# Geometric Analysis of Reachability and Observability for Impulsive Systems on Complex Field

Authors: Shouwei Zhao; Jitao Sun; Hai Lin
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Nowadays, quantum systems have become one of the focuses of the ongoing research and they
are typical complex systems, whose state variables are defined on the complex field. In this paper,
the issue of reachability and observability is addressed for a class of linear impulsive systems on
complex field, for simplicity, complex linear impulsive systems. This kind of time-driven impulsive
systems allows free impulsive instants, which leads to the limitation of using traditional definitions of
reachability and observability directly. New notations about the span reachable set and unobservable
set are proposed. Sufficient and necessary conditions for span reachability and observability of such
systems are established. Moreover, the explicit characterization of span reachable set and unobservable
set is presented by geometric analysis. It is pointed out that the geometric conditions are equivalent to
the algebraic ones in known results for special cases. Numerical examples are also presented to show
the effectiveness of the proposed methods.


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## Body

## 1. Introduction

Recent years have witnessed growing interest in investigating the control theory of hybrid systems and most progress has been made in the stability and stabilization of hybrid systems, see [1–5] and the references therein. Impulsive dynamical systems are an important class of hybrid systems which exhibit continuous evolutions described by ordinary differential equations and abrupt changes at some instants or impulses. Examples of these systems include evolution processes, optimal control models in economics, stimulated neural networks, frequency-modulated systems, and some motions of missiles or aircrafts. In view of both theoretical and practical significance, much attention has been paid on the analysis and synthesis of impulsive systems, or impulsive control systems, see [6–11] and the references therein.Closely related to the pole assignment, structural decomposition, quadratic optimal control and observer design, the controllability, reachability, and observability play a significant role in the control theory and engineering [12–14]. The controllability and observability of various hybrid systems have been extensively investigated using different approaches such as geometric analysis [7–9], algebraic characterization [10, 11], functional analysis [15, 16], and differential geometric method [3]. Particularly, research efforts have been made on the controllability and observability for impulsive systems. By proposing algebraic rank conditions, the state controllability and observability of linear time-varying impulsive systems were investigated in [10, 11]. For impulsive functional differential systems, the controllability is considered with the help of fixed-point theorems [15, 16]. References [7–9] presented the geometric analysis of reachability, controllability and observability for (switched) impulsive systems. Geometric analysis is effective in providing easily verifiable conditions for the controllability and observability based on the explicit characterization of controllable and observable sets in terms of invariant sets of systems. Hence, it provides an effective and simple method to investigate the fundamental properties of hybrid systems.However, in the above-mentioned works, the state space of the considered systems is alwaysn-dimensional real vector space, that is, ℝn, except few reports on the issue of controllability for complex systems [10, 17]. Nowadays, control of complex systems, especially quantum systems, has attracted considerable attention [18–22]. It should be noticed that quantum system models are typical complex dynamical systems whose states evolve in Banach (Hilbert) space on the field of complex number, which are much more complicated than real systems. In view of this, complex dynamics systems have many potential applications ranging from science to engineering. Therefore, it is important and necessary to study the control theory of a special class of complex dynamical systems, complex linear impulsive systems. This motivates us to consider the reachability and observability of complex linear impulsive systems by geometric analysis. The impulsive system considered in the current paper has uncertainty in the impulsive instants which can be regarded as time-driven impulsive systems. This kind of more general systems exists in many practical applications [7]. Due to the novel properties of reachable set and unobservable set for this kind of systems, traditional geometric analysis may be limited to characterize them. Hence, new concepts on the reachable set and unobservable set are introduced. Based on these definitions, we generalize the geometric analysis approach for reachability and observability to complex linear impulsive systems. Specifically, sufficient and necessary criteria for reachability and observability are derived, and explicit characterization of reachable set and unobservable set is proposed consequently. Moreover, it is proved that the span reachable set and unobservable set with free impulsive times are invariant subspaces of the complex impulsive system.The rest of this paper is organized as follows. In Section2, the complex linear impulsive systems to be dealt with are formulated and the solution expression for such systems is presented. In Sections 3 and 4, based on the geometric characterization of reachable set and unobservable set for complex linear impulsive systems, sufficient and necessary conditions for state reachability and state observability of complex linear impulsive systems are derived, respectively. Moreover, examples are discussed to illustrate the effectiveness of the proposed methods. Finally, some conclusions are drawn in Section 5.


