Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

Control Theory and Technology

http://link.springer.com/journal/11768

Non-Zenoness of piecewise affine dynamicalsystems and affine complementarity systems

with inputsLe Quang THUAN†

Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Vietnam

Received 17 May 2013; revised 20 November 2013; accepted 21 November 2013

Abstract:In the context of continuous piecewise affine dynamical systems and affine complementarity systems with inputs, we study

the existence of Zeno behavior, i.e., infinite number of mode transitions in a finite-length time interval, in this paper. The mainresult reveals that continuous piecewise affine dynamical systems with piecewise real-analytic inputs do not exhibit Zeno behavior.Applied the achieved result to affine complementarity systems with inputs, we also obtained a similar conclusion. A direct benefitof the main result is that one can apply smooth ordinary differential equations theory in a local manner for the analysis ofcontinuous piecewise affine dynamical systems with inputs.

Keywords: Piecewise affine systems; Zeno behavior; Hybrid systems; Affine complementarity systems

DOI 10.1007/s11768-014-0074-5

1 Introduction

Analysis, simulation and design of hybrid dynamicalsystems become considerably complicated when thereare infinitely many mode transitions in a finite time in-terval. Such behavior is called Zeno behavior in the lit-erature [1,2]. To the best of our knowledge, the earliestwork goes back to the eighties when references [3, 4]studied Zeno behavior in the setting of piecewise an-alytic systems. With the increasing attention to hybridsystems, the study of Zeno behavior received consider-

able interest in the past few years [5–11].In this paper, we focus on piecewise affine dynam-

ical systems with inputs. Piecewise affine dynamicalsystems are a special kind of finite-dimensional, non-linear input/state/output systems, with the distinguish-ing feature that the functions representing the systemsdifferential equations and output equations are piece-wise affine functions. Any piecewise affine system canbe considered as a collection of finite-dimensional lin-ear input/state/output systems, together with a partitionof the product of the state space and input space into

†Corresponding author.E-mail: lethuan2004@yahoo.com.

2014 South China University of Technology, Academy of Mathematics and Systems Science, CAS, and Springer-Verlag

36 L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

polyhedral regions. Each of these regions is associatedwith one particular affine system from the collection.Depending on the region in which the state and inputvector are contained at a certain time, the dynamicsis governed by the affine system associated with thatregion. Thus, the dynamics switches if the state-inputvector changes from one polyhedral region to another.Any piecewise affine systems is therefore also a hybridsystem.

This note aims at providing conditions guaranteeingthe absence of Zeno behavior for a class of piecewiseaffine dynamical systems. More specifically, we showthe absence of Zeno behavior for continuous piecewiseaffine systems with the presence of a large class of ex-ternal inputs. This result is an extension of the recentresults in [12]. Similar conditions were already given forvarious subclasses of piecewise affine systems. Refer-ences [13–15] have provided such conditions for lin-ear passive complementarity systems, [16,17] for linearcomplementarity systems with singleton property, [18]for conewise linear systems, [19] for well-posed bi-modal piecewise linear systems, [20] for piecewise ana-lytic systems. Conditions for presence of Zeno behaviorhave been addressed in [21,22] for linear relay systems.Closely related to piecewise affine dynamical systems,differential variational systems were another subclassof hybrid systems for which Zeno behavior has beenstudied [23,24].

The organization of the paper is as follows. In Sec-tion 2, we introduce continuous piecewise affine dy-namical systems with inputs and its alternative repre-sentations. This will be followed by stating and prov-ing the main result of non-Zenoness of piecewise affinedynamical systems with inputs in Section 3. Section 4presents some applications of what we have developedin Section 3 to affine complemenatarity systems. Finally,conclusions are addressed in Section 5.

2 Continuous piecewise affine dynamicalsystems

In this section we will introduce the class of contin-uous piecewise affine dynamical systems that will beconsidered in the paper. To give a precise definition, webegin by recalling the so-called piecewise affine func-tions and its properties. A function ψ : Rν → R� is saidto be affine if there exist a matrix F ∈ R�×ν and a vectorg ∈ R� such that ψ(x) = Fx + g for all x ∈ Rν. It iscalled piecewise affine if there exists a finite family of

affine functions ψi : Rν → R� with i = 1, 2, . . . , r suchthat ψ(x) ∈ {ψ1(x), ψ2(x), . . . , ψr(x)} for all x ∈ Rν. Theaffine function ψi for i = 1, 2, . . . , r is called a selectionfunction corresponding to ψ.

In this paper, we consider the dynamical systems ofthe form

x(t) = f (x(t),u(t)), (1)

where x ∈ Rn is the state, u ∈ Rm is the input, andf : Rn × Rm → R

n is a continuous piecewise affinefunction. We call such a system continuous piecewiseaffine dynamical system (CPAS) with inputs.

Definition 1 A solution to system (1) for the ini-tial state x0 and locally integrable input u is under-stood in the sense that an absolutely continuous func-tion x : R → Rn such that x(0) = x0 and the pair (x, u)satisfies (1) for almost everywhere t ∈ R.

Note that the representation (1) describes the systemat hand in an implicit way via the component functions.Alternatively, a more explicit representation of (1) canbe obtained by invoking the well-known properties ofcontinuous piecewise affine functions. To do this, weuse the notion of polyhedral subdivision which is de-fined as follows. A finite collection P of polyhedra inR� is a polyhedral subdivision of R� if the union of all

polyhedra in P is equal to R�, each polyhedron in P isof full dimension, i.e., �, and the intersection of any twopolyhedra inP is either empty or a common proper faceof both polyhedra.

Since f is a continuous piecewise affine functionon Rn × Rm, one can find a polyhedral subdivisionΞ = {Ξi|1 � i � r} of Rn × Rm and a finite family ofaffine functions { fi|1 � i � r} such that f coincides fion Ξi; see for instance [25, Proposition 4.2.1]. Supposethat fi(x, u) = Aix + Biu + ei, and Ξi has the form

Ξi = {(x, u) ∈ Rn ×Rm|Cix +Diu + di � 0},

where Ai ∈ Rn×n, Bi ∈ Rn×m, ei ∈ Rn,Ci ∈ Rri×n,Di ∈ Rri×m, and di ∈ Rri . For these notations, system(1) can be rewritten in the explicit form as follows:

x(t) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

A1x(t) + B1u(t) + e1, if (x(t), u(t)) ∈ Ξ1,

A2x(t) + B2u(t) + e2, if (x(t), u(t)) ∈ Ξ2,...

