denotes where ℎ  () ( = 1, 2, . . ., ) are convex functions on the R  . is a reset map, and  is a forced transition set.Consider the following region : and   () ( = 1, 2, . . ., ) are sub-differentiable functions on R  .Furthermore, we assume that sub-differential   () is a convex hull of many finite points.
For hybrid time set  = {  }  =0 , where  is interval sequence.For  < , it has   = [  ,    ], for all ,   ≤    =  +1 .(   ) are the points at which discrete transitions take place, ( +1 ) are the points after discrete transitions take place; that is, On the other hand, we assume that the discrete transition does not occur infinite times within the limited time.The set  is a forced transition set; that is, the discrete transition must happen for every point in .Without generality, we assume that the set  contains the forced transition set  and the set  contains countable transition points.For discussing easily, we still denote by (   ) ( = 0, 1, 2, . . .,  − 1).In addition, in order to describe the uncertainy in the hybrid differential system and to determine whether discrete transition will happen for every point  in the set we assume that it can prevent the system from death cycle.Obviously, the points which are in  −1 () \  may not be jump.Let Since the point  ∈ R  satisfies max which is equivalent to so the set  can be denoted by the following formula: Because   () ( = 1, 2, . . ., ) are sub-differentiable, so () is also sub-differentiable; since   () is a convex hull of many finite points, the sub-differential of () is also a convex hull of many finite points, marking Define matrix  = ( 1 , . . .,   )  .
According to [20], we get the following Proposition 11 immediately.
Before we state Theorem 13, we construct the following inequality system: where  ∈ R  is a variable.
The Lemma 12( 2) is sufficient to show that the changes is possible (() ∩   () ̸ = 0) for continuous section in , when discrete transition point (or jump point) (()∩ = 0) will be not in  after the jump.The set  satisfies constraint qualification 1 or 2; then   () = { ∈ R  |  ≤ 0}.We set (, ) = () + () in Proposition 9; then the set  is viable under the hybrid system  if and only if the following formula is satisfied: where  is a fixed point in  \  −1 ().Consider the expressions of the set  and   (); the above expression is equivalent to Obviously, the above equation is equivalent to following solvable system: In ( 22), we set that  = () + () substitute into  ≤ 0; then we can obtain (18).Also, we can obtain ( 22) by substituting  = () + () into (18).This shows that the system ( 18) is equivalent to the system (22).This completes the proof.
In Lemma 12(2), we noticed that when discrete transition point (or jump point) after the jump (() ∩  = 0) will be not in , then the changes are possible (() ∩   () ̸ = 0) for continuous section in .
are equivalent, and also are equivalent to the linear programming problem (P) in which the optimal solution is zero.This completes the proof.

Example
We provide here an example that better illustrates the class of hybrid systems where our theoretical framework is relevant.Consider the differential inclusion  = (, , , ), where  () = co { 1 () ,  2 ()} ,  ∈ R ( We can easily conclude that () is a sub-differentiable function, and the set  is a quarter of the unit circle.
(2) Viability discrimination of the point