## 2. Preliminaries

Consider the complex linear time-varying impulsive system described by(2.1)ẋ(t)=A(t)x(t)+B(t)u(t),t≠tk,Δx(tk)=Ekx(tk)+Fkuk,y(t)=C(t)x(t)+D(t)u(t),x(t0+)=x0,
where k=1,2,…,A(t),B(t),C(t), and D(t) are known n×n, n×m, p×n, and p×m continuous-time complex-valued matrices, x∈ℂn is the state vector, u∈ℂm is the control input, Ek and Fk are complex n×n and n×m constant matrices, respectively, y∈ℂp is the output, J=[t0,+∞), Δx(tk)=x(tk+)-x(tk-), where x(tk+)=limh→0+x(tk+h), x(tk)=x(tk-)=limh→0+x(tk-h) with discontinuity points t0<t1<t2<⋯<tk<⋯,limk→∞tk=∞, which implies that the solution of system (2.1) is left-continuous at tk. It should be noticed that the impulsive instants tk can be chosen freely in this paper. We know that x(t):ℝ→ℂn, and ℂn is a Banach space on the complex field ℂ. A(t):ℝ→𝔘 where 𝔘=𝔏(ℂn,ℂn) is the bounded ℂn-linear continuous map. Hence complex impulsive system (2.1) is a special differential-difference equation in Banach space defined on the complex number field ℂ. Let A*=A̅⊤ be the conjugated transpose of the complex matrix A. ∏i=k-11Ai stands for the matrix product Ak-1Ak-2⋯A1.Corresponding to system (2.1), consider the following complex differential equation:(2.2)ẋ(t)=A(t)x(t).
Suppose that X(t) is the fundamental solution matrix of system (2.2). Then X(t,s):=X(t)X-1(s), (t,s∈J) is the transition matrix associated with the matrix A(t). It is clear that X(t,t)=I, X(t,τ)X(τ,s)=X(t,s) and X(t,s)=X-1(s,t). Now we present the solution expression of complex impulsive system (2.1) which was proved in [10] using ordinary differential equations theory in the complex field.Lemma 2.1 (see [10]).
Fort∈(tk-1,tk],  k=1,2,…, the solution of system (2.1) is given by
(2.3)x(t)=X(t,tk-1)[∏j=k-11(I+Ej)X(tj,tj-1)x0+∑i=1k-1∏j=k-1i(I+Ej)X(tj,tj-1)×∫ti-1tiX(ti-1,s)B(s)u(s)ds+∑i=2k-1∏j=k-1i(I+Ej)X(tj,tj-1)Fi-1ui-1+Fk-1uk-1]+∫tk-1tX(t,s)B(s)u(s)ds.In the remainder of this paper, we focus our attention on the reachability of time-invariant version of system (2.1) with respect to the continuous-time input u(t) and observability with respect to the continuous-time output y(t). The complex linear impulsive system is given by(2.4)ẋ(t)=Ax(t)+Bu(t),t≠tk,x(tk+)=Ex(tk),y(t)=Cx(t),x(t0+)=x0,
where A,B,C,E are known n×n, n×m, p×n, n×n constant complex matrices. Let 𝒯={t1,t2,….} be a countable set of impulse times.Remark 2.2.
System (2.4) is a class of more general linear impulsive systems in the complex field with time-driven impulsive behavior. The system parameter matrices are all complex matrices. It should be noticed that the impulse times could be chosen freely, allowing for a richer interaction between the continuous-time dynamics and the impulsive effects. Hence, with inherent uncertainties, system (2.4) has interesting features in reachability and observability different from that of common impulsive systems. This motivates our current work.Given an initial timet0 and final time tf, Lemma 2.1 gives the solution of (2.4) as follows:(2.5)x(tf)=eAhM{∏m=M-11EeAhmx(t0)+∑m=1M-2[∏j=M-1m+1(EeAhj)E∫tm-1tmeA(tm-s)Bu(s)ds]+E∫tM-2tM-1eA(tM-1-s)Bu(s)ds∏m=M-11EeAhmx(t0)}+∫tM-1tMeA(tM-s)Bu(s)ds,
where tf=tM and hi=ti-ti-1, i=1,2,…,M. In the subsequent, we proceed to investigate the reachability and observability criteria of complex linear impulsive system (2.4).


