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Controllability and observability for impulsive systems in complex fields
Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
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Controllability and observability for impulsive systems in complex fieldsI Shouwei Zhao, Jitao Sun ∗ Department of Mathematics, Tongji University, 200092, China
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Article history: Received 19 October 2008 Accepted 10 March 2009 Keywords: Impulsive systems Complex dynamical systems Controllability Observability
abstract Since the classical example of complex system is the quantum system which is one of the foci of ongoing research, in this paper, the issue of controllability and observability for a class of time-varying impulsive systems defined in complex fields, to be brief, complex time-varying impulsive systems, is addressed. Several sufficient and necessary conditions for state controllability and observability of such systems are established. Meanwhile, corresponding criteria for complex linear time-invariant impulsive systems are also obtained. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Hybrid systems have been receiving increasing interest in the control community recently and most progress has been made in the stability and stabilization of hybrid systems [1]. Impulsive dynamical systems which exhibit continuous evolution typically described by Ordinary Differential Equations (ODEs) and instantaneous state jumps or impulses are a special class of hybrid systems. Since many evolution processes, optimal control models in economics, stimulated neural networks, frequency-modulated systems and some motions of missiles or aircrafts are characterized by impulsive dynamical behavior, the study of impulsive systems is of great importance. Nowadays, there has been increasing interest in the analysis and synthesis of impulsive systems, or impulsive control systems, due to their significance both in theory and applications, see [2–8] and the references therein. It is well known that controllability and observability play a central role throughout the history of modern control theory and engineering because they have close connections to pole assignment, structural decomposition, quadratic optimal control and observer design, etc. The controllability concept has been studied extensively in the context of finite-dimensional linear systems, nonlinear systems, infinite-dimensional systems, n-dimensional systems, and hybrid systems using different kinds of approaches (e.g. [4–15]). In particular, most efforts have been focused on the problem of controllability and observability for various kinds of impulsive systems using different approaches. The geometric analysis of reachability, controllability and observability for (switched) impulsive systems in terms of invariant subspaces were presented in [4,5]. By proposing the rank condition, Guan et al. [6] and Zhao and Sun [7] investigated the sufficient and necessary conditions for state controllability and observability of different kinds of linear time-varying impulsive system. The controllability of impulsive functional differential systems with the help of some fixed-point theorems was discussed in [8–10]. However, the common setting adopted in above-mentioned works is always that of a state space which is Rn except a few reports on the issue of controllability for complex systems [14]. Nowadays, research on the control theory of quantum systems has attracted considerable attention [15–18]. And quantum systems are a class of complex dynamical systems which take values in Banach (Hilbert) space in a complex field. More abstract than the real system, the study on control I This work is supported by the NSF of China under Grant 60874027.
∗
Corresponding author. Tel.: +86 21 6598 3241-1307; fax: +86 21 65982341. E-mail address: [email protected] (J. Sun).
1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.03.009
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theory of dynamics take values in complex fields, for simplicity, complex dynamics, has many potential applications in science and engineering. Due to these reasons, it is important and necessary to study the control theory of another class of complex dynamical systems. To the best of our knowledge, there is no result about the control theory of complex impulsive dynamical systems. Inspired by [6] and [7], in this paper, we consider the fundamental concepts of controllability and observability of complex linear time-varying impulsive systems by an algebraic approach. The main difficulty is to investigate the conditions for controllability and observability of complex impulsive systems in the context of complex matrices. Explicit characterization for controllability and observability of this kind of system in terms of the rank conditions is presented. And the corresponding conditions for complex linear impulsive systems are also discussed by an algebraic method. This paper is organized as follows. In Section 2, the complex linear time-varying impulsive systems to be dealt with are formulated and several new results about the variation of parameters for such systems are presented. Several sufficient and necessary conditions for state controllability and state observability of complex linear time-varying impulsive systems and corresponding complex linear time-invariant impulsive systems are established in Sections 3 and 4, respectively. Finally, some conclusions are drawn in Section 5. 2. Preliminaries We consider the complex linear time-varying impulsive systems described by x˙ (t ) = A(t )x(t ) + B(t )u(t ),
∆x = Ek x(tk ) + Fk uk ,
t 6= tk ,
t = tk ,
(1)
y(t ) = C (t )x(t ) + D(t )u(t ), x(t0+ ) = x0
where k = 1, 2, . . ., and A(t ), B(t ), C (t ) and D(t ) are known n × n, n × m, p × n and p × m continuous complex matrices, x ∈ Cn is the state vector, u ∈ Cm is the control input, Ek , Fk are n × n, n × m complex constant matrices, respectively, y ∈ Cp is the output, J = [t0 , ∞), ∆x(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 (1) is left-continuous at tk . We know that x(t ) : R → Cn , and Cn is a Banach space on C. A(t ) : R → U where U = L(Cn , Cn ) is the bounded Cn -linear continuous map. Hence the complex impulsive system (1) is the differential equation in Banach Q1 space defined in complex field C. Let A∗ = A¯ > be the conjugated transpose of the complex matrix A and i=k−1 Ai stands for the matrix product Ak−1 Ak−2 · · · A1 . Corresponding to the complex system (1), consider the complex differential equation x˙ (t ) = A(t )x(t ).
(2)
According to the ordinary differential equation theory, suppose that X (t ) is the fundamental solution matrix of system (2). Then X (t , s) := X (t )X −1 (s) (t , s ∈ J ) are the transition matrices 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 system (1). Lemma 1. For any t ∈ (tk−1 , tk ], k = 1, 2, . . ., the solution of complex system (1) is
(
1 Y
x(t ) = X (t , tk−1 )
(I + Ej )X (tj , tj−1 )x0 +
j=k−1
+
k−1 Y i X
k−1 Y i X
) (I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 +
Z
Z
t
X (t , s)B(s)u(s)ds,
t ∈ [t0 , t1 ],
t0
which leads to x(t1 ) = X (t1 , t0 )x0 +
Z
t1 t0
X (t1 , s)B(s)u(s)ds.
ti
X (ti−1 , s)B(s)u(s)ds
t
X (t , s)B(s)u(s)ds. tk−1
Proof. From the ordinary variation of parameters we have
Z
ti−1
i=1 j=k−1
i=2 j=k−1
x(t ) = X (t , t0 )x0 +
(I + Ej )X (tj , tj−1 )
(3)
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
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Since ∆x(tk ) = Ek x(tk ) + Fk uk ,
x(t1 ) = (I + E1 ) X (t1 , t0 )x0 + +
Z
t1
X (t1 , s)B(s)u(s)ds + F1 u1
t0
= (I + E1 )X (t1 , t0 ) x0 +
Z
t1
X (t0 , s)B(s)u(s)ds + F1 u1 .
t0
Moreover, for t ∈ (t1 , t2 ], x(t ) = X (t , t1 )x(t1+ ) +
t
Z
X (t , s)B(s)u(s)ds t1
Z = X (t , t1 ) (I + E1 )X (t1 , t0 ) x0 +
t1
X (t0 , s)B(s)u(s)ds + F1 u1
Z
t
X (t , s)B(s)u(s)ds.
+ t1
t0
For t = t2 and t = t2+ , we have
x(t2 ) = X (t2 , t1 ) (I + E1 )X (t1 , t0 ) x0 +
t1
Z
X (t0 , s)B(s)u(s)ds + F1 u1
X (t2 , s)B(s)u(s)ds,
t1
t0
x(t2+ ) = (I + E2 )X (t2 , t1 )(I + E1 )X (t1 , t0 ) x0 +
+ (I + E2 )X (t2 , t1 )F1 u1 + (I + E2 )
t2
Z +
Z
t2
Z
t1
X (t0 , s)B(s)u(s)ds
t0
X (t2 , s)B(s)u(s)ds + F2 u2 .
t1
Hence, for any t ∈ (t2 , t3 ], x(t ) = X (t , t2 )x(t2+ ) +
t
Z
X (t , s)B(s)u(s)ds t2 t1
Z = X (t , t2 ) (I + E2 )X (t2 , t1 )(I + E1 )X (t1 , t0 ) x0 +
X (t0 , s)B(s)u(s)ds
t0
+ (I + E2 )X (t2 , t1 )
Z
t2
X (t1 , s)B(s)u(s)ds + (I + E2 )X (t2 , t1 )F1 u1 + F2 u2
Z
t
X (t , s)B(s)u(s)ds.