Arx(t) + Bru(t) + er, if (x(t), u(t)) ∈ Ξr.

(2)

In this form, the continuity of the function f is equivalent

L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014 37

to the validity of the following implication:

(x, u) ∈ Ξi ∩ Ξ j ⇒ Aix + Biu + ei = Ajx + Bju + ej (3)

for all i, j ∈ {1, 2, . . . , r}. Since a continuous piecewiseaffine function must be globally Lipschitz continuous(see, [25, Proposition 4.2.2]), it follows from the theoryof the first-order ordinary differential equations that sys-tem (1), or equivalently system (2), must admit a uniquesolution for each initial state x0 and locally integral inputu, which is denoted by xu(t; x0).

3 Non-Zenoness of CPASs with piecewisereal-analytic inputs

It has been known that every continuous piecewiseaffine system is a hybrid system. In the hybrid systemsliterature, the occurrence of an infinite number of modetransitions within a finite time interval is called Zenobehavior. In hybrid systems literature, Zeno behaviorrefers to the possibility of infinitely many mode tran-sitions in a finite time interval. The presence of Zenobehavior causes difficulties in computer simulation aswell as in theoretical analysis of basic issues such aswell-posedness, controllability, stabilizability, etc. It istherefore important to investigate under what conditionsZeno behavior occur. In recent years, with the increas-ing attention to hybrid systems, the study of Zeno be-havior has received considerable interest. However, theexisting theoretical results are quite limited. The lack oftheoretical results has often led researchers to imposenon-Zeno assumptions on hybrid systems for furtheranalysis. Because of its importance, the non-Zenonessissue will be studied in the context of continuous piece-wise affine systems with inputs. For this purpose, wefirst define several notions of non-Zeno behavior forcontinuous piecewise affine dynamical systems with in-puts.

Definition 2 The continuous piecewise affine sys-tem (1) is called to have� the forward non-Zeno property with an input u if for

any x0 ∈ Rn and t∗ ∈ R, there exist an ε > 0 and anindex i ∈ {1, 2, . . . , r} such that (xu(t; x0), u(t)) ∈ Ξi for allt ∈ [t∗, t∗ + ε).� the backward non-Zeno property with an input u if

for any x0 ∈ Rn and t∗ ∈ R, there exist ε > 0 and anindex i ∈ {1, 2, . . . , r} such that (xu(t; x0), u(t)) ∈ Ξi for allt ∈ (t∗ − ε, t∗].� the non-Zeno property with an input u if it has both

the forward and the backward non-Zeno property withthis input.

In this paper, we restrict our consideration to theclass of inputs which are piecewise real-analytic onany compact time interval. A vector-valued functionu : [a, b] → Rm is said to be real-analytic if it is a re-striction of a real-analytic function defined on an openneighborhood of the interval [a, b]. It is called piecewisereal-analytic if there exists a finite partition of the inter-val [a, b] by the points a = t0 < t1 < . . . < tk = b suchthat the restriction of u on [ti, ti+1] is real-analytic forall 0 � i � k − 1. From now on, we will use ‘piecewisereal-analytic’ standing for ‘piecewise real-analytic on anycompact time interval’.

Note that the class of piecewise real-analytic func-tions is quite large. It covers the class of step functions,the class of Bohl function and piecewise Bohl functions,the class of piecewise polynomial functions, or morespecially the class of real-analytic functions, etc. Theseclasses are often used in analysis and control of dynam-ical systems with inputs; see for instance [26,27].

The following theorem is the main result of this paperregarding to the non-Zenoness of the continuous piece-wise affine dynamical systems with inputs. It shows theabsence of Zeno behavior in such a system with piece-wise real-analytic inputs.

Theorem 1 The continuous piecewise affine dynam-ical system (2) has the non-Zeno property with piece-wise real-analytic inputs.

Proof Associated with system (2), we consider thereverse-time system defined as

xrt =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

−A1xrt − B1urt − e1, if (xrt,urt) ∈ Ξ1,

−A2xrt − B2urt − e2, if (xrt,urt) ∈ Ξ2,...

−Arxrt − Brurt − er, if (xrt,urt) ∈ Ξr,

(4)

where xrt(t) := x(−t) and urt(t) := u(−t) for all t ∈ R. Itis clear that the reverse-time system (4) is also a contin-uous piecewise affine system. Moreover, if u is piece-wise real-analytic then so is urt. The following proposi-tion relates the backward non-Zeno property of system(2) to the forward non-Zeno property of the associatedreverse-time system (4). Its proof is straightforward andhence it is omitted.

Proposition 1 The continuous piecewise affine dy-namical system (2) has the backward non-Zeno propertywith piecewise real-analytic inputs if and only if its the

38 L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

associated reverse-time system (4) has the forward non-Zeno property with piecewise real-analytic inputs.

In the light of this proposition, in order to prove The-orem 1, it suffices to show that every continuous piece-wise affine system of form (2) has the forward non-Zenoproperty with piecewise real-analytic inputs. In what fol-lows, we will prove the latter statement. Let u be agiven piecewise real-analytic input, x0 ∈ Rn and t∗ ∈ R.We need to prove that there exist an ε > 0 and an in-dex i ∈ {1, 2, . . . , r} such that (xu(t; x0),u(t)) ∈ Ξi for allt ∈ [t∗, t∗ + ε). To do so, first note that we only need toshow for the case t∗ = 0. The result of the general casescan be obtained by letting x0 = xu(t∗; x0), u(t) = u(t+ t∗),and considering x0 and u in the role of x0 and u, respec-tively.

Clearly, at t∗ = 0 one has either u is real-analytic or uis not real-analytic. Next, we will deal with each case ofthem.

Case 1 This case addresses the case in which u isreal-analytic at the time t∗ = 0. To do so, we need to in-troduce some nomenclature and some auxiliary results.

For an ordered tuple a = (a1, a2, . . . , ak), we write a � 0if a = 0 or the first non-zero component is positive. Ifa � 0 and a � 0 then we write a > 0. Sometimes,we also use the symbols ‘�’ and ‘<’ with the obviousmeanings. For a finite collection of n-dimensional vec-tors z = (z1, z2, . . . , zk), we write z � (>)0 if for eachj ∈ {1, . . . ,n} it holds that (z1

j , z2j , . . . , z

kj ) � (>)0 where

the subscript j denotes the jth component of the corre-sponding vector. The same notations are also used forinfinite sequences.