## 3. Geometric Analysis of Reachability

In this section, the main purpose is to characterize the geometric properties of reachability of complex linear impulsive system (2.4) and establish the equivalence between algebraic criteria in known results [10] and the geometric ones obtained here. To discuss the geometric property of reachable set for complex impulsive system (2.4), we first introduce the concept of invariant subspace of complex linear systems.Consider the following complex linear system:(3.1)ẋ(t)=Ax(t)+Bu(t),y(t)=Cx(t),x(t0+)=x0.
For the complex constant matrix B∈ℂn×m, denote ImB as the range of B spanned by the columns of B, that is, ℬ≜ImB={y∣y=Bu,∀u∈ℂm}. For a given matrix A∈ℂn×n and a linear space 𝒲⊆ℂn, let 〈A∣𝒲〉 be the minimal A-invariant subspace containing 𝒲, that is, 〈A∣𝒲〉=∑i=0n-1Ai𝒲. For simplicity, we denote 〈A∣B〉=〈A∣ImB〉. By [13], for any complex matrices A and B, we have {x:x=∫t0teA(t-τ)Bu(τ)dτ,forsomep.c.v.functionu,t>t0}=〈A∣B〉, which is the reachable subspace of complex linear system (3.1). Moreover, the equivalence between the algebraic condition for controllability and the geometric one is given as follows. rank (BAB⋯An-1B)=n is equivalent to Im(B)+AIm(B)+⋯+An-1Im(B)=ℂn.In view of the special structure of the system considered here, definitions about the reachability are introduced first. The state space for complex impulsive system (2.4) is denoted by 𝒳.Definition 3.1 (Reachable set with fixed final time and fixed impulse times).
For complex linear impulsive system (2.4), a nonzero state xf is said to be reachable from zero with fixed final time and fixed impulse times, if given t0,tf>t0 and a set of impulse times 𝒯,there exists a piecewise continuous input u(t),t∈[t0,tf],such that the system is driven from x(t0)=0tox(tf)=xf.The set of reachable states with fixed final time tf and fixed impulse times 𝒯 is denoted by ℛfixed(t0,tf,𝒯).Definition 3.2 (Reachable set with free final time and free impulse times).
For system (2.4), a nonzero state xf is said to be reachable from zero with free final time and free impulse times, if given t0, there exists tf>t0,a set of impulse times 𝒯 and a piecewise continuous input u(t),t∈[t0,tf], such that the system is driven from x(t0)=0 to x(tf)=xf. The set of reachable states with free final time tf and free impulse times 𝒯 is denoted by ℛfree.From the definitions, we obtainℛfree=⋃𝒯⋃tf>t0ℛfixed(t0,tf,𝒯). Given an impulse times set 𝒯, by (2.5) and Definition 3.1, ℛfixed(𝒯) is given by(3.2)Rfixed=eAhM{∏m=M-11EeAhmx(t0)+∑m=1M-2[∏j=M-1m+1(EeAhj)E〈A∣B〉]+E〈A∣B〉}+〈A∣B〉.
In [7], it was pointed out that for real impulsive systems, the reachable set does not necessarily constitute a subspace. Thus, for complex linear impulsive system (2.4), ℛfree may be a subset instead of subspace of the state space. This fact will be clarified in the following example.Example 3.3.
Consider complex linear impulsive system (2.4) with
(3.3)A=[0010000000000000],B=(01+2i00),E=[0000001001+i001000].