+
t1
t2
By repeating the same procedure, we can easily deduce the general result (3) for k = 1, 2, . . .. This completes the proof. 3. Controllability Subsequently, we proceed to investigate the controllability criteria of the complex impulsive system (1) using an algebraic method. The definition and some notations are introduced first. Definition 1. The complex time-varying impulsive system (1) is called state controllable on [t0 , tf ] (tf > t0 ), if given any initial state x0 ∈ Cn there exists a piecewise continuous input signal u(t ) : [t0 , tf ] → Cm such that the corresponding solution of (1) satisfies x(tf ) = 0. For tf ∈ (tk−1 , tk ] and X (t , s) being the transition matrix of (2), denote the following k n × n matrices:
Φ0 := I ,
Φi :=
i Y
X (tj−1 , tj )(I + Ej )−1 ,
i = 1, 2, . . . , k − 1,
j=1
Wi := W (ti−1 , ti ) =
ti
Z
X (ti−1 , s)B(s)B∗ (s)X (ti−1 , s)∗ ds,
i = 1, 2, . . . , k − 1,
ti−1
Wk := W (tk−1 , tf ) =
tf
Z
X (tk−1 , s)B(s)B∗ (s)X (tk−1 , s)∗ ds,
(4)
tk−1
W (Φi−1 , ti−1 , ti ) :=
Z
ti
Φi−1 X (ti−1 , s)B(s)B∗ (s)X (ti−1 , s)∗ Φi∗−1 ds,
i = 1, . . . , k − 1,
ti−1
W (Φk−1 , tk−1 , tf ) :=
Z
tf
Φk−1 X (tk−1 , s)B(s)B∗ (s)X (tk−1 , s)∗ Φk∗−1 ds,
tk−1
Vi := Φi Fi ,
i = 1, 2, . . . , k − 1.
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Now we present the sufficient and necessary conditions for controllability of the complex impulsive systems (1). Theorem 1. For complex impulsive systems (1), if one of the following conditions holds, then the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). (i) There exists at least a l ∈ {1, 2, . . . , k − 1} and a m × n complex matrix Fl0 such that Fl Fl0 = I (identity matrix). (ii) There exists at least an invertible matrix Wl (l ∈ {1, 2, . . . , k}) defined in (4) such that rank (Wl ) = n. Proof. First, we consider the case (i). Without loss of generality, suppose that there exists a l ∈ {1, 2, . . . , k − 2} and a m × n complex matrix Fl0 satisfying Fl Fl0 = I. Then given an n × 1 initial state x0 , we design the following control law
u(t ) =
−Fl0 0,
1 Y
(I + Ej )X (tj , tj−1 )x0 ,
t = tl ,
j =l
(5)
t ∈ [t0 , tf ] \ tl ,
which implies that the control is piecewise continuous on [t0 , tf ]. By Lemma 1 and (5), the corresponding solution of (1) yields
( x(tf ) = X (tf , tk−1 )
1 Y
(I + Ej )X (tj , tj−1 )x0 +
j=k−1
" = X (tf , tk−1 )
1 Y
l+1 Y
(I + Ej )X (tj , tj−1 )Fl (−Fl ) 0
(I + Ej )X (tj , tj−1 ) −
j=k−1
) (I + Ej )X (tj , tj−1 )x0
j=l
j=k−1 1 Y
1 Y
# (I + Ej )X (tj , tj−1 ) x0 = 0.
j=k−1
Hence by the definition of controllability, the complex impulsive system (1) is controllable on [t0 , tf ]. Next, we consider case (ii). Without loss of generality, suppose that there exists a l ∈ {1, 2, . . . , k − 2} such that the complex matrix W (tl−1 , tl ) is invertible. For an initial state x0 , choose u(t ) =
−B(t )∗ X (tl−1 , t )∗ Wl−1 0,
1 Y
(I + Ej )X (tj , tj−1 )x0 ,
t ∈ (tl−1 , tl ),
(6)
j=l−1
t ∈ [t0 , tf ] \ (tl−1 , tl ).
Thus applying (6) into (3) yields that
" x(tf ) = X (tf , tk−1 )
1 Y
(I + Ej )X (tj , tj−1 )x0 −
j=k−1
Z
tl
×
l Y
(I + Ej )X (tj , tj−1 )
j=k−1
X (tl−1 , s)B(s)B(s) X (tl−1 , s) dsW ∗
∗
−1
(tl−1 , tl )
tl−1
# (I + Ej )X (tj , tj−1 )x0
j =l −1
" = X (tf , tk−1 )
1 Y
1 Y
(I + Ej )X (tj , tj−1 ) −
j=k−1
1 Y
# (I + Ej )X (tj , tj−1 ) x0 = 0.
j=k−1
It follows that the complex impulsive system (1) is controllable on [t0 , tf ]. This completes the proof.
Theorem 2. Assume that (I + Ej ), j = 1, 2, . . . , k − 1 are invertible. If the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]), then rank{W (Φ0 , t0 , t1 ), . . . , W (Φk−2 , tk−2 , tk−1 ), W (Φk−1 , tk−1 , tf ), V1 , . . . , Vk−1 } = n,
(7)
where W (·, ·, ·), Φi and Vi are defined in (4). Proof. Suppose that the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]) while rank{W (Φ0 , t0 , t1 ), . . . , W (Φk−2 , tk−2 , tk−1 ), W (Φk−1 , tk−1 , tf ), V1 , . . . , Vk−1 } < n. Then there exists a nonzero n × 1 complex vector x1 such that 0 = x∗1 Vi = x∗1 Φi Fi ,
i = 1, 2, . . . , k − 1,
0 = x1 W (Φi−1 , ti−1 , ti )x1 = x1 Φi−1 ∗
∗
Z
ti
X (ti−1 , s)B(s)B∗ (s)X ∗ (ti−1 , s)dsΦi∗−1 x1 ,
ti−1
0 = x∗1 W (Φk−1 , tk−1 , tf )x1 = x∗1 Φk−1
Z
tf tk−1
X (tk−1 , s)B(s)B∗ (s)X ∗ (tk−1 , s)dsΦk∗−1 x1 .
(8)
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
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The integrands of (8) are nonnegative continuous functions kx∗1 Φi−1 X (ti−1 , s)B(s)k2 , i = 1, 2, . . . , k − 1 and it follows that
x∗1 Φi−1 X (ti−1 , t )B(t ) = 0, x∗1 Φk−1 X (tk−1 , t )B(t ) = 0,
t ∈ (ti−1 , ti ), i = 1, 2, . . . , k − 1, t ∈ (tk−1 , tf ].
(9)
Since the complex impulsive system (1) is controllable on [t0 , tf ], then there exists a piecewise control input u(t ) such that for initial state x0 = x1 ,
"
1 Y
x(tf ) = X (tf , tk−1 )
(I + Ej )X (tj , tj−1 )x1 +
j=k−1
+
k−1 Y i X
(I + Ej )X (tj , tj−1 )
ti
X (ti−1 , s)B(s)u(s)ds
ti−1
i=1 j=k−1
#
k −1 Y i X
Z
(I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 +
Z
tf
X (tf , s)B(s)u(s)ds = 0
tk−1
i=2 j=k−1
which implies that
"
1 Y
−X (tf , tk−1 )
(I + Ej )X (tj , tj−1 )x1 = X (tf , tk−1 )
k−1 Y i X
j=k−1
+
k−1 Y i X
# (I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 +
Z
k−1 X
tf
ti
X (ti−1 , s)B(s)u(s)ds
ti−1
X (tf , s)B(s)u(s)ds.
tk−1
[X (tj−1 , tj )(I + Ej )−1 ]X (tk−1 , tf ) from left yields that # k−1 X Φi−1 Fi−1 ui−1 + Φk−1 Fk−1 uk−1 X (ti−1 , s)B(s)u(s)ds +
Multiplying both sides of the above equation by
−x 1 =
Z
i=1 j=k−1
i=2 j=k−1
"
(I + Ej )X (tj , tj−1 )
Φi−1
ti
Z
ti−1
i=1
+ Φk−1
tf
Z
Qk−1 j =1
i=2
X (tk−1 , s)B(s)u(s)ds.
tk−1
Moreover, by multiplying x∗1 to both sides of the above equality from left, from (8) and (9) we have ∗
−x 1 x 1 =
" k−1 X
x1 Φi−1 ∗
ti
Z
X (ti−1 , s)B(s)u(s)ds +
ti−1
i =1
k−1 X
# x1 Φi Fi ui
+ x∗1 Φk−1
∗
Z
tf
X (tk−1 , s)B(s)u(s)ds = 0.
tk−1
i=1
This contradicts with the assumption that x1 6= 0 and we conclude that (7) holds. This completes the proof.
Next we apply Theorems 1 and 2 to the case of complex linear time-invariant impulsive systems. First a claim is presented [11]: for a complex matrix A, there exist scalar functions β0 (t ), β1 (t ), . . . , βn−1 (t ) such that eAt =
n−1 X
βj (t )Aj .
(10)
j =0
Theorem 3. For complex impulsive system (1), assume that A(t ) = A, B(t ) = B are n × n, n × m complex constant matrices, then the following sufficient and necessary conditions hold. (i) If rank(B, AB, . . . , An−1 B) = n
(11)
then the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). (ii) Suppose that AEi = Ei A, Fi = F (complex constant matrix) and (I + Ei ) are invertible. If the complex system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]), then rank(E˜ 1 B˜ , E˜ 2 B˜ , . . . , E˜ k−1 B˜ , B˜ , E˜ 2 F˜ , . . . , E˜ k−1 F˜ , F˜ ) = n
(12)
where B˜ = (B, AB, . . . , An−1 B), F˜ = (F , AF , . . . , An−1 F ), E˜ i =
Qi
j=k−1
(I + Ej ), i = 1, 2, . . . , k − 1.
