ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR

PERIODIC IMPULSIVE CONTROL SYSTEMS

E. V. Goncharova

Institute for System Dynamics and Control Theory, Siberian Branch of Russian Academy of Sciences

134 Lermontov Street, Irkutsk, Russia

A. I. Ovseevich

Institute for Problems in Mechanics, Russian Academy of Sciences

101 Vernadsky Avenue, Moscow, Russia

Keywords:

Linear periodic dynamic systems, impulsive control, reachable sets, shapes of convex bodies.

Abstract:

We study linear periodic control systems with a bounded total impulse of control. The main result is an asymp-

totic formula for the reachable set, which, at the same time, reveals the structure of the attractor — the set of

all limit shapes of the reachable sets. The attractor is shown to be parameterized by a (ﬁnite-dimensional) toric

ﬁbre bundle over a circle. The ﬁbre of the bundle can be described via the Floquet multipliers (monodromy

matrix) of the linear system. Moreover, the limit dynamics of shapes of reachable sets can be parametrized by

an explicit curve on the toric bundle.

1 INTRODUCTION

One of the fundamental notions of control theory is

that of reachable sets which provide a visible bound

for control capabilities. In general, these sets have a

complicated shape and dynamics. There is, however,

a kind of problems where the behavior of reachable

sets is well-understood.

Namely, it turns out that the reachable sets of lin-

ear control systems have simple limit properties as

time evolves to inﬁnity provided that a suitable time-

dependent matrix scaling is applied. This kind of re-

sults was found for the ﬁrst time in (Ovseevich, 1991),

where time-invariant linear control systems with geo-

metric bounds on control were studied. It was shown

that in this setup there is a single limit shape of the

reachable sets, shape being the set regarded up to an

arbitrary nondegenerate linear transform.

At present the scope of this phenomena is not yet

clear cut. It is very likely that there is a natural exten-

sion of these results to general time-dependent linear

systems. Moreover, a similar phenomena was discov-

ered for some nonlinear stochastic dynamic systems

(Dolgopyat et al., 2004).

The purpose of the study is to develop the asymp-

totic theory of the reachable sets to linear impulsive

control systems. A motivation to address impulsive

control systems is also due to the perceived relevance

of the impulsive control theory for hybrid systems

whose state evolution is dictated by the interaction of

conventional time-driven dynamics and event-driven

dynamics (see, e.g., (Aubin, 2000; Branicky et al.,

1998; Miller and Rubinovich, 2003)).

In this paper, we study the periodic linear control

systems with a bounded total impulse of control. The

main result is an asymptotic formula for the reachable

sets (see (3)), that, in particular, reveals the structure

of attractor — the set of all limit shapes of the reach-

able sets.

It would be extremely interesting to understand

the limit behavior of reachable sets for a general lin-

ear system. Unfortunately, the nature of the present

methods is computational and it looks like new ideas

are needed in order to grasp the limit dynamics of the

reachable sets.

2 PROBLEM STATEMENT

Consider a linear control system on the time interval

[0,T]

˙x(t) = A(t)x(t)+ B(t)u(t), x(0) = 0, (1)

131

V. Goncharova E. and I. Ovseevich A. (2008).

ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR PERIODIC IMPULSIVE CONTROL SYSTEMS.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 131-136

DOI: 10.5220/0001490001310136

 SciTePress

under the constraint on the total impulse of control u:

h f(t),u(t)dti ≤ 1, (2)

where x(t) ∈ V = R

, u(t) ∈ W = R

, A(t), B(t) are

matrices of appropriate dimensions, U(t) is a given

central symmetric convex body in W, f is an arbitrary

continuous function such that f(t) ∈ U

◦

(t), andU

◦

(t)

is the polar of set U(t).

Assume that the Kalman type condition of com-

plete controllability holds, namely, for any vector

u ∈ W and a time moment T, function Φ(T,t)B(t)u

does not vanish identically in any interval of time t.