It is clear that 〈A∣B〉=Im(B). For the case t0<t1<tf,
(3.4)Rfixed(t0,tf,T)=[(1+i)(1+2i)h1001+2i(1+i)(1+2i)000].
For any even number k≥2 and t0<t1<⋯<tk<tf, it yields that
(3.5)Rfixed(t0,tf,T)=[(1+i)k-1(1+2i)000(1+i)k-2(1+2i)1+2i(1+i)k(1+2i)hk000(1+i)k-2(1+2i)hk-10].
Therefore, when the final time and the impulse times are fixed, the system can reach at most a three-dimensional complex subspace of the state space. It follows that when only two impulse times are required, that is, 0<t1<t2<tf, ℛfree can be characterized as follows:
(3.6)Rfree=⋃h1,h2>0Rfixed(t0,tf,T)=⋃h1,h2>0Im⁡[(1+i)(1+2i)000(1+2i)1+2i2i(1+2i)h2000(1+2i)h10].
For a vector given by x=[abcd]T(a,b,c,d≠0), it can be represented by the linear combination of elements in ℛfree while it is not included in ℛfree. It should be noticed that the subspace spanned by the reachable set with free impulse times is the entire complex state space.Example3.3 motivates us to present a new concept, span reachability, for complex impulsive system (2.4).Definition 3.4 (Span Reachability).
For complex impulsive system (2.4), the subspace spanned by the elements of ℛfree is denoted by ℛspan. A complex impulsive system for which ℛspan=ℂn is said to be span reachable.In the following, the explicit construction of span reachable set is proposed and its property is discussed. Denote the following subspaces sequences:(3.7)W0=〈A∣B〉,Wm=〈A∣EWm-1〉,m≥1,Vm=∑i=0mWi,m=0,1,2,….
It is clear that 𝒱0⊆𝒱1⊆⋯⊆𝒱m-1⊆𝒱m, dim𝒱m<∞. If there exists an integer m>0 such that 𝒱m=𝒱m+1, by the construction of 𝒱m, it is easy to verify 𝒱m=𝒱m+1=𝒱m+2=…. This implies that the sequence {𝒱m,m=1,2,…} converges to 𝒱n. For the proof of the main results, a Lemma is presented first. The proof is similar to that of Lemma  2 in [12]. Thus, we omit it here.Lemma 3.5.
Given a complex matrixA∈ℂn×n, for almost  T∈ℝ, one has  〈A∣𝒲〉=〈eAT∣𝒲〉.Theorem 3.6.
For complex linear impulsive system (2.4), one has
(3.8)Rspan⁡=Vn.Proof.
For anyxf∈ℛfree, by (3.2) and the property of invariant subspace, we have xf∈𝒱n. Then ℛspan⊆𝒱n. Next we prove the reverse inclusion. From Lemma 3.5, there exists h>0 such that sequence (3.7) can be redefined as follows:
(3.9)W0=〈A∣B〉,Wm=〈eAh∣EWm-1〉,m≥1.
Using the property of invariant subspace, (3.9) can be rewritten as
(3.10)W0=〈A∣B〉,Wm=eAh〈eAh∣EWm-1〉,m≥1,
which implies that 𝒱n has the following form:
(3.11)Vn=〈A∣B〉+∑m=1n∑lm,…,l2,l1∈{1,2,…,n}[eAh]lmE⋯[eAh]l1E〈A∣B〉.
Denote an impulse times set to be {l1h,l2h,…,lnh}. It is easy to get that 〈A∣B〉+∑m=1n[eAh]lmE⋯[eAh]l1E〈A∣B〉⊆ℛfixed. Hence, we obtain 𝒱n⊆ℛfree. Since any element of ℛspan can be expressed as a linear combination of elements from ℛfree, we conclude that 𝒱n⊆ℛspan. This completes the proof.Remark 3.7.