Proof. We first prove (i) by contradiction. If (11) holds whereas the system is not controllable, then from (ii) in Theorem 1 it follows that Wi and W (tk−1 , tf ) i = 1, 2, . . . , k − 1, are not invertible. Thus there exists a nonzero complex vector xα which satisfies 0 = x∗α W (t0 , t1 )xα =
Z
t1 t0
x∗α eA(t0 −s) BB∗ [eA(t0 −s) ]∗ xα ds =
Z
t1 t0
kx∗α eA(t0 −s) Bk2 ds
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S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
which leads to x∗α eA(t0 −t ) B = 0, t ∈ [t0 , t1 ]. When t = t0 , differentiating this equality j times and evaluating the results at t = t0 give (−1)j x∗α Aj B = 0, j = 0, 1, . . . , n − 1. Therefore, we have x∗α (B, AB, . . . , An−1 B) = 0 which means that rank condition (11) does not hold. This leads to a contradiction. Therefore, if (i) holds, the complex impulsive system (1) with A(t ) = A, B(t ) = B is controllable on [t0 , tf ]. Next we prove (ii) also by contradiction. If the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]) while (12) does not hold, that is, rank(E˜ 1 B˜ , E˜ 2 B˜ , . . . , E˜ k−1 B˜ , B˜ , E˜ 2 F˜ , . . . , E˜ k−1 F˜ , F˜ ) < n. By multiplying E˜ 1−1 from the left, the above equation is equivalent to
( rank B˜ , (I + E1 )−1 B˜ , . . . ,
k−1 Y
(I + Ej ) B, (I + E1 ) F , . . . , −1 ˜
−1 ˜
j=1
k−1 Y
) (I + Ej ) F
−1 ˜
j =1
which implies that there exists a nonzero complex vector xα such that x∗α B˜ = 0,
x∗α
i Y
(I + Ej )−1 B˜ = 0,
x∗α
i Y
j =1
(I + Ej )−1 F˜ = 0,
i = 1, 2, . . . , k − 1.
j=1
It follows from the definitions of B˜ and F˜ that for l = 0, 1, . . . , n − 1, x∗α Al B = 0,
x∗α
i Y
(I + Ej )−1 Al B = 0,
x∗α
i Y
j=1
(I + Ej )−1 Al F = 0,
i = 1, 2, . . . , k − 1.
(13)
j =1
Since AEi = Ei A and (I + Ei ) are invertible, it is easy to get (I + Ei )−1 A(I + Ei ) = (I + Ei )−1 (I + Ei )A = A which means that (I + Ei )−1 A = A(I + Ei )−1 . Hence we have from (10) and (13) x∗α W (Φ0 , t0 , t1 ) =
Z
t1
x∗α eA(t0 −s) BB∗ [eA(t0 −s) ]∗ ds
t0 t1 n−1
Z
X
= t0
βl (t0 − s)x∗α Al BB∗ [eA(t0 −s) ]∗ ds = 0
l=0 ti
Z
x∗α W (Φi−1 , ti−1 , ti ) =
ti−1
n−1 X
ti
Z
x∗α Φi−1 eA(ti−1 −s) BB∗ [eA(ti−1 −s) ]∗ Φi∗−1 ds
=
ti−1 l=0
x∗α W (Φk−1 , tk−1 , tf ) =
Z
tk−1
Z =
tf
tf
βl (t0 − s)x∗α
x∗α Vi = x∗α Φi F = x∗α
j =1
(I + Ej )−1 Al BB∗ [eA(ti−1 −s) ]∗ Φi∗−1 ds = 0,
j=1
x∗α Φk−1 eA(tk−1 −s) BB∗ [eA(tk−1 −s) ]∗ Φk∗−1 ds n−1 X
tk−1 l=0 i Y
i −1 Y
βl (t0 − s)x∗α
(I + Ej )−1 eA(t0 −ti ) F =
k−1 Y
(I + Ej )−1 Al BB∗ [eA(tk−1 −s) ]∗ Φk∗−1 ds = 0,
j =1 n −1 X l =0
βl (t0 − ti )x∗α
i Y
(I + Ej )−1 Al F = 0,
i = 1, . . . , k − 1.
j =1
Therefore, rank{W (Φ0 , t0 , t1 ), W (Φ1 , t1 , t2 ), . . . , W (Φk−1 , tk−1 , tf ), V1 , . . . , Vk−1 } < n which contradicts with Theorem 2. So we know that (12) holds. This completes the proof.
4. Observability In this section, our objective is to explicitly characterize the observability criteria of complex impulsive system (1) and the reduced complex time-invariant impulsive systems. Definition 2. The system (1) is said to be observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]), if any initial state x0 ∈ Cn can be uniquely determined by the corresponding system input u(t ) and output y(t ), for t ∈ [t0 , tf ].
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
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From (1) and Lemma 1, we can get the output
(
1 Y
y(t ) = C (t )x(t ) + D(t )u(t ) = C (t )X (t , tk−1 )
(I + Ej )X (tj , tj−1 )x0
j=k−1
+
k −1 Y i X
(I + Ej )X (tj , tj−1 )
+
X (ti−1 , s)B(s)u(s)ds
ti−1
i=1 j=k−1 k−1 Y i X
ti
Z
) (I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 + C (t )
Z
t
X (t , s)B(s)u(s)ds + D(t )u(t ).
tk−1
i=2 j=k−1
It is easy to see from the Definition 2 that the observability of system (1) is equivalent to the observability of zero-input response y(t ) given by
y(t ) =
C (t )X (t , t )x , 0 0 1 Y C (t )X (t , ti−1 ) (I + Ej )X (tj , tj−1 )x0 ,
t ∈ (t0 , t1 ], t ∈ (ti−1 , ti ], i = 2, . . . , k − 1, (14)
j=i−1 1
Y (I + Ej )X (tj , tj−1 )x0 , C (t )X (t , tk−1 )
t ∈ (tk−1 , tf ].
j=k−1
For subsequent discussion, we denote the n × n matrix M (t0 , tf ) as follows: M (t0 , tf ) =
k−1 X
M (ti−1 , ti ) + M (tk−1 , tf ), M (t0 , t1 ) =
Z
X ∗ (s, t0 )C (s)∗ C (s)X (s, t0 )ds,
t0
i=1
"
ti
Z
M (ti−1 , ti ) =
t1
ti−1
(I + Ej )X (tj , tj−1 )
X ∗ (s, ti−1 )C (s)∗ C (s)X (s, ti−1 )
j=i−1
1 Y
×
#∗
1 Y
(15)
(I + Ej )X (tj , tj−1 )ds,
i = 1, 2, . . . , k − 1,
j=i−1
M (tk−1 , tf ) =
tf
Z
tk−1
"
1 Y
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, tk−1 )C (s)∗ C (s)X (s, tk−1 )
j=k−1
1 Y
(I + Ej )X (tj , tj−1 )ds.
j=k−1
Now we present the sufficient and necessary condition for the observability of complex impulsive systems (1). Theorem 4. The complex impulsive system (1) is observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]) if and only if n × n complex matrix M (t0 , tf ) defined in (15) is invertible. Proof. Multiplying both sides of (14) respectively, by X ∗ (t , t0 )C (t )∗ and [ the left and integrating with respect to t from t0 to tf , we have
Z
t1
X ∗ (s, t0 )C (s)∗ y(s)ds +
t0
k−1 Z X
ti−1
i =2
"
tf
Z +
tk−1
Z
t1
=
1 Y
"
ti
1 Y
j=i−1
(I + Ej )X (tj , tj−1 )]∗ X ∗ (t , ti−1 )C (t )∗ from
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, ti−1 )C (s)∗ y(s)ds
j=i−1
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, tk−1 )C (s)∗ y(s)ds
j=k−1
X ∗ (s, t0 )C (s)∗ C (s)X (s, t0 )ds +
t0
k−1 Z X i =2
× C (s)X (s, ti−1 )
1 Y
"
ti ti−1
(I + Ej )X (tj , tj−1 )ds +
Z
× C (s) C (s)X (s, tk−1 )
tf
"
(I + Ej )X (tj , tj−1 )ds =
j=k−1
(I + Ej )X (tj , tj−1 )
X ∗ (s, ti−1 )C (s)∗
j=i−1
1
Y
#∗
1 Y
tk−1
j=i−1
∗
Q1
1 Y
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, tk−1 )
j=k−1
" k−1 X
# M (ti−1 , ti ) + M (tk−1 , tf ) x0 .
(16)
i=1
It is easy to see that the left-hand side of (16) depends on y(t ), t ∈ [t0 , tf ]. So if M (t0 , tf ) is invertible then the initial state x(t0 ) = x0 is uniquely determined by the corresponding complex system output y(t ), for t ∈ [t0 , tf ].