For a vector-valued function w which is infinitelydifferentiable at t = 0, we denote wk the kth deriva-tive of w at t = 0, i.e., wk = w(k)(0) and denotewk = col(w0,w1, . . . ,wk). Also for each i ∈ {1, 2, . . . , r}we define the set

Yi,w := {x | Tki

⎡⎢⎢⎢⎢⎢⎣

x

wk

⎤⎥⎥⎥⎥⎥⎦ + ek

i � 0,∀k},

where the matrices Tki and ek

i are defined by

Tki =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Ci Di 0 · · · 0

CiAi CiBi Di · · · 0...

......

...

CiAki CiAk−1

i Bi CiAk−2i Bi · · · Di

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

eki =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

di

Ciei...

CiAk−1i ei

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

Finally, for a given matrix M the notation rowl(M) standsfor the lth-row of M.

For each input w real-analytic at t = 0, we will use thesetsYi,w to characterize the forward non-Zeno propertyof system (2) at time t = 0 with this input. For this pur-pose, we first state the following lemma that relates thesets Ξi, Yi,w, and the behavior of solutions.

Lemma 1 For each input w real-analytic at t = 0, thefollowing statements are equivalent:

1) The state x0 belongs to Yi,w.2) There exists an ε > 0 such that (xw(t; x0),w(t)) ∈ Ξi

for all t ∈ [0, ε].Proof 1) ⇒ 2): Let xw(t; x0) denote the solution

to equation x(t) = Aix(t) + Biw(t) + ei for the initialstate x0 and the input w, and let denote yw(t; x0) :=Cixw(t; x0) + Diw(t) + di. It can be seen that yw(t; x0) isreal-analytic at t = 0, and moreover at this point one has

col(y(0),w(0; x0), . . . , y(k),w(0; x0)) = Tki

⎡⎢⎢⎢⎢⎢⎣

x0

wk

⎤⎥⎥⎥⎥⎥⎦ + ek

i

for all k � 1 where y(�),w(0; x0) denotes the �th-derivativeof yw(t; x0) at t = 0. Since x0 ∈ Yi,w, i.e.,

col(y(0),w(0; x0), . . . , y(k),w(0; x0)) � 0

for all k � 1, we obtain that for each l ∈ {1, 2, . . . , ri} onlyone of the following two cases is possible:

i) The first case is that rowl(y(k),w(0; x0)) = 0 for all k �0. Then, the analyticity of yw(t; x0)) at t = 0 guaranteesthe existence of an εl > 0 such that rowl(yw(t; x0)) = 0for all t ∈ [0, εl].

ii) The second case is that

rowl(y(0),w(0; x0), . . . , y(k),w(0; x0), . . .) > 0. (5)

In this case, we first consider the case where thefirst element is positive, i.e., rowl(y(0),w(0; x0)) > 0.Then, due to continuity there exists εl > 0 such thatrowl(yw(t; x0)) > 0 for all t ∈ [0, εl]. Now, supposethat rowl(yw(0; x0)) = 0. Then, it follows from (5) thatthere exists kl � 1 such that rowl(y( j),w(0; x0)) = 0 forall 1 � j � kl − 1 and rowl(y(kl),w(0; x0)) > 0. Sincerowl(y(kl),w(0; x0)) > 0, there exists εl > 0 such that

L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014 39

rowl(y(kl),w(t; x0)) > 0 for all t ∈ [0, εl] due to conti-nuity rowl(y(kl),w(t; x0)). This means that the functionrowl(y(kl−1),w(t; x0)) strictly increases on [0, εl]. Hence,

rowl(y(kl−1),w(t; x0)) > rowl(y(kl−1),w(0; x0)) = 0

for all t ∈ [0, εl]. By repeating this argument, we finallyobtain rowl(yw(t; x0)) > 0 for all t ∈ [0, εl].

Finally, we can conclude that there exists εl > 0such that rowl(yw(t; x0)) � 0 for all t ∈ [0, εl] forboth cases. Define ε := min{ε1, . . . , εri}. Then, we haveyw(t; x0) = Cixw(t; x0) + Diw(t) + di � 0 for all t ∈ [0, ε]and hence (xw(t; x0),w(t)) ∈ Ξi for all t ∈ [0, ε]. Thisimplies that xw(t; x0) is a solution on the interval [0, ε]of system (1) for the initial state x0 and input w. Due tothe uniqueness of solution, we have xw(t; x0) = xw(t; x0)for all t ∈ [0, ε], and hence, (xw(t; x0),w(t)) ∈ Ξi for allt ∈ [0, ε].

2) ⇒ 1): Suppose that (xw(t; x0),w(t)) ∈ Ξi for allt ∈ [0, ε]. Then, we have

Cixw(t; x0) +Diw(t) + di � 0, (6)xw(t; x0) = Aixw(t; x0) + Biw(t) + ei (7)

for all t ∈ [0, ε]. Let xwi (t; x0) be the solution to system

(7) for the initial state x0 and the input w. Then, wehave xw(t; x0) = xw

i (t; x0) for all t ∈ [0, ε] and xwi (t; x0)

is real-analytic at t = 0 due to the real-analyticity of wat this point. Thus, this solution can be expressed inconvergent Taylor series around t = 0 as

xw(t; x0) = xwi (t; x0) = xw

i (0; x0) + txwi (0; x0)

+t2

2x(2),w

i (0; x0) + . . . (8)

for all t � 0 sufficiently small. In views of (6) and (8)together with noticing that xw

i (0; x0) = x0, x(k),wi (0; x0) =

Aki x0 +Ak−1

i Biw0 + . . .+Ak−1i ei + Biwk−1 for all k � 1, we

obtain

0 � (Cix0 +Diw0 + di) + t(CiAix0 + CiBiw0

+Diw1 + Ciei) + . . . +tk

k!(CiAk

i x0 + CiAk−1i Biw0

+ . . . +Diwk + CiAk−1i ei) + . . .

for all t � 0 sufficiently small. It is easy to see that thisinequality implies

Tki

⎡⎢⎢⎢⎢⎢⎣

x0

wk

⎤⎥⎥⎥⎥⎥⎦ + ek

i � 0

for all k � 1, and hence x0 ∈ Yi,w.

The following lemma characterizes for the forwardnon-Zeno property of a system of form (2) with piece-wise real-analytic inputs at every analytic point of inputsin terms of the sets Yi,·, 1 � i � r.