From Definition3.4 and Theorem 3.6, it can be found that if 𝒱n=ℂn, system (2.4) is span reachable. For fixed final time and impulse times, if 〈A∣B〉=ℂn,which implies that ℛfixed constitutes the entire space, then we know that rank(B,AB,…,An-1B)=n. From Theorem  3 in [10], the above condition indicates that system (2.4) is controllable. Hence, when reduced to linear systems, the algebraic condition (3.11) in [10] and the geometric criterion 〈A∣B〉=ℂn are equivalent in checking the reachability and controllability of system (2.4). When reduced to complex linear impulsive systems with fixed impulse times and AE=EA, simple computation follows that the conditions for reachability and controllability are equivalent. While in this paper, we consider a more general system with time-driven impulses, and a new concept, span reachability is introduced. Hence, the derived conditions in this paper and the known literature [10] cannot be compared directly.The concept of invariant subspace is fundamental to a geometric analysis of linear time-invariant systems. The invariance facilitates the investigation of system control problems such as disturbance decoupling, output stabilization, output regulation, and structure stability. Hence, we develop the invariant subspace characterization of the span reachable setℛspan for complex linear time-driven impulsive systems. A follow-up question is that whether the span reachable set ℛspan is an invariant subspace of system (2.4). The invariant subspace of complex impulsive systems (2.4) is defined as follows.Definition 3.8.
For complex impulsive system (2.4) with u(t)≡0, 𝒱 is an invariant subspace if for any initial time t0 and any set of impulse times 𝒯, x(0)∈𝒱 implies x(t)∈𝒱, ∀t≥t0.Generalizing Lemma  4.2 in [7] to the complex case, we conclude that for complex linear impulsive systems, 𝒱 is an invariant subspace if and only if A𝒱⊂𝒱, E𝒱⊂𝒱. Now, for system (2.4), we relate ℛspan to the infimal invariant subspace 〈A,E∣B〉 containing ImB.Theorem 3.9.
For complex linear impulsive system (2.4), one has
(3.12)Rspan⁡=Vn=〈A,E∣B〉.Proof.
First, we prove that𝒱n⊇〈A,E∣B〉. From (3.7), it is obvious that ImB⊆𝒲0⊆𝒱n, A𝒲0=A〈A∣B〉⊆〈A∣B〉=𝒲0, A𝒲m=A〈A∣E𝒲m-1〉⊆〈A∣E𝒲m-1〉=𝒲m, m≥1; E𝒲m⊆〈A∣E𝒲m〉=𝒲m+1⊆𝒱n, m≥0. Thus 𝒱n is an invariant subspace containing ImB. Since 〈A,E∣B〉 is the infimal one, we obtain that 𝒱n⊇〈A,E∣B〉.
Next, we prove that𝒱n⊆〈A,E∣B〉. Since 〈A,E∣B〉 is the infimal invariant subspace containing ImB, we get ImB⊆〈A,E∣B〉, AiImB⊆〈A,E∣B〉, i=1,…,n-1. Then 𝒲0⊆〈A,E∣B〉 and E𝒲0⊆〈A,E∣B〉. By the same reasoning, the fact that AiE𝒲0⊆〈A,E∣B〉 implies that 𝒲1⊂〈A,E∣B〉,  i=0,…,n-1. Similarly, for m>1, 𝒲m⊆〈A,E∣B〉, which means that ∑i=0n𝒲i=𝒱n⊆〈A,E∣B〉. The proof is completed.Example 3.10.
Consider complex linear impulsive system (2.4) with the same coefficient matrices as that in Example 3.3. Now we modify the matrix E as follows
(3.13)E=[0001000011+i000(1+i)00].
Using the construction proposed in (3.7), we have 𝒲0=[0(1+2i)00]T, 𝒲1=〈A∣E𝒲0〉=[000(1+2i)]T, 𝒲2=〈A∣E𝒲1〉=[(1+2i)000]T and 𝒲3=〈A∣E𝒲2〉=[00(1+2i)0]T. It can be easily verified that 𝒱4 spans the entire complex state space ℂ4. Hence, system (2.4) with the above matrices is span reachable. Moreover, form this example, we can find that the explicit construction (3.7) helps us to derive the span reachable set easily.