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S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
Next we consider the necessary part. If the complex matrix M (t0 , tf ) is not invertible, then there exists a nonzero vector xα such that x∗α M (t0 , tf )xα = 0. Since M (ti−1 , ti )(i = 1, . . . , k − 1) and M (tk−1 , tf ) are positive semidefinite matrices, we have x∗α M (ti−1 , ti )xα = 0,
i = 1, 2, . . . , k − 1,
x∗α M (tk−1 , tf )xα = 0.
(17)
If let initial state x(t0 ) = xα , it follows from (14) and (15) that
Z
tf
y(s)∗ y(s)ds =
k−1 Z X
t0
i =1 t1
Z =
t0
ti
y(s)∗ y(s)ds +
tf
Z
y(s)∗ y(s)ds
tk−1
ti−1
x∗α X ∗ (s, t0 )C ∗ (s)C (s)X (s, t0 )xα +
k−1 Z X i=2
× C (s) C (s)X (s, ti−1 ) ∗
1 Y
ti ti−1
" x∗α
(I + Ej )X (tj , tj−1 )xα ds +
#∗ (I + Ej )X (tj , tj−1 )
1 Y
X ∗ (s, ti−1 )
j=i−1
Z
tf
tk−1
j=i−1
× X ∗ (s, tk−1 )C (s)∗ C (s)X (s, tk−1 )
1 Y
" x∗α
1 Y
#∗ (I + Ej )X (tj , tj−1 )
j=k−1
(I + Ej )X (tj , tj−1 )xα ds
j=k−1
=
x∗α
" k−1 X
# M (ti−1 , ti ) + M (tk−1 , tf ) x∗α = 0.
i=1
which implies that
0 = y(t ) =
R tf t0
ky(s)k2 ds = 0. Thus by (14),
C (t )X (t , t )x , 0 0 1 Y C (t )X (t , ti−1 ) (I + Ej )X (tj , tj−1 )x0 ,
t ∈ (t0 , t1 ], t ∈ (ti−1 , ti ], i = 2, . . . , k − 1,
j=i−1 1
Y (I + Ej )X (tj , tj−1 )x0 , C (t )X (t , tk−1 )
t ∈ (tk−1 , tf ].
j=k−1
From Definition 2, the system (1) is not observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). This contradicts with the assumption of observability. This completes the proof. For the impulsive system (1), when A(t ) = A, C (t ) = C are complex constant matrices, the complex impulsive system becomes a complex linear time-invariant impulsive system. We have a more concise result than Theorem 4. Denote
C CA S := ...
,
S˜ :=
CAn−1
where Eˆ i =
Q1
j =i
S S Eˆ1
.. .
S Eˆ k−1 ,
(18)
(I + Ej ), i = 1, 2, . . . , k − 1.
Theorem 5. If the complex impulsive system (1) has complex constant coefficient matrices A and C , then the following conclusions hold. (i) If rank (S ) = n, then the complex linear impulsive system (1) is observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). (ii) Assume that AEi = Ei A, i = 1, 2, . . . , k − 1. If the complex linear system (1) is observable, then rank (S˜ ) = n. Proof. Sufficiency: If rank(S ) = n while the complex system (1) is not observable, then by Theorem 4 the matrix M (t0 , tf ) is not invertible which implies that there exists a nonzero vector xα satisfying x∗α M (t0 , tf )xα = 0. Since the matrices M (ti−1 , ti ) are non-negative definite, we obtain x∗α M (t0 , t1 )xα
Z
t1
=
[C eA(s−t0 ) xα ]∗ [C eA(s−t0 ) xα ]ds = 0.
t0
This shows that C eA(t −t0 ) xα = 0,
t ∈ (t0 , t1 ].
(19)
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
1521
Clearly, when t = t0 we have Cxα = 0. Differentiating (19) j times and evaluating the results at t = t0 yields that CAj xα = 0,
j = 0, 1, . . . n − 1.
(20)
Hence we deduce that Sxα = 0 for xα 6= 0. It follows that rank(S ) < n which leads to a contradiction with the assumption that rank(S ) = n. The proof of the sufficiency part is completed. Necessity: If otherwise, assume that the complex impulsive system (1) is observable while rank(S˜ ) < n, then there exists ˜ α = 0 which reduces from (18) to a vector xα 6= 0 satisfying Sx CAl xα = 0,
CAl
1 Y (I + Ej )xα = 0,
l = 0, 1, . . . , n − 1, i = 1, . . . , k − 1.
(21)
j =i
From (21), (10) and the fact that AEi = Ei A, we obtain M (t0 , t1 )xα =
t1
Z
[eA(s−t0 ) ]∗ C ∗
t0
M (ti−1 , ti )xα =
n −1 X
βl (s − t0 )CAl xα ds = 0,
l =0
"
ti
Z
ti−1
×
1 Y
#∗ (I + Ej )e
A(tj−1 −tj )
[eA(s−ti−1 ) ]∗ C ∗
n −1 X
j=i−1
1 Y
βl (s − t0 )CAl
l =0
(I + Ej )xα ds = 0,
i = 2, . . . , k − 1,
j=i−1
M (tk−1 , tf )xα =
Z
tf tk−1
"
1 Y
j=k−1
#∗ (I + Ej )e
A(tj−1 −tj )
[eA(s−tk−1 ) ]∗ C ∗
n−1 X l=0
βl (s − t0 )CAl
1 Y
(I + Ej )xα ds = 0.
j=k−1
So M (t0 , tf )xα = 0. Because xα 6= 0 the matrix M (t0 , tf ) is not invertible. Hence the complex linear impulsive system (1) is not observable from Theorem 4, and it contradicts with the assumption of observability. This completes the proof. 5. Conclusion In this paper, the issue on the controllability and observability criteria for a class of complex linear time-varying impulsive systems has been addressed for the first time. Several sufficient and necessary conditions for state controllability and observability of such systems have been established respectively. Moreover, the corresponding criteria for controllability and observability of complex linear impulsive systems are also derived. References [1] H. Lin, P.J. Antsaklis, Switching stabilizability for continuous-time uncertain switched linear systems, IEEE Trans. Automat. Control 52 (4) (2007) 633–646. [2] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [3] Y. Zhang, J.T. Sun, Stability of impulsive functional differential equations, Nonlinear Anal. TMA 68 (12) (2008) 3665–3678. [4] E.A. Medina, D.A. Lawrence, Reachability and observability of linear impulsive systems, Automatica 44 (2008) 1304–1309. [5] G.M. Xie, L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Trans. Automat. Control 49 (6) (2004) 960–966. [6] Z.H. Guan, T.H. Qian, X.H. Yu, Controllability and observability of linear time-varying impulsive systems, IEEE Trans. Circuits Syst.-I 49 (8) (2002) 1198–1208. [7] S.W. Zhao, J.T. Sun, Controllability and observability for a class of time-varying impulsive systems, Nonlinear Anal. RWA 10 (3) (2008) 1370–1380. [8] N. Abada, M. Benchohra, H. Hammouche, Existence and controllability results for impulsive partial functional differential inclusions, Nonlinear Anal. 69 (2008) 2892–2909. [9] S. Sivasundaram, J. Uvahb, Controllability of impulsive hybrid integro-differential systems, Nonlinear Anal. Hybrid Syst. 2 (4) (2008) 1003–1009. [10] B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal. 60 (2005) 1533–1552. [11] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, second edition, Springer-Verlag, New York, 1979. [12] G.M. Xie, L. Wang, Controllability and observability of a class of linear impulsive systems, J. Math. Anal. Appl. 304 (2005) 336–355. [13] Z.J. Ji, L. Wang, X.X. Guo, On controllability of switched linear systems, IEEE Trans. Automat. Control 53 (3) (2008) 796–801. [14] H.J. Sussmann, V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1) (1972) 95–116. [15] R.B. Wu, T.J. Tarn, C.W. Li, Smooth controllability of infinite-dimensional quantum-mechanical systems, Phys. Rev. A 73 (2006) 012719. [16] M. Mirrahimi, R. Van Handel, Stabilizing feedback controls for quantum systems, SIAM J. Control Optim. 46 (2) (2007) 445–467. [17] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [18] S. Kuang, S. Cong, Lyapunov control methods of closed quantum systems, Automatica 44 (1) (2008) 98–108.