Lemma 2 Let w be real-analytic at time t = 0. Then,the following statements are equivalent:

1) The system (2) has the forward non-Zeno propertywith the input w at time t = 0, i.e., for any x0 ∈ Rn thereexist ε > 0 and an index i such that

(xw(t; x0),w(t)) ∈ Ξi for all t ∈ [0, ε].

2) The following equality holds

r⋃

i=1Yi,w = R

n.

Proof If the first statement holds, then for anyx0 ∈ Rn there exist ε > 0 and an index i ∈ {1, 2, . . . , r}such that (xu(t; x0),w(t)) ∈ Ξi for all t ∈ [0, ε]. Due toLemma 1, we obtain x0 ∈ Yi,w and hence Rn ⊆ r⋃

i=1Yi,u.

Since the reverse inclusion is evident, the second state-ment holds.

Note that the converse implication is obvious.In general, a given vector in Rn × Rm may be con-

tained in more than one of the sets Ξi. We define foreach (x0, y0) of Rn ×Rm the set I(x0, y0) as

I(x0, y0) := {i ∈ {1, 2, . . . , r}|(x0, y0) ∈ Ξi}.The following lemma is the last auxiliary result that willbe used in the proof of Theorem 1. Its proof was pre-sented in [12].

Lemma 3 [12] Let (x0, y0) ∈ Rn × Rm be given. Forany polynomial p in t with its coeficients are in Rn ×Rm

and p(0) = (x0, y0), there exist i ∈ I(x0, y0) and ε > 0such that p(t) ∈ Ξi for all t ∈ [0, ε].

With all these preparations, we are now in a positionto prove that system (2) has the forward non-Zeno prop-erty at time t∗ = 0 with the input u. In views of Propo-sition 1 and Lemma 2, it is enough to show that the

equalityr⋃

i=1Yi,u = R

n holds. The inclusionr⋃

i=1Yi,u ⊆ Rn

is obvious. To prove the inverse inclusion, we take anyx0 ∈ Rn, and denote I0 = I(x0,u0) for brevity. It followsfrom (3) that Aix0 + Biu0 + ei = Ajx0 + Bju0 + ej for alli, j ∈ I0. Let i ∈ I0 and define x1 := Aix0 + Biu0 + ei.Lemma 3 ensures that the set

I1 := {i ∈ I0|(x0,u0) + t(x1,u1) ∈ Ξi

for all sufficiently small t > 0}

40 L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

is non-empty. Since I1 ⊆ I0 and Ai(x0+tx1)+Bi(u0+tu1)+ei = Aj(x0+tx1)+Bj(u0+tu1)+ej for all i, j ∈ I1 and for allsufficiently small t > 0, one has Aix1+Biu1 = Ajx1+Bju1

for all i, j ∈ I1. Let us define x2 := Aix1 + Biu1 for somei ∈ I1. The set

I2 := {i ∈ I0|(x0,u0) + t(x1,u1) + t2(x2,u2) ∈ Ξi

for all sufficiently small t > 0}

is non-empty due to Lemma 3. Now, we claim thatI2 ⊆ I1. Indeed, for any i ∈ I2 we have Cix0+Diu0+di � 0because I2 ⊆ I0. Moreover, if rowl(Cix0 +Diu0 + di) = 0for some index 1 � l � ri, then we must haverowl(Cix1 + Diu1) � 0. Hence, it must hold thatCi(x0 + tx1) + Di(u0 + tu1) + di � 0 for all sufficientlysmall t > 0. Thus, it follows that i ∈ I1 and thenAix2 + Biu2 = Ajx2 + Bju2 for all i, j ∈ I2. Next, wedefine x3 := Aix2 + Biu2 for some i ∈ I2 and

I3 := {i ∈ I0|(x0,u0) + t(x1,u1) + t2(x2,u2) + t3(x3,u3)∈ Ξi for all sufficiently small t > 0}.

Due to Lemma 3, the set I3 is non-empty. By similararguments, we can show that I3 ⊆ I2 and Aix3 + Biu3 =

Ajx3 + Bju3 for all i, j ∈ I3. Continuing this process, wecan construct a infinite sequence of non-empty indexsets {Ij}∞j=1 such that

I0 ⊇ I1 ⊇ I2 ⊇ . . . ⊇ Ij ⊇ . . . .

Since I0 is a finite set, this sequence has to be stationary.i.e., there is a k0 such that

I0 ⊇ I1 ⊇ I2 ⊇ . . . ⊇ Ik0 = Ik0+�

for all � � 0. Then, for any i ∈ Ik0 we claim that x0 ∈ Yi,u.This claim can be proved by contradiction as follows.Assume that x0 � Yi,u, there exists k∗ � k0 such that

Tk∗i

⎡⎢⎢⎢⎢⎢⎣

x0

uk∗

⎤⎥⎥⎥⎥⎥⎦ + ek∗

i < 0. Going back to the construction, we

can find a sequence of elements x1, x2, . . . , xk∗ such that(x0,u0)+ t(x1,u1)+ . . .+ tk∗ (xk∗ ,uk∗ ) ∈ Ξk∗ for all t ∈ [0, ε)for some ε > 0, or more specifically the element

(x0,u0) + t(Ak∗x0 + Bk∗u0 + ek∗ , u1) + . . .

is in Ξk∗ for all t ∈ [0, ε). Thus, we have

Ck∗x0 +Dk∗u0 + dk∗ + t(Ck∗Ak∗x0 + Ck∗Bk∗u0

+Ck∗ek∗ +Dk∗u1) + . . . � 0

for all t ∈ [0, ε). This implies that Tk∗i

⎡⎢⎢⎢⎢⎢⎣

x0

uk∗

⎤⎥⎥⎥⎥⎥⎦+ ek∗

i � 0. This

is a contradiction, and hence x0 ∈ Yi,u.Case 2 In this case we deal with the case in which u

is not real-analytic at the time t∗ = 0. Since u is piecewisereal-analytic, there exists δ > 0 such that u is analyticon [0, δ]. By the definition of the real-analytic functionson a compact time interval, there exists a real-analyticfunction u on (α, β) containing [0, δ] such that u = u on[0, δ]. Applying what we have just obtained in Case 1 tothe function u, we can find an ε ∈ (0, δ) and an index isuch that (xu(t; x0), u(t)) ∈ Ξi for all t ∈ [0, ε]. Noticingthat xu(t; x0) = xu(t; x0) and u(t)) = u(t) for all t ∈ [0, ε),we conclude that the system has the forward non-Zenoproperty at t∗ = 0 with the input u.