## 4. Geometric Characterization of Observability

In this section, we present the geometric characterization of the unobservability of complex linear impulsive system (2.4). For convenience, the unobservable set of complex linear systems and its geometric property are introduced first. For a matrix C∈ℂm×n, let 𝒦 be the kernel of C, that is, 𝒦≜KerC={x∈ℂn∣Cx=0}. Given a matrix A∈ℂn×n and a linear space ℳ⊆ℂn, the largest A-invariant subspace contained in ℳ is given by 〈ℳ∣A〉:=ℳ∩A-1ℳ∩A-2ℳ∩⋯∩A-n+1ℳ which is the unobservable subspace for complex system (3.1) when ℳ=𝒦, where A-1ℳ denotes the inverse image of subspace ℳ. Also we have Ker(MA)=A-1Ker(M) [13]. We introduce the following definitions of unobservability.Definition 4.1 (Unobservable set with finite intervals and fixed impulse times).
For complex impulsive system (2.4), a state x0∈𝒳 is said to be unobservable on [t0,tf] with fixed impulse times, if given tf>t0, impulse times set 𝒯 and x0=x(t0), the output y(t) is identically equal to zero for all t∈[t0,tf]. The set of unobservable states with finite interval and fixed impulse times is denoted by 𝒬fixed.Definition 4.2 (Unobservable set with free impulse times).
For complex impulsive system (2.4), a state x0∈𝒳 is said to be unobservable on [t0,tf] with free impulse time, if given t0,x0=x(t0) yields a response y(t) that is identically equal to zero for all t≥t0 and all impulse times sets 𝒯. The set of these unobservable states is denoted by 𝒬free. System (2.4) is observable if 𝒬free={0}.By the above definitions, we have𝒬free=⋂𝒯⋂tf>t0𝒬fixed(tf,𝒯).It is easy to see from Definitions4.1 and 4.2, the observability of complex linear impulsive system (2.4) is equivalent to that of zero-input complex impulsive system. In this way, given an impulse times set 𝒯 and x0∈𝒳, the output y(t) is given by(4.1)y(t)={CeA(t-t0)x0,t∈(t0,t1],CeA(t-tm-1)∏j=m-11EeAhjx0,t∈(tm-1,tm],2≤m≤M.
Denote the following subspace sequences:(4.2)O0=〈K∣A〉,Om=〈E-1Om-1∣A〉,m≥1,Pm=⋂i=0mOi,m=0,1,2,….
Similar to the discussion about 𝒱m, the sequence {𝒫m,m=0,1,…} converges to 𝒫n.Theorem 4.3.
For complex linear impulsive system (2.4), one has  𝒬free=𝒫n.Proof.
For an initial statex0∈𝒫n and a given impulsive times set 𝒯∩(t0,tf)={t0,t1,…,tM}, it is obvious that from x0∈𝒪0=Ker(C)∩A-1Ker(C)∩⋯∩A-(n-1)Ker(C), we have CAkx0=0, k=0,1,…,n-1 which implies that CeA(t-t0)x0=0, t∈(t0,t1]. Since x0∈𝒪1=〈E-1〈𝒦∣A〉∣A〉, the definition of the largest invariant subspace implies that x0∈⋂k=0n-1A-kE-1〈𝒦∣A〉. Then EAkx0∈〈𝒦∣A〉, k=0,1,…,n-1. From the property of matrix exponent, it follows that EeAh1x0∈〈𝒦∣A〉, which means that CeA(t-t1)EeAh1x0=0,t∈(t1,t2]. By the same reasoning, we get CeA(t-tm-1)∏j=m-11(EeAhj)x0=0, t∈(tm-1,tm], 2≤m≤M. It means that the output y(t)≡0, t∈[t0,tf]. From Definition 4.2, we conclude that x0∈𝒬free and 𝒫n⊆𝒬free.
On the other hand, ifx0∈𝒬free, then for any impulse times set 𝒯,
(4.3)0=y(t)={CeA(t-t0)x0,t∈(t0,t1],CeA(t-tm-1)∏j=m-11EeAhjx0,t∈(tm-1,tm],2≤m≤M.
The first equation in (4.3) shows that x0∈〈Ker(C)∣A〉=𝒪0. If m=2, (4.3) becomes CeA(t-t1)EeAh1x0=0, t∈(t1,t2], then it follows from the definition of unobservable subspace that
(4.4)x0∈Ker⁡(CeA(t-t1)EeAh1)=〈Ker⁡(CeA(t-t1)E)∣A〉=〈E-1Ker⁡(CeA(t-t1))∣A〉=〈E-1〈K∣A〉∣A〉.