Contents lists available at ScienceDirect
Nonlinear Analysis: Real World Applications journal homepage: www.elsevier.com/locate/nonrwa
Controllability and observability for impulsive systems in complex fieldsI Shouwei Zhao, Jitao Sun ∗ Department of Mathematics, Tongji University, 200092, China
article
info
Article history: Received 19 October 2008 Accepted 10 March 2009 Keywords: Impulsive systems Complex dynamical systems Controllability Observability
abstract Since the classical example of complex system is the quantum system which is one of the foci of ongoing research, in this paper, the issue of controllability and observability for a class of time-varying impulsive systems defined in complex fields, to be brief, complex time-varying impulsive systems, is addressed. Several sufficient and necessary conditions for state controllability and observability of such systems are established. Meanwhile, corresponding criteria for complex linear time-invariant impulsive systems are also obtained. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction Hybrid systems have been receiving increasing interest in the control community recently and most progress has been made in the stability and stabilization of hybrid systems [1]. Impulsive dynamical systems which exhibit continuous evolution typically described by Ordinary Differential Equations (ODEs) and instantaneous state jumps or impulses are a special class of hybrid systems. Since many evolution processes, optimal control models in economics, stimulated neural networks, frequency-modulated systems and some motions of missiles or aircrafts are characterized by impulsive dynamical behavior, the study of impulsive systems is of great importance. Nowadays, there has been increasing interest in the analysis and synthesis of impulsive systems, or impulsive control systems, due to their significance both in theory and applications, see [2–8] and the references therein. It is well known that controllability and observability play a central role throughout the history of modern control theory and engineering because they have close connections to pole assignment, structural decomposition, quadratic optimal control and observer design, etc. The controllability concept has been studied extensively in the context of finite-dimensional linear systems, nonlinear systems, infinite-dimensional systems, n-dimensional systems, and hybrid systems using different kinds of approaches (e.g. [4–15]). In particular, most efforts have been focused on the problem of controllability and observability for various kinds of impulsive systems using different approaches. The geometric analysis of reachability, controllability and observability for (switched) impulsive systems in terms of invariant subspaces were presented in [4,5]. By proposing the rank condition, Guan et al. [6] and Zhao and Sun [7] investigated the sufficient and necessary conditions for state controllability and observability of different kinds of linear time-varying impulsive system. The controllability of impulsive functional differential systems with the help of some fixed-point theorems was discussed in [8–10]. However, the common setting adopted in above-mentioned works is always that of a state space which is Rn except a few reports on the issue of controllability for complex systems [14]. Nowadays, research on the control theory of quantum systems has attracted considerable attention [15–18]. And quantum systems are a class of complex dynamical systems which take values in Banach (Hilbert) space in a complex field. More abstract than the real system, the study on control I This work is supported by the NSF of China under Grant 60874027.
∗
Corresponding author. Tel.: +86 21 6598 3241-1307; fax: +86 21 65982341. E-mail address: [email protected] (J. Sun).
1468-1218/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2009.03.009
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S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
theory of dynamics take values in complex fields, for simplicity, complex dynamics, has many potential applications in science and engineering. Due to these reasons, it is important and necessary to study the control theory of another class of complex dynamical systems. To the best of our knowledge, there is no result about the control theory of complex impulsive dynamical systems. Inspired by [6] and [7], in this paper, we consider the fundamental concepts of controllability and observability of complex linear time-varying impulsive systems by an algebraic approach. The main difficulty is to investigate the conditions for controllability and observability of complex impulsive systems in the context of complex matrices. Explicit characterization for controllability and observability of this kind of system in terms of the rank conditions is presented. And the corresponding conditions for complex linear impulsive systems are also discussed by an algebraic method. This paper is organized as follows. In Section 2, the complex linear time-varying impulsive systems to be dealt with are formulated and several new results about the variation of parameters for such systems are presented. Several sufficient and necessary conditions for state controllability and state observability of complex linear time-varying impulsive systems and corresponding complex linear time-invariant impulsive systems are established in Sections 3 and 4, respectively. Finally, some conclusions are drawn in Section 5. 2. Preliminaries We consider the complex linear time-varying impulsive systems described by x˙ (t ) = A(t )x(t ) + B(t )u(t ),
∆x = Ek x(tk ) + Fk uk ,
t 6= tk ,
t = tk ,
(1)
y(t ) = C (t )x(t ) + D(t )u(t ), x(t0+ ) = x0
where k = 1, 2, . . ., and A(t ), B(t ), C (t ) and D(t ) are known n × n, n × m, p × n and p × m continuous complex matrices, x ∈ Cn is the state vector, u ∈ Cm is the control input, Ek , Fk are n × n, n × m complex constant matrices, respectively, y ∈ Cp is the output, J = [t0 , ∞), ∆x(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 (1) is left-continuous at tk . We know that x(t ) : R → Cn , and Cn is a Banach space on C. A(t ) : R → U where U = L(Cn , Cn ) is the bounded Cn -linear continuous map. Hence the complex impulsive system (1) is the differential equation in Banach Q1 space defined in complex field C. Let A∗ = A¯ > be the conjugated transpose of the complex matrix A and i=k−1 Ai stands for the matrix product Ak−1 Ak−2 · · · A1 . Corresponding to the complex system (1), consider the complex differential equation x˙ (t ) = A(t )x(t ).
(2)
According to the ordinary differential equation theory, suppose that X (t ) is the fundamental solution matrix of system (2). Then X (t , s) := X (t )X −1 (s) (t , s ∈ J ) are the transition matrices 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 system (1). Lemma 1. For any t ∈ (tk−1 , tk ], k = 1, 2, . . ., the solution of complex system (1) is
(
1 Y
x(t ) = X (t , tk−1 )
(I + Ej )X (tj , tj−1 )x0 +
j=k−1
+
k−1 Y i X
k−1 Y i X
) (I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 +
Z
Z
t
X (t , s)B(s)u(s)ds,
t ∈ [t0 , t1 ],
t0
which leads to x(t1 ) = X (t1 , t0 )x0 +
Z
t1 t0
X (t1 , s)B(s)u(s)ds.
ti
X (ti−1 , s)B(s)u(s)ds
t
X (t , s)B(s)u(s)ds. tk−1
Proof. From the ordinary variation of parameters we have
Z
ti−1
i=1 j=k−1
i=2 j=k−1
x(t ) = X (t , t0 )x0 +
(I + Ej )X (tj , tj−1 )
(3)
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
1515
Since ∆x(tk ) = Ek x(tk ) + Fk uk ,
x(t1 ) = (I + E1 ) X (t1 , t0 )x0 + +
Z
t1
X (t1 , s)B(s)u(s)ds + F1 u1
t0
= (I + E1 )X (t1 , t0 ) x0 +
Z
t1
X (t0 , s)B(s)u(s)ds + F1 u1 .
t0
Moreover, for t ∈ (t1 , t2 ], x(t ) = X (t , t1 )x(t1+ ) +
t
Z
X (t , s)B(s)u(s)ds t1
Z = X (t , t1 ) (I + E1 )X (t1 , t0 ) x0 +
t1
X (t0 , s)B(s)u(s)ds + F1 u1
Z
t
X (t , s)B(s)u(s)ds.
+ t1
t0
For t = t2 and t = t2+ , we have
x(t2 ) = X (t2 , t1 ) (I + E1 )X (t1 , t0 ) x0 +
t1
Z
X (t0 , s)B(s)u(s)ds + F1 u1
X (t2 , s)B(s)u(s)ds,
t1
t0
x(t2+ ) = (I + E2 )X (t2 , t1 )(I + E1 )X (t1 , t0 ) x0 +
+ (I + E2 )X (t2 , t1 )F1 u1 + (I + E2 )
t2
Z +
Z
t2
Z
t1
X (t0 , s)B(s)u(s)ds
t0
X (t2 , s)B(s)u(s)ds + F2 u2 .
t1
Hence, for any t ∈ (t2 , t3 ], x(t ) = X (t , t2 )x(t2+ ) +
t
Z
X (t , s)B(s)u(s)ds t2 t1
Z = X (t , t2 ) (I + E2 )X (t2 , t1 )(I + E1 )X (t1 , t0 ) x0 +
X (t0 , s)B(s)u(s)ds
t0
+ (I + E2 )X (t2 , t1 )
Z
t2
X (t1 , s)B(s)u(s)ds + (I + E2 )X (t2 , t1 )F1 u1 + F2 u2
Z
t
X (t , s)B(s)u(s)ds.