To summary, in both of cases we always obtain thatthe system has the forward non-Zeno property with theinput u at time t∗. Since the piecewise real-analytic func-tion u is arbitrarily taken, we conclude that the systemhas the forward non-Zeno property with piecewise real-analytic inputs. The proof of Theorem 1 is completed.

4 Affine complementarity systems with ex-ternal inputs

In recent years, differential complementarity prob-lems are the main investigated object of several pa-pers for many fundamental issues; see for instance[14,16,17,27,28] and the references therein. In such is-sues, the issue whether or not existence of Zeno behav-ior has been attracted the attention of many researchersbecause of its importance in analysis of such systems.

In the remainder of this paper, a special class of differ-ential complementarity problems called affine comple-mentarity systems will be considered for Zeno property.Our main contribution on this issue is that we showthat some large classes of affine complementarity sys-tems with piecewise real-analytic inputs do not exhibitZeno behavior. This result is established based on ap-plying the results what we have just developed in theprevious section, and based on the piecewise affine re-formulation of the affine complementarity systems withsingleton property. In what follows, we will present indetail this result. The proceed begins with recalling theformulation of the linear complementarity problem andthen we formally introduce the so-called affine comple-mentarity systems.

L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014 41

4.1 Affine complementarity systems

In mathematical programming literature, the so-calledlinear complementarity problem has received much at-tention; see for instance [29] for a survey. This problemtakes as data a vector q ∈ Rm and a matrix M ∈ Rm×m,and asks whether it is possible to find a vector z ∈ Rm

such that

z � 0, q +Mz � 0 and zT(q +Mz) = 0. (9)

As usual, the notation LCP(q,M) will be used to referthis problem, the notation SOL(q,M) is standing for theset of all its solution, and the notation

0 � z ⊥ (q +Mz) � 0

indicates the relations in (9). One of the main results onthe linear complementarity problems that will be usedlater is as follows [29, Theorem 3.3.7]: the problemLCP(q,M) admits a unique solution for each q if andonly if M is a P-matrix, i.e., all its principal minors ispositive; see for instance [25,29] for further detail.

An affine complementarity system, roughly speaking,is a special case of differential complementarity prob-lems [30] which is described by standard state spaceequations of input/state/output affine systems togetherwith complementarity conditions as in the linear com-plementarity problem (9). Every affine complementaritysystem is arisen in such a way that one first takes a stan-dard input/state/output affine system and then selectsa number of pairs of input and output to impose foreach these pairs complementarity relation at each timet. This process results in a differential-algebraic systemwith inequality constraints of the form

x(t) = Ax(t) + Bu(t) + Ez(t) + g, (10a)w(t) = Cx(t) +Du(t) + Fz(t) + e, (10b)0 � z(t) ⊥ w(t) � 0, (10c)

where x ∈ Rn is the state, u ∈ Rm is the external in-put (unconstrained), the input z ∈ Rp and the outputw ∈ Rp play the role as the complementarity variables,and all the matrices are of appropriate sizes. In partic-ular, when g = 0 and e = 0, system (10) is known as alinear complementarity system with inputs. Such a sys-tem was well-studied in [27] for well-posedness, i.e.,existence and uniqueness of solutions, with consideredinputs to be bounded piecewise Bohl functions, and wasrecently investigated in [28] for controllability.

The results we obtain in this section can be seen as

the generalizations of the results of Shen and Pang [16]from linear to affine as well as from without inputs to theappearance of external inputs. Before presenting themin detail, we motivate that there are several problems inelectrical engineering, mechanical systems, economics,etc. can be modeled as systems of form (10); see forinstance [27,31,32]. Let us give an example of systemsthat may be recast into the affine complementarity sys-tems with external inputs.

Example 1 As an illustrated example, we considerthe state saturation system in Fig. 1 (a) with the PL rela-tion is as in Fig. 1 (b).

(a)

(b)

Fig. 1 Cascaded systems with a PL connection. (a) Cascadedsystems, and (b) the graph of PL connection.

This is an affine complementarity system given by

x1(t) = A1x1(t) + B1u(t),x2(t) = A2x2(t) + B2(λ1(t) − λ2(t) − B2,

y1(t) = λ1(t) − C1x1(t) − F1u(t) − 1,y2(t) = λ2(t) − C1x1(t) − F1u(t) + 1,0 � y(t) ⊥ λ(t) � 0.

4.2 Non-Zenoness of affine complementarity sys-tems with external inputs

Suppose that the class of affine complementarity sys-tems is considered in this section having the followingproperties:

S1) The solution set SOL(q, F) to the problemLCP(q,F) is non-empty for all q ∈ Rm.

S2) E and F have the F-singleton property in the sensethat the sets ESOL(q, F) and FSOL(q,F) are singleton foreach q ∈ Rm.Then, due to assumption S1) and the F-singleton prop-

42 L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

erty of E, ESOL(Cx+Du+ e, F) is known to be a contin-uous piecewise linear function in Cx+Du+ e and henceto be continuous piecewise affine in (x, u) ∈ Rn×Rm. Inviews of this, for each initial state x0 and an integrableinput u, there exist a unique absolutely continuous statetrajectory xu(t; x0), integrable trajectories zu(t; x0) andwu(t; x0) such that xu(0; x0) = x0 and the triple

(xu(t; x0), zu(t; x0),wu(t; x0)) (11)

satisfies relations (10) for almost all t ∈ R. We call sucha triple (11) solution to the affine complementarity sys-tem (10) for the initial state x0 and the input u. Notethat xu(t; x0) and wu(t; x0) are uniquely determined, butzu(t; x0) may not be unique for each x0 and input u.

Next, we will exactly give definitions of non-Zeno be-havior for system (10). In the linear complementaritysystems literature, there has existed two types of non-Zenoness notions [16,17] named strong non-Zenonessand weak non-Zenoness. These notions can extend tothe context of affine complementarity systems with in-puts.