Repeating the same process, we obtain x0∈𝒪i, i∈{0,1,…,n}. This means that x0∈𝒫n and 𝒬free⊆𝒫n. The proof is completed.From Definition4.2 and Theorem 4.3, we can see that if 𝒫n={0}, system (2.4) is observable. Similarly, we aim to show the invariance of the unobservable set with free impulse times 𝒬free. Denote the supremal invariant subspace of system (2.4) contained in 𝒦 to be 〈𝒦∣A,E〉.Theorem 4.4.
For complex linear impulsive system (2.4), one has
(4.5)Qfree=Pn=〈K∣A,E〉.Proof.
First, we prove that𝒫n⊇〈𝒦∣A,E〉. Given any x0∈〈𝒦∣A,E〉, since 〈𝒦∣A,E〉 is the largest invariant subspace contained in KerC, we have Aix0∈〈𝒦∣A,E〉⊆KerC, i=0,…,n-1, which means that x0∈𝒪0=⋂i=0n-1A-iKerC. Furthermore, AjEAix0∈〈𝒦∣A,E〉⊆KerC, i,j=0,…,n-1, which means that x0∈𝒪1=⋂i=0n-1A-iE-1𝒪0. By the same deduction, x0∈𝒪m indicates that x0∈𝒫n by the definition of 𝒫n, m>1. Then we have 𝒫n⊇〈𝒦∣A,E〉.
Next, we prove that𝒫n⊆〈𝒦∣A,E〉. Given any x0∈𝒫n, x0∈𝒪0=⋂i=0n-1A-iKerC⊆KerC, then 𝒫n⊆KerC. Moreover, since 𝒪m are A-invariant subspaces, we have Ax0∈A𝒪m⊆𝒪m, m=0,1,…,n. It is clear that Ax0∈⋂m=0n𝒪m=𝒫n. On the other hand, the sequence {𝒫m} converges to 𝒫n, which implies that x0∈𝒫n=𝒫n+1 and x0∈𝒪m, m=1,2,…,n+1. Thus Ex0∈E𝒪m=⋂i=0n-1EA-iE-1𝒪m-1⊆EE-1𝒪m-1=𝒪m-1, m=1,2,…,n+1. This shows that Ex0∈𝒪m, m=0,1,…,n. In conclusion, we have Ex0∈𝒫n and Ax0∈𝒫n. It means that 𝒫n is an invariant subspace contained in 𝒦 with respect to matrices A and E. Since 〈𝒦∣A,E〉 is the largest one, we conclude that 𝒫n⊂〈𝒦∣A,E〉. This completes the proof.Remark 4.5.
When the systems in this paper and [10] are reduced to complex linear systems, if 〈Ker(C)∣A〉={0}, from [13], we know that (〈Ker(C)∣A〉)⊥=〈A⊤∣C⊤〉=ℂn, that is, rank(S)=n, which means that system (2.4) is observable, where S=[C⊤A⊤C⊤⋯(An-1)⊤C⊤]⊤.
Thus the geometric condition is equivalent to the algebraic one in Theorem  5(i) in [10] for the observability of system (2.4). When reduced to complex linear impulsive systems with AE=EA, simple computation follows that the algebraic condition in [10] and the geometric one here for the observability are equivalent.Example 4.6.
Consider complex linear impulsive system (2.4) with
(4.6)A=[0010000000000000],E=[0001100101000000],C=[0100].
It is easy to get that 𝒪0=ker(C)=[1+i0000001+i0001+i]. Simple computations from (4.2) yield that 𝒪1=⋂i,j=13ker(CAiEAj)=[001+i00001+i] and 𝒪2=[1+i0001+i0001+i000]. Then we have 𝒪0∩𝒪1∩𝒪2=𝒬free={0}, which means that the system is observable.


## 5. Conclusion

In this paper, the reachability and observability have been investigated for a class of time-driven complex linear impulsive systems which allow free impulsive times. It has been shown that traditional geometric approach may be not sufficient to study the reachability and observability for such systems. Hence, a new geometric analysis method is developed. New concepts of the reachability and observability have been introduced. Sufficient and necessary conditions for the span reachability and observability of such systems have been established. Moreover, geometric properties of span reachable set and unobservable set have been studied. The equivalence between the algebraic conditions in known results [10] and the geometric ones obtained here has been established. Numerical examples have been provided to show the explicit construction of the reachable subspace and unobservable subspace and easily-verifiable conditions for the reachability and observability of complex linear impulsive systems.



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*Source: 876120-2012-02-20.xml*