+
t1
t2
By repeating the same procedure, we can easily deduce the general result (3) for k = 1, 2, . . .. This completes the proof. 3. Controllability Subsequently, we proceed to investigate the controllability criteria of the complex impulsive system (1) using an algebraic method. The definition and some notations are introduced first. Definition 1. The complex time-varying impulsive system (1) is called state controllable on [t0 , tf ] (tf > t0 ), if given any initial state x0 ∈ Cn there exists a piecewise continuous input signal u(t ) : [t0 , tf ] → Cm such that the corresponding solution of (1) satisfies x(tf ) = 0. For tf ∈ (tk−1 , tk ] and X (t , s) being the transition matrix of (2), denote the following k n × n matrices:
Φ0 := I ,
Φi :=
i Y
X (tj−1 , tj )(I + Ej )−1 ,
i = 1, 2, . . . , k − 1,
j=1
Wi := W (ti−1 , ti ) =
ti
Z
X (ti−1 , s)B(s)B∗ (s)X (ti−1 , s)∗ ds,
i = 1, 2, . . . , k − 1,
ti−1
Wk := W (tk−1 , tf ) =
tf
Z
X (tk−1 , s)B(s)B∗ (s)X (tk−1 , s)∗ ds,
(4)
tk−1
W (Φi−1 , ti−1 , ti ) :=
Z
ti
Φi−1 X (ti−1 , s)B(s)B∗ (s)X (ti−1 , s)∗ Φi∗−1 ds,
i = 1, . . . , k − 1,
ti−1
W (Φk−1 , tk−1 , tf ) :=
Z
tf
Φk−1 X (tk−1 , s)B(s)B∗ (s)X (tk−1 , s)∗ Φk∗−1 ds,
tk−1
Vi := Φi Fi ,
i = 1, 2, . . . , k − 1.
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S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
Now we present the sufficient and necessary conditions for controllability of the complex impulsive systems (1). Theorem 1. For complex impulsive systems (1), if one of the following conditions holds, then the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). (i) There exists at least a l ∈ {1, 2, . . . , k − 1} and a m × n complex matrix Fl0 such that Fl Fl0 = I (identity matrix). (ii) There exists at least an invertible matrix Wl (l ∈ {1, 2, . . . , k}) defined in (4) such that rank (Wl ) = n. Proof. First, we consider the case (i). Without loss of generality, suppose that there exists a l ∈ {1, 2, . . . , k − 2} and a m × n complex matrix Fl0 satisfying Fl Fl0 = I. Then given an n × 1 initial state x0 , we design the following control law
u(t ) =
−Fl0 0,
1 Y
(I + Ej )X (tj , tj−1 )x0 ,
t = tl ,
j =l
(5)
t ∈ [t0 , tf ] \ tl ,
which implies that the control is piecewise continuous on [t0 , tf ]. By Lemma 1 and (5), the corresponding solution of (1) yields
( x(tf ) = X (tf , tk−1 )
1 Y
(I + Ej )X (tj , tj−1 )x0 +
j=k−1
" = X (tf , tk−1 )
1 Y
l+1 Y
(I + Ej )X (tj , tj−1 )Fl (−Fl ) 0
(I + Ej )X (tj , tj−1 ) −
j=k−1
) (I + Ej )X (tj , tj−1 )x0
j=l
j=k−1 1 Y
1 Y
# (I + Ej )X (tj , tj−1 ) x0 = 0.
j=k−1
Hence by the definition of controllability, the complex impulsive system (1) is controllable on [t0 , tf ]. Next, we consider case (ii). Without loss of generality, suppose that there exists a l ∈ {1, 2, . . . , k − 2} such that the complex matrix W (tl−1 , tl ) is invertible. For an initial state x0 , choose u(t ) =
−B(t )∗ X (tl−1 , t )∗ Wl−1 0,
1 Y
(I + Ej )X (tj , tj−1 )x0 ,
t ∈ (tl−1 , tl ),
(6)
j=l−1
t ∈ [t0 , tf ] \ (tl−1 , tl ).
Thus applying (6) into (3) yields that
" x(tf ) = X (tf , tk−1 )
1 Y
(I + Ej )X (tj , tj−1 )x0 −
j=k−1
Z
tl
×
l Y
(I + Ej )X (tj , tj−1 )
j=k−1
X (tl−1 , s)B(s)B(s) X (tl−1 , s) dsW ∗
∗
−1
(tl−1 , tl )
tl−1
# (I + Ej )X (tj , tj−1 )x0
j =l −1
" = X (tf , tk−1 )
1 Y
1 Y
(I + Ej )X (tj , tj−1 ) −
j=k−1
1 Y
# (I + Ej )X (tj , tj−1 ) x0 = 0.
j=k−1
It follows that the complex impulsive system (1) is controllable on [t0 , tf ]. This completes the proof.
Theorem 2. Assume that (I + Ej ), j = 1, 2, . . . , k − 1 are invertible. If the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]), then rank{W (Φ0 , t0 , t1 ), . . . , W (Φk−2 , tk−2 , tk−1 ), W (Φk−1 , tk−1 , tf ), V1 , . . . , Vk−1 } = n,
(7)
where W (·, ·, ·), Φi and Vi are defined in (4). Proof. Suppose that the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]) while rank{W (Φ0 , t0 , t1 ), . . . , W (Φk−2 , tk−2 , tk−1 ), W (Φk−1 , tk−1 , tf ), V1 , . . . , Vk−1 } < n. Then there exists a nonzero n × 1 complex vector x1 such that 0 = x∗1 Vi = x∗1 Φi Fi ,
i = 1, 2, . . . , k − 1,
0 = x1 W (Φi−1 , ti−1 , ti )x1 = x1 Φi−1 ∗
∗
Z
ti
X (ti−1 , s)B(s)B∗ (s)X ∗ (ti−1 , s)dsΦi∗−1 x1 ,
ti−1
0 = x∗1 W (Φk−1 , tk−1 , tf )x1 = x∗1 Φk−1
Z
tf tk−1
X (tk−1 , s)B(s)B∗ (s)X ∗ (tk−1 , s)dsΦk∗−1 x1 .
(8)
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
1517
The integrands of (8) are nonnegative continuous functions kx∗1 Φi−1 X (ti−1 , s)B(s)k2 , i = 1, 2, . . . , k − 1 and it follows that
x∗1 Φi−1 X (ti−1 , t )B(t ) = 0, x∗1 Φk−1 X (tk−1 , t )B(t ) = 0,
t ∈ (ti−1 , ti ), i = 1, 2, . . . , k − 1, t ∈ (tk−1 , tf ].
(9)
Since the complex impulsive system (1) is controllable on [t0 , tf ], then there exists a piecewise control input u(t ) such that for initial state x0 = x1 ,
"
1 Y
x(tf ) = X (tf , tk−1 )
(I + Ej )X (tj , tj−1 )x1 +
j=k−1
+
k−1 Y i X
(I + Ej )X (tj , tj−1 )
ti
X (ti−1 , s)B(s)u(s)ds
ti−1
i=1 j=k−1
#
k −1 Y i X
Z
(I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 +
Z
tf
X (tf , s)B(s)u(s)ds = 0
tk−1
i=2 j=k−1
which implies that
"
1 Y
−X (tf , tk−1 )
(I + Ej )X (tj , tj−1 )x1 = X (tf , tk−1 )
k−1 Y i X
j=k−1
+
k−1 Y i X
# (I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 +
Z
k−1 X
tf
ti
X (ti−1 , s)B(s)u(s)ds
ti−1
X (tf , s)B(s)u(s)ds.
tk−1
[X (tj−1 , tj )(I + Ej )−1 ]X (tk−1 , tf ) from left yields that # k−1 X Φi−1 Fi−1 ui−1 + Φk−1 Fk−1 uk−1 X (ti−1 , s)B(s)u(s)ds +
Multiplying both sides of the above equation by
−x 1 =
Z
i=1 j=k−1
i=2 j=k−1
"
(I + Ej )X (tj , tj−1 )
Φi−1
ti
Z
ti−1
i=1
+ Φk−1
tf
Z
Qk−1 j =1
i=2
X (tk−1 , s)B(s)u(s)ds.
tk−1
Moreover, by multiplying x∗1 to both sides of the above equality from left, from (8) and (9) we have ∗
−x 1 x 1 =
" k−1 X
x1 Φi−1 ∗
ti
Z
X (ti−1 , s)B(s)u(s)ds +
ti−1
i =1
k−1 X
# x1 Φi Fi ui
+ x∗1 Φk−1
∗
Z
tf
X (tk−1 , s)B(s)u(s)ds = 0.
tk−1
i=1
This contradicts with the assumption that x1 6= 0 and we conclude that (7) holds. This completes the proof.
Next we apply Theorems 1 and 2 to the case of complex linear time-invariant impulsive systems. First a claim is presented [11]: for a complex matrix A, there exist scalar functions β0 (t ), β1 (t ), . . . , βn−1 (t ) such that eAt =
n−1 X
βj (t )Aj .
(10)
j =0
Theorem 3. For complex impulsive system (1), assume that A(t ) = A, B(t ) = B are n × n, n × m complex constant matrices, then the following sufficient and necessary conditions hold. (i) If rank(B, AB, . . . , An−1 B) = n
(11)
then the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). (ii) Suppose that AEi = Ei A, Fi = F (complex constant matrix) and (I + Ei ) are invertible. If the complex system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]), then rank(E˜ 1 B˜ , E˜ 2 B˜ , . . . , E˜ k−1 B˜ , B˜ , E˜ 2 F˜ , . . . , E˜ k−1 F˜ , F˜ ) = n
(12)
where B˜ = (B, AB, . . . , An−1 B), F˜ = (F , AF , . . . , An−1 F ), E˜ i =
Qi
j=k−1
(I + Ej ), i = 1, 2, . . . , k − 1.