To formally define the so-called strong non-Zenonessconcept, we introduce the fundamental triple of indexsets for a given initial state x0, input u and a solution(xu(t; x0), zu(t; x0),wu(t; x0)) as follows:

αu(t; x0) = {i ∈ {1, 2, . . . , p}|zui (t; x0) > 0 = wu

i (t; x0)},βu(t; x0) = {i ∈ {1, 2, . . . , p}|zu

i (t; x0) = 0 = wui (t; x0)},

γu(t; x0) = {i ∈ {1, 2, . . . , p}|zui (t; x0) = 0 < wu

i (t; x0)}.Definition 3 The affine complementarity system

(10) with an input u is called to have� the strong forward non-Zeno property if for any ini-

tial state x0 ∈ Rn and t∗ ∈ R there exists ε > 0 and atriple of index sets (α+, β+, γ+) such that

(αu(t; x0), βu(t; x0), γu(t; x0)) = (α+, β+, γ+)

for all t ∈ (t∗, t∗ + ε].� the strong backward non-Zeno property if for any

initial state x0 ∈ Rn and t∗ ∈ R there exists ε > 0 and atriple of index sets (α−, β−, γ−) such that

(αu(t; x0), βu(t; x0), γu(t; x0)) = (α−, β−, γ−)

for all t ∈ [t∗ − ε, t∗).� the strong non-Zeno property if it has both the strong

forward and backward non-Zeno property with this in-put.

In contrast, to define the modes in weak non-

Zenoness we consider an affine differential algebraicsystem characterized by a pair of disjoint index sets(θ, θc) whose union is {1, 2, . . . , p}:

x(t) = Ax(t) + Bu(t) + Ez(t) + g, (12a)0 = (Cx(t) +Du(t) + Fz(t) + e)θ = wθ(t), (12b)0 = zθc (t). (12c)

Definition 4 The affine complementarity system(10) with an input u is called to have� the weak forward non-Zeno property if for any initial

state x0 ∈ Rn, t∗ ∈ R and solution

(xu(t; x0), zu(t; x0),wu(t; x0))

there exists ε > 0 and a partition (θ, θc) of the set{1, 2, . . . , p} such that (xu(t; x0), zu(t; x0),wu(t; x0)) sat-isfies the differential algebraic system (12) for all t ∈(t∗, t∗ + ε].� the weak backward non-Zeno property if for any

initial state x0 ∈ Rn, t∗ ∈ R and solution

(xu(t; x0), zu(t; x0),wu(t; x0))

there exists ε > 0 and a partition (θ, θc) of the set{1, 2, . . . , p} such that (xu(t; x0), zu(t; x0),wu(t; x0)) sat-isfies the differential algebraic system (12) for all t ∈[t∗ − ε, t∗).� the weak non-Zeno property if it has both the weak

forward and backward non-Zeno property with this in-put.

It is clear that strong non-Zeno property implies weaknon-Zeno property. In the case that the matrix F is aP-matrix, the following theorem shows the absence ofstrong Zeno behavior, and hence of weak Zeno behav-ior, in the affine complementarity system (10).

Theorem 2 If F is a P-matrix then the affine comple-mentarity system (10) has the strong non-Zeno propertywith piecewise real-analytic inputs.

A proof of this theorem will be presented in the nextsubsection. As you will see later in that proof, the sameconclusion still holds if we replace the assumption of P-matrix property of F by a more general assumption thatthe linear complement problem LCP(Cx+Du+e, F) hasa unique solution for all x ∈ Rn and u ∈ Rm.

When the problem LCP(Cx+Du+e,F) does not haveunique solution for some x ∈ Rn and u ∈ Rm, assump-tions S1) and S2) do not guarantee for the absence ofstrong Zeno behaviors. The following is an illustratedexample in which assumptions S1) and S2) hold, but

L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014 43

strong non-Zenoness fails, which is taken from [16]:Example 2 Consider the linear complementarity

system as

x(t) = Ax(t) + b f Tz(t),w(t) = f cTx(t) + f f Tz(t),0 � z(t) ⊥ w(t) � 0,

where A ∈ Rn×n, b, c ∈ Rn with n � 2, and f is a non-zero m-vector. This system is of form (10) with u(t) = 0for all t.

However, assumptions S1) and S2) do guarantee forthe absence of weak Zeno behavior. This will be shownin the theorem below.

Theorem 3 Under assumptions S1) and S2), theaffine complementarity system (10) has the weak non-Zeno property with piecewise real-analytic inputs.

In the next two subsections below, we will presentthe proofs of Theorems 2 and 3.

4.3 Proof of Theorem 2

The proof of this theorem is accomplished in threesteps. In the first one we reformulate the affine comple-mentarity system (10) in terms of input/state/output con-tinuous piecewise affine system with the pair of comple-mentarity variables z,w playing the role of non-negativeoutputs. In the second step we derive an auxiliary resultrelated to sign partitions of components of the outputsin an input/state/output affine system. Finally, we ap-ply this auxiliary result together with Theorem 1 to thesystem obtained in the first step to finalize the proof.

Before proceeding the plan, we need to introducesome more notations. For a matrix Q ∈ Rn×m, QT

stands for its transpose, Q−1 stands for its inverseif this exists, imQ refers to its image, i.e., the set{Qx|x ∈ Rm}. We write Qij for the (i, j)th element of Q.For α ⊆ {1, 2, . . . ,n} and β ⊆ {1, 2, . . . ,m}, Qαβ denotesthe submatrix (Qjk)i∈α, j∈β. If α = {1, 2, . . . ,n}, we writeQ·β standing for Qαβ. The notation Qα· is also defined ina similar way.

We are now ready to present the proof. Since F is aP-matrix, the problem LCP(Cx+Du+ e, F) has a uniquesolution for each x ∈ Rn and u ∈ Rm. Thus, due to(10b) and (10c), for each t there exists an index setθt ⊆ {1, 2, . . . , p} such that

zθt (t) = −F−1θtθt

(Cθt·x(t) +Dθt·u(t) + eθt ) � 0, (13a)wθt (t) = 0, (13b)zθc

t(t) = 0, (13c)

wθct(t) = (Cθc

t · − Fθctθt F

−1θtθt

Cθt·)x(t)

+(Dθct ·−Fθc

tθt F−1θtθt

Dθt·)u(t) + eθct� 0,

(13d)

where θct denotes the complementary of θt in {1, 2,

. . . , p}. Note that θt is an arbitrary subset of the set{1, 2, . . . , p}, so that there are 2p different sets θt, saysθ1, θ2, . . . , θ2p .

For a subset θ of {1, 2, . . . , p}, we denote Πθ the per-mutation matrix of order p×p withΠθ(y) = col(yθ, yθc ),and define

Tθ =

⎡⎢⎢⎢⎢⎢⎣−F−1

θθ 0

−FθcθF−1θθ I

⎤⎥⎥⎥⎥⎥⎦ .

Corresponding to each index set θi, we construct a poly-hedron Ξθi of Rn ×Rm in such a way that

Ξθi = {(x, u) ∈ Rn ×Rm|TθiΠθi Cx + TθiΠθi Du

+

⎡⎢⎢⎢⎢⎢⎣F−1θiθi

eθi

eθci

⎤⎥⎥⎥⎥⎥⎦ � 0}.