Proof. We first prove (i) by contradiction. If (11) holds whereas the system is not controllable, then from (ii) in Theorem 1 it follows that Wi and W (tk−1 , tf ) i = 1, 2, . . . , k − 1, are not invertible. Thus there exists a nonzero complex vector xα which satisfies 0 = x∗α W (t0 , t1 )xα =
Z
t1 t0
x∗α eA(t0 −s) BB∗ [eA(t0 −s) ]∗ xα ds =
Z
t1 t0
kx∗α eA(t0 −s) Bk2 ds
1518
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
which leads to x∗α eA(t0 −t ) B = 0, t ∈ [t0 , t1 ]. When t = t0 , differentiating this equality j times and evaluating the results at t = t0 give (−1)j x∗α Aj B = 0, j = 0, 1, . . . , n − 1. Therefore, we have x∗α (B, AB, . . . , An−1 B) = 0 which means that rank condition (11) does not hold. This leads to a contradiction. Therefore, if (i) holds, the complex impulsive system (1) with A(t ) = A, B(t ) = B is controllable on [t0 , tf ]. Next we prove (ii) also by contradiction. If the complex impulsive system (1) is controllable on [t0 , tf ] (tf ∈ (tk−1 , tk ]) while (12) does not hold, that is, rank(E˜ 1 B˜ , E˜ 2 B˜ , . . . , E˜ k−1 B˜ , B˜ , E˜ 2 F˜ , . . . , E˜ k−1 F˜ , F˜ ) < n. By multiplying E˜ 1−1 from the left, the above equation is equivalent to
( rank B˜ , (I + E1 )−1 B˜ , . . . ,
k−1 Y
(I + Ej ) B, (I + E1 ) F , . . . , −1 ˜
−1 ˜
j=1
k−1 Y
) (I + Ej ) F
−1 ˜
j =1
which implies that there exists a nonzero complex vector xα such that x∗α B˜ = 0,
x∗α
i Y
(I + Ej )−1 B˜ = 0,
x∗α
i Y
j =1
(I + Ej )−1 F˜ = 0,
i = 1, 2, . . . , k − 1.
j=1
It follows from the definitions of B˜ and F˜ that for l = 0, 1, . . . , n − 1, x∗α Al B = 0,
x∗α
i Y
(I + Ej )−1 Al B = 0,
x∗α
i Y
j=1
(I + Ej )−1 Al F = 0,
i = 1, 2, . . . , k − 1.
(13)
j =1
Since AEi = Ei A and (I + Ei ) are invertible, it is easy to get (I + Ei )−1 A(I + Ei ) = (I + Ei )−1 (I + Ei )A = A which means that (I + Ei )−1 A = A(I + Ei )−1 . Hence we have from (10) and (13) x∗α W (Φ0 , t0 , t1 ) =
Z
t1
x∗α eA(t0 −s) BB∗ [eA(t0 −s) ]∗ ds
t0 t1 n−1
Z
X
= t0
βl (t0 − s)x∗α Al BB∗ [eA(t0 −s) ]∗ ds = 0
l=0 ti
Z
x∗α W (Φi−1 , ti−1 , ti ) =
ti−1
n−1 X
ti
Z
x∗α Φi−1 eA(ti−1 −s) BB∗ [eA(ti−1 −s) ]∗ Φi∗−1 ds
=
ti−1 l=0
x∗α W (Φk−1 , tk−1 , tf ) =
Z
tk−1
Z =
tf
tf
βl (t0 − s)x∗α
x∗α Vi = x∗α Φi F = x∗α
j =1
(I + Ej )−1 Al BB∗ [eA(ti−1 −s) ]∗ Φi∗−1 ds = 0,
j=1
x∗α Φk−1 eA(tk−1 −s) BB∗ [eA(tk−1 −s) ]∗ Φk∗−1 ds n−1 X
tk−1 l=0 i Y
i −1 Y
βl (t0 − s)x∗α
(I + Ej )−1 eA(t0 −ti ) F =
k−1 Y
(I + Ej )−1 Al BB∗ [eA(tk−1 −s) ]∗ Φk∗−1 ds = 0,
j =1 n −1 X l =0
βl (t0 − ti )x∗α
i Y
(I + Ej )−1 Al F = 0,
i = 1, . . . , k − 1.
j =1
Therefore, rank{W (Φ0 , t0 , t1 ), W (Φ1 , t1 , t2 ), . . . , W (Φk−1 , tk−1 , tf ), V1 , . . . , Vk−1 } < n which contradicts with Theorem 2. So we know that (12) holds. This completes the proof.
4. Observability In this section, our objective is to explicitly characterize the observability criteria of complex impulsive system (1) and the reduced complex time-invariant impulsive systems. Definition 2. The system (1) is said to be observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]), if any initial state x0 ∈ Cn can be uniquely determined by the corresponding system input u(t ) and output y(t ), for t ∈ [t0 , tf ].
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
1519
From (1) and Lemma 1, we can get the output
(
1 Y
y(t ) = C (t )x(t ) + D(t )u(t ) = C (t )X (t , tk−1 )
(I + Ej )X (tj , tj−1 )x0
j=k−1
+
k −1 Y i X
(I + Ej )X (tj , tj−1 )
+
X (ti−1 , s)B(s)u(s)ds
ti−1
i=1 j=k−1 k−1 Y i X
ti
Z
) (I + Ej )X (tj , tj−1 )Fi−1 ui−1 + Fk−1 uk−1 + C (t )
Z
t
X (t , s)B(s)u(s)ds + D(t )u(t ).
tk−1
i=2 j=k−1
It is easy to see from the Definition 2 that the observability of system (1) is equivalent to the observability of zero-input response y(t ) given by
y(t ) =
C (t )X (t , t )x , 0 0 1 Y C (t )X (t , ti−1 ) (I + Ej )X (tj , tj−1 )x0 ,
t ∈ (t0 , t1 ], t ∈ (ti−1 , ti ], i = 2, . . . , k − 1, (14)
j=i−1 1
Y (I + Ej )X (tj , tj−1 )x0 , C (t )X (t , tk−1 )
t ∈ (tk−1 , tf ].
j=k−1
For subsequent discussion, we denote the n × n matrix M (t0 , tf ) as follows: M (t0 , tf ) =
k−1 X
M (ti−1 , ti ) + M (tk−1 , tf ), M (t0 , t1 ) =
Z
X ∗ (s, t0 )C (s)∗ C (s)X (s, t0 )ds,
t0
i=1
"
ti
Z
M (ti−1 , ti ) =
t1
ti−1
(I + Ej )X (tj , tj−1 )
X ∗ (s, ti−1 )C (s)∗ C (s)X (s, ti−1 )
j=i−1
1 Y
×
#∗
1 Y
(15)
(I + Ej )X (tj , tj−1 )ds,
i = 1, 2, . . . , k − 1,
j=i−1
M (tk−1 , tf ) =
tf
Z
tk−1
"
1 Y
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, tk−1 )C (s)∗ C (s)X (s, tk−1 )
j=k−1
1 Y
(I + Ej )X (tj , tj−1 )ds.
j=k−1
Now we present the sufficient and necessary condition for the observability of complex impulsive systems (1). Theorem 4. The complex impulsive system (1) is observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]) if and only if n × n complex matrix M (t0 , tf ) defined in (15) is invertible. Proof. Multiplying both sides of (14) respectively, by X ∗ (t , t0 )C (t )∗ and [ the left and integrating with respect to t from t0 to tf , we have
Z
t1
X ∗ (s, t0 )C (s)∗ y(s)ds +
t0
k−1 Z X
ti−1
i =2
"
tf
Z +
tk−1
Z
t1
=
1 Y
"
ti
1 Y
j=i−1
(I + Ej )X (tj , tj−1 )]∗ X ∗ (t , ti−1 )C (t )∗ from
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, ti−1 )C (s)∗ y(s)ds
j=i−1
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, tk−1 )C (s)∗ y(s)ds
j=k−1
X ∗ (s, t0 )C (s)∗ C (s)X (s, t0 )ds +
t0
k−1 Z X i =2
× C (s)X (s, ti−1 )
1 Y
"
ti ti−1
(I + Ej )X (tj , tj−1 )ds +
Z
× C (s) C (s)X (s, tk−1 )
tf
"
(I + Ej )X (tj , tj−1 )ds =
j=k−1
(I + Ej )X (tj , tj−1 )
X ∗ (s, ti−1 )C (s)∗
j=i−1
1
Y
#∗
1 Y
tk−1
j=i−1
∗
Q1
1 Y
#∗ (I + Ej )X (tj , tj−1 )
X ∗ (s, tk−1 )
j=k−1
" k−1 X
# M (ti−1 , ti ) + M (tk−1 , tf ) x0 .
(16)
i=1
It is easy to see that the left-hand side of (16) depends on y(t ), t ∈ [t0 , tf ]. So if M (t0 , tf ) is invertible then the initial state x(t0 ) = x0 is uniquely determined by the corresponding complex system output y(t ), for t ∈ [t0 , tf ].