Note that, due to (13a) and (13d), we have⎡⎢⎢⎢⎢⎢⎣

zθi (t)

wθci(t)

⎤⎥⎥⎥⎥⎥⎦ = TθiΠθi Cx(t) + TθiΠθi Du(t) +

⎡⎢⎢⎢⎢⎢⎣F−1θiθi

eθi

eθci

⎤⎥⎥⎥⎥⎥⎦

for all i = 1, 2, . . . , 2p and for all t such that θi = θt.Due to this equality and the fact that F is a P-matrix, wecan verify that Σ = {Ξθ1 , . . . , Ξθ2p } forms a polyhedralsubdivision ofRn×Rm. Moreover, subsystem (10a) canbe rewritten as a continuous piecewise affine system ofthe form

x(t) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

A1x(t) + B1u(t) + g1, if (x(t), u(t)) ∈ Ξθ1 ,...

A2m x(t) + B2m u(t) + g2m , if (x(t), u(t)) ∈ Ξθ2p ,

(14)

where the matrices Ai, Bi, gi defined as

Ai = A − E·θi [F−1θiθi

0]Πθi C,

Bi = B − E·θi [F−1θiθi

0]Πθi D,

gi = g − E·θi [F−1θiθi

0]Πθi e.

It also can be seen from (13) that z and w are non-negative and piecewise affine as

z(t) = (Πθi )T

⎡⎢⎢⎢⎢⎢⎣−F−1

θiθiCθi·

0

⎤⎥⎥⎥⎥⎥⎦ x(t) + (Πθi )T

44 L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

×⎡⎢⎢⎢⎢⎢⎣−F−1

θiθiDθi·

0

⎤⎥⎥⎥⎥⎥⎦ u(t) + (Πθi )T

⎡⎢⎢⎢⎢⎢⎣eθi

0

⎤⎥⎥⎥⎥⎥⎦ , (15)

w(t) = (Πθi )T

⎡⎢⎢⎢⎢⎢⎣

0

Cθci · − Fθc

i θi F−1θiθi

Cθi

⎤⎥⎥⎥⎥⎥⎦ x(t) + (Πθi )T

×⎡⎢⎢⎢⎢⎢⎣

0

Dθci · − Fθc

i θi F−1θiθi

Dθi

⎤⎥⎥⎥⎥⎥⎦ u(t) + (Πθi )T

⎡⎢⎢⎢⎢⎢⎣

0

eθci

⎤⎥⎥⎥⎥⎥⎦ ,

(16)

whenever (x(t), u(t)) ∈ Ξθi . Therefore, system (10) isreformulated as the system concluding (14), (15) and(16). To obtain a further analysis of the sign of the com-ponents of z and w, we derive the following auxiliarylemma.

Lemma 4 Consider the input/state/output affine sys-tem

x(t) = Kx(t) + Lu(t) + q, x(0) = x0, (17a)y(t) =Mx(t) +Nu(t) + p. (17b)

Let x and u be real-analytic on [0, ε] such that on thistime interval they satisfy (17a) and (17b) and y(t) � 0.Then, there exist an δ > 0 and an index set of compo-nents of the output y such that yθ(t) > 0 and yθc (t) = 0for all t ∈ (0, δ].

Proof As assumed, x is the solution to system (17a)for the initial state x0 and the input u on [0, ε]. Due tothe uniqueness of solutions to system (17a), we can seethat the output y does not change the values on [0, ε]if we replace u by its a real-analytic extension u on anopen neighborhood of [0, ε], and consider xu(t; x0) on[0, ε] in stead of x. Thus, if we can prove that the de-sired conclusion holds when x and u are real-analytic atthe time t = 0, then we are done. Note that the laterassumption implies that the output y is real-analytic att = 0. Therefore, in principle, by checking the sign ofthe derivatives of the rows of y(t) at time t = 0 we canform the index set θ and find a δ > 0 as we claimed.

With this lemma, we can finalize the proof of The-orem 2. Let us first prove for the strong forward non-Zeno property. Let u be a piecewise real-analytic in-put, x0 ∈ Rn and t∗ ∈ R. Without loss of generality,we can assume that t∗ = 0. Note that the solutionsxu(t; x0), zu(t; x0) and wu(t; x0) of system (10) is unique,and they satisfy (14), (15) and (16). By Theorem 1, sys-tem (14) has the forward non-Zeno property with piece-wise real-analytic inputs. Thus, there exist an ε > 0 andan index i such that xu(t; x0) and u(t) are real-analytic

and satisfy

xu(t; x0) = Aixu(t; x0) + Biu(t) + gi

on the closed time interval [0, ε]. Automatically, zu(t;x0) � 0 and wu(t; x0) � 0 and they satisfy (15) and (16)on [0, ε] with θi, respectively. Now, using Lemma 4, wecan find δ > 0 and a triple of index sets (α+, β+, γ+) suchthat

(αu(t; x0), βu(t; x0), γu(t; x0)) = (α+, β+, γ+)

for all t ∈ (0, δ], i.e., system (10) has the strong forwardnon-Zeno property.

By a similar argument, we also can prove that system(10) has the strong backward non-Zeno property, andhence system (10) has the strong non-Zeno propertywith piecewise real-analytic inputs.

4.4 Proof of Theorem 3

We begin with a technical lemma which will be em-ployed later on. This can be seen as a generalization ofLemma 3.

Lemma 5 Let Σ = {Σ1, Σ2, . . . , Σν} be a polyhedralsubdivision of Rn ×Rm.

1) For any x0 ∈ Rn, u0 ∈ Rm, the set⋃

i∈I(x0,u0)Σi con-

tains a neighborhood of (x0,u0) in Rn ×Rm.2) For any two functions x : R→ Rn and u : R→ Rm

which are real-analytic on [0, δ], there exist an ε ∈ (0, δ)and an � ∈ I(x(0),u(0)) such that p(t) := (x(t),u(t)) ∈ Σ�for all t ∈ [0, ε].

Proof Note that one can identity the product spaceR

n × Rm with Rn+m. Thus, the first statement followsfrom [12, Lemma 4.4].