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S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
Next we consider the necessary part. If the complex matrix M (t0 , tf ) is not invertible, then there exists a nonzero vector xα such that x∗α M (t0 , tf )xα = 0. Since M (ti−1 , ti )(i = 1, . . . , k − 1) and M (tk−1 , tf ) are positive semidefinite matrices, we have x∗α M (ti−1 , ti )xα = 0,
i = 1, 2, . . . , k − 1,
x∗α M (tk−1 , tf )xα = 0.
(17)
If let initial state x(t0 ) = xα , it follows from (14) and (15) that
Z
tf
y(s)∗ y(s)ds =
k−1 Z X
t0
i =1 t1
Z =
t0
ti
y(s)∗ y(s)ds +
tf
Z
y(s)∗ y(s)ds
tk−1
ti−1
x∗α X ∗ (s, t0 )C ∗ (s)C (s)X (s, t0 )xα +
k−1 Z X i=2
× C (s) C (s)X (s, ti−1 ) ∗
1 Y
ti ti−1
" x∗α
(I + Ej )X (tj , tj−1 )xα ds +
#∗ (I + Ej )X (tj , tj−1 )
1 Y
X ∗ (s, ti−1 )
j=i−1
Z
tf
tk−1
j=i−1
× X ∗ (s, tk−1 )C (s)∗ C (s)X (s, tk−1 )
1 Y
" x∗α
1 Y
#∗ (I + Ej )X (tj , tj−1 )
j=k−1
(I + Ej )X (tj , tj−1 )xα ds
j=k−1
=
x∗α
" k−1 X
# M (ti−1 , ti ) + M (tk−1 , tf ) x∗α = 0.
i=1
which implies that
0 = y(t ) =
R tf t0
ky(s)k2 ds = 0. Thus by (14),
C (t )X (t , t )x , 0 0 1 Y C (t )X (t , ti−1 ) (I + Ej )X (tj , tj−1 )x0 ,
t ∈ (t0 , t1 ], t ∈ (ti−1 , ti ], i = 2, . . . , k − 1,
j=i−1 1
Y (I + Ej )X (tj , tj−1 )x0 , C (t )X (t , tk−1 )
t ∈ (tk−1 , tf ].
j=k−1
From Definition 2, the system (1) is not observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). This contradicts with the assumption of observability. This completes the proof. For the impulsive system (1), when A(t ) = A, C (t ) = C are complex constant matrices, the complex impulsive system becomes a complex linear time-invariant impulsive system. We have a more concise result than Theorem 4. Denote
C CA S := ...
,
S˜ :=
CAn−1
where Eˆ i =
Q1
j =i
S S Eˆ1
.. .
S Eˆ k−1 ,
(18)
(I + Ej ), i = 1, 2, . . . , k − 1.
Theorem 5. If the complex impulsive system (1) has complex constant coefficient matrices A and C , then the following conclusions hold. (i) If rank (S ) = n, then the complex linear impulsive system (1) is observable on [t0 , tf ] (tf ∈ (tk−1 , tk ]). (ii) Assume that AEi = Ei A, i = 1, 2, . . . , k − 1. If the complex linear system (1) is observable, then rank (S˜ ) = n. Proof. Sufficiency: If rank(S ) = n while the complex system (1) is not observable, then by Theorem 4 the matrix M (t0 , tf ) is not invertible which implies that there exists a nonzero vector xα satisfying x∗α M (t0 , tf )xα = 0. Since the matrices M (ti−1 , ti ) are non-negative definite, we obtain x∗α M (t0 , t1 )xα
Z
t1
=
[C eA(s−t0 ) xα ]∗ [C eA(s−t0 ) xα ]ds = 0.
t0
This shows that C eA(t −t0 ) xα = 0,
t ∈ (t0 , t1 ].
(19)
S. Zhao, J. Sun / Nonlinear Analysis: Real World Applications 11 (2010) 1513–1521
1521
Clearly, when t = t0 we have Cxα = 0. Differentiating (19) j times and evaluating the results at t = t0 yields that CAj xα = 0,
j = 0, 1, . . . n − 1.
(20)
Hence we deduce that Sxα = 0 for xα 6= 0. It follows that rank(S ) < n which leads to a contradiction with the assumption that rank(S ) = n. The proof of the sufficiency part is completed. Necessity: If otherwise, assume that the complex impulsive system (1) is observable while rank(S˜ ) < n, then there exists ˜ α = 0 which reduces from (18) to a vector xα 6= 0 satisfying Sx CAl xα = 0,
CAl
1 Y (I + Ej )xα = 0,
l = 0, 1, . . . , n − 1, i = 1, . . . , k − 1.
(21)
j =i
From (21), (10) and the fact that AEi = Ei A, we obtain M (t0 , t1 )xα =
t1
Z
[eA(s−t0 ) ]∗ C ∗
t0
M (ti−1 , ti )xα =
n −1 X
βl (s − t0 )CAl xα ds = 0,
l =0
"
ti
Z
ti−1
×
1 Y
#∗ (I + Ej )e
A(tj−1 −tj )
[eA(s−ti−1 ) ]∗ C ∗
n −1 X
j=i−1
1 Y
βl (s − t0 )CAl
l =0
(I + Ej )xα ds = 0,
i = 2, . . . , k − 1,
j=i−1
M (tk−1 , tf )xα =
Z
tf tk−1
"
1 Y
j=k−1
#∗ (I + Ej )e
A(tj−1 −tj )
[eA(s−tk−1 ) ]∗ C ∗
n−1 X l=0
βl (s − t0 )CAl
1 Y
(I + Ej )xα ds = 0.
j=k−1
So M (t0 , tf )xα = 0. Because xα 6= 0 the matrix M (t0 , tf ) is not invertible. Hence the complex linear impulsive system (1) is not observable from Theorem 4, and it contradicts with the assumption of observability. This completes the proof. 5. Conclusion In this paper, the issue on the controllability and observability criteria for a class of complex linear time-varying impulsive systems has been addressed for the first time. Several sufficient and necessary conditions for state controllability and observability of such systems have been established respectively. Moreover, the corresponding criteria for controllability and observability of complex linear impulsive systems are also derived. References [1] H. Lin, P.J. Antsaklis, Switching stabilizability for continuous-time uncertain switched linear systems, IEEE Trans. Automat. Control 52 (4) (2007) 633–646. [2] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. [3] Y. Zhang, J.T. Sun, Stability of impulsive functional differential equations, Nonlinear Anal. TMA 68 (12) (2008) 3665–3678. [4] E.A. Medina, D.A. Lawrence, Reachability and observability of linear impulsive systems, Automatica 44 (2008) 1304–1309. [5] G.M. Xie, L. Wang, Necessary and sufficient conditions for controllability and observability of switched impulsive control systems, IEEE Trans. Automat. Control 49 (6) (2004) 960–966. [6] Z.H. Guan, T.H. Qian, X.H. Yu, Controllability and observability of linear time-varying impulsive systems, IEEE Trans. Circuits Syst.-I 49 (8) (2002) 1198–1208. [7] S.W. Zhao, J.T. Sun, Controllability and observability for a class of time-varying impulsive systems, Nonlinear Anal. RWA 10 (3) (2008) 1370–1380. [8] N. Abada, M. Benchohra, H. Hammouche, Existence and controllability results for impulsive partial functional differential inclusions, Nonlinear Anal. 69 (2008) 2892–2909. [9] S. Sivasundaram, J. Uvahb, Controllability of impulsive hybrid integro-differential systems, Nonlinear Anal. Hybrid Syst. 2 (4) (2008) 1003–1009. [10] B. Liu, Controllability of impulsive neutral functional differential inclusions with infinite delay, Nonlinear Anal. 60 (2005) 1533–1552. [11] W.M. Wonham, Linear Multivariable Control: A Geometric Approach, second edition, Springer-Verlag, New York, 1979. [12] G.M. Xie, L. Wang, Controllability and observability of a class of linear impulsive systems, J. Math. Anal. Appl. 304 (2005) 336–355. [13] Z.J. Ji, L. Wang, X.X. Guo, On controllability of switched linear systems, IEEE Trans. Automat. Control 53 (3) (2008) 796–801. [14] H.J. Sussmann, V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations 12 (1) (1972) 95–116. [15] R.B. Wu, T.J. Tarn, C.W. Li, Smooth controllability of infinite-dimensional quantum-mechanical systems, Phys. Rev. A 73 (2006) 012719. [16] M. Mirrahimi, R. Van Handel, Stabilizing feedback controls for quantum systems, SIAM J. Control Optim. 46 (2) (2007) 445–467. [17] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [18] S. Kuang, S. Cong, Lyapunov control methods of closed quantum systems, Automatica 44 (1) (2008) 98–108.