For the second statement, we observe that the con-clusion does not change if we replace x and u by analyti-cally extension on an open neighborhood of [0, δ]. Thus,without loss of generality we can assume that x and uare real-analytic at t = 0. If x(t) = 0 and u(t) = 0 for allt ∈ [0, ε], then the conclusion is trivial. Therefore, weconsider for the case this does not hold. We prove thisstatement by contradiction. We assume that the claimdoes not hold. Then, for each i ∈ I(x(0),u(0)), one of thefollowing statements holds:

a) There exists ε > 0 such that p(t) � Σi for all t ∈ (0, ε).b) There exists an infinite sequence of positive scalars

tk all distinct and converging to 0 as k → ∞ such thatx(t2k−1) ∈ Σi and p(t2k) � Σi for all k � 1.Note that the latter statement must hold for at least one

L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014 45

index i ∈ I(x(0), u(0)). Otherwise, there would exist apositive number ε such that

p(t) �⋃

i∈I(x(0),u(0))Σi

for all t ∈ (0, ε). However, the set on the right hand sidecontains a neighbourhood U of (x(0), u(0)) due to thefirst statement of Lemma 3. This leads to a contradictionsince p(t) belongs to U for all sufficiently small t > 0 dueto continuity.

Since b) holds for some i ∈ I(x(0), u(0)), for everyk � 1 there exists an index �k ∈ {1, . . . ,mi} such that

row�k (Cix(t2k) +Diu(t2k) + di) < 0.

Denote Λ = {�k|k � 1}. Then, note that Λ is a finite set,there exists an index �∗ ∈ Λ such that

row�∗(Cix(t2k) +Diu(t2k) + di) < 0

for infinitely many k’s. Without loss of generality, wemay assume that

row�∗(Cix(t2k) +Diu(t2k) + di) < 0 (18)

for all k � 1. Then, for every k � 1, due to (18) andthe fact that row�∗ (Cix(t2k−1)+Diu(t2k−1)+di) � 0, thereexists μk ∈ [t2k−1, t2k) such that

row�∗ (Cix(μk) +Diu(μk) + di) = 0.

Since the μk’s are all distinct and row�∗(Cix(t)+Diu(t)+di) is a non-zero real-analytic function in t with real coef-ficients with finitely many roots on a finite time interval,we obtain row�∗(Cix(t)+Diu(t)+ di) = 0 for all t. This iscontradiction with (18) and hence the second statementholds.

Proof of Theorem 3 To prove this theorem, we firstalso reformulate the affine complementarity system (10)in terms of continuous piecewise affine systems. Due toS1) and S2), it is known that ESOL(Cx + Du + e,F) iscontinuous piecewise linear in Cx+Du+ e, and hence itis continuous piecewise affine in (x, u) onRn×Rm. Thus,one can find a polyhedral subdivision Σ = {Σ1, . . . , Σr}of Rn × Rm and the matrices Pi,Qi, hi, 1 � i � r, suchthat

ESOL(Cx +Du + e, F) = Pix +Qiu + hi,

whenever (x, u) ∈ Σi. Because of this, subsystem (10a)can be reformulated as a continuous piecewise affine

system with the polyhedral subdivision Σ,

x(t) = Aix(t) + Biu(t) + ei if (x(t), u(t)) ∈ Σi, (19)

where Ai = (A+Pi), Bi = B+Qi, and ei = g+hi. By a sim-ilar argument, FSOL(Cx +Du + e,F) is also continuouspiecewise affine in (x,u) on Rn × Rm. Thus, there exista polyhedral subdivision Δ = {Δ1, . . . , Δk} and matricesPi, Qi, hi, 1 � i � k, such that

FSOL(Cx +Du + e, F) = Pix + Qiu + hi,

whenever (x, u) ∈ Δi. In views of this, w in (10b) can berewritten as

w(t) = Cix(t) + Diu(t) + ei if (x(t), u(t)) ∈ Δi, (20)

where Ci = C + Pi, Di = D + Qi, and ei = e + hi.Now, let u be a piecewise real-analytic input, x0 ∈ Rn

and t∗ ∈ R. Then, the state solution xu(t; x0) is alsothe solution to system (19) for the initial x0 and theinput u. By Theorem 1, there exist ε1 > 0 and an in-dex i such that the pair xu(t; x0), u(t) satisfies (19) for allt ∈ [t∗, t∗ + ε1]. By Lemma 5, we can find δ > 0 such that(xu(t; x0), u(t)) ∈ Δτ ∩ Σ� for all t ∈ [0, δ]. Thus, we have

xu(t; x0) = A�xu(t; x0) + B�u(t) + e�, xu(0; x0) = x0

and 0 � wu(t; x0) = Cτxu(t; x0) + Dτu(t) + eτ for allt ∈ [0, δ]. Now, Lemma 4 implies the existence of ε > 0and an index set θ ⊆ {1, 2, . . . , p} such that

xu(t; x0) = Axu(t; x0) + Bu(t) + Ezu(t; x0) + g, (21a)0 = wu

θ(t; x0), (21b)0 = zu

θc (t; x0) (21c)

for all t ∈ (0, ε]. This means system (10) has the forwardweak non-Zeno property with piecewise real-analyticinputs.

By a similar argument, we can prove that the systemhas the weak backward non-Zeno property, hence theweak non-Zeno property at time t∗. The proof is com-pleted.

5 Conclusions

In this paper, we proved that continuous piecewiseaffine dynamical systems with piecewise real-analyticinputs do not exhibit Zeno behavior. This is a general-ization of recent results of the paper [12]. Using the re-sult which had just obtained, we also proved that some

46 L. Q. Thuan / Control Theory Tech, Vol. 12, No. 1, pp. 35–47, February 2014

classes of affine complementarity systems with piece-wise real-analytic inputs do not have Zeno behavior.The absence of Zeno behavior considerably simplifiesthe analysis of piecewise affine dynamical systems. Thisopens new possibilities in studying fundamental system-theoretic problems like controllability and observabilityfor these systems. Also the ideas employed in this noteare akin to be extended for possibly discontinuous butwell-posed (in the sense of existence and uniquenessof solutions) piecewise affine dynamical systems withinputs.

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Le Quang THUAN was born in Binh Dinh,Vietnam, in 1980. He received his B.S.and M.S. degrees in Mathematics fromQuy Nhon University, Vietnam, in 2002and 2005, respectively. Since 2002, he hasserved as a lecturer at the Mathematics De-partment of the Quy Nhon University. In2008, he joined the Systems, Control andApplied Analysis Group, the Department of

Mathematics, the University of Groningen, The Netherlands, where hereceived his Ph.D. degree in 2013. E-mail: lethuan2004@yahoo.com.