Under these assumptions the reachable sets D (T) to

system (1), (2) are central symmetric convex bodies.

The problem addressed is to study the limit be-

havior of the reachable sets D (T) as T → ∞. The

reachable sets are regarded as elements of the metric

space B of central symmetric convex bodies with the

Banach-Mazur distance ρ:

ρ(Ω

,Ω

) = log(t(Ω

,Ω

)t(Ω

,Ω

)),

where t(Ω

,Ω

) = inf{t ≥ 1 : tΩ

⊃ Ω

The general linear group GL(V) naturally acts on the

space B by isometries. The factorspace S is called the

space of shapes of central symmetric convex bodies,

where the shape ShΩ ∈ S of a convex body Ω ∈ B

is the orbit ShΩ = {CΩ : detC 6= 0} of the point Ω

with respect to the action of GL(V). The Banach-

Mazur factormetric makes S into a compact metric

space. The convergence of the reachable sets D (T)

and their shapes is understood in the sense of the

Banach-Mazur metric. For two asymptotically equal

functions with values in the space of convex bodies

or the space of their shapes, the following notations

are used: Ω

(T) ∼ Ω

(T), if ρ(Ω

(T),Ω

(T)) → 0

as T → ∞, and similarly ShΩ

(T) ∼ ShΩ

(T), if

ρ(ShΩ

(T),ShΩ

(T)) → 0 as T → ∞. The conver-

gence of convex bodies may be also understood in the

sense of convergence of their support functions. Re-

mind that the support function of a convex compact

set is given by formula: H

Ω

(ξ) = sup

x∈Ω

hx,ξi, where

ξ ∈ V

∗

, and uniquely deﬁnes the set Ω. The equiva-

lence of the two deﬁnitions of convergence of convex

bodies — in the terms of convergence of their support

functions and in the sense of the Banach-Mazur met-

ric — is established by the following lemma (Figurina

and Ovseevich, 1999):

Lemma 1. A sequence Ω

∈ B converges to Ω ∈ B

in the sense of the Banach-Mazur metric if and only

if the corresponding sequence of the support func-

tions H

(ξ) = H

Ω

(ξ) converges to the support func-

tion H

Ω

(ξ) pointwise and is uniformly bounded on the

unit sphere in the dual space V

∗

We address the periodic case, when the con-

stituents A, B, and U of control system (1), (2) are

supposed to be continuous and periodic in t. To ﬁx

ideas, the period is assumed to be 1. It would be in-

teresting to understand the limit behavior of reach-

able sets for a general linear system. This prob-

lem, however, seems rather difﬁcult, since already the

time-invariant case is nontrivial, and, say, for quasi-

periodic systems it is not clear how to prove the cor-

responding natural conjectures.

3 ASYMPTOTIC BEHAVIOR OF

SHAPES OF THE REACHABLE

SETS

We study the limit behavior as T → +∞ of the curve

T 7→ ShD (T) under different assumptions on the

spectrum of the monodromy matrix. At the heart of

the considerations below there is an explicit formula

for the support function of the reachable set:

Lemma 2. The support function of the reachable set

D (T) to system (1), (2) is given by

D (T)

(ξ) = sup

t∈[0,T]

U(t)

(B(t)

∗

Φ(T,t)

∗

ξ), (3)

where Φ(t,s) is the fundamental matrix of linear sys-

tem ˙x = A(t)x.

Stable Case. Let the system (1) be asymptotically

stable, i.e.

Φ(T,t) = o(1) as T − t → +∞,

and o(1) is uniformly small. It is easy to estab-

lish the stability criterion: system (1) is asymptoti-

cally stable iff the spectrum of the monodromy matrix

M = Φ(1,0) is contained in the open unit disk of the

complex plane.

Let us show that the curve T 7→ D (T) is asymptot-

ically periodic as T → ∞. In other words, there exists

such a continuous periodic curve f : R/Z → B that

D (T) ∼ f(T) as T → +∞. Informally speaking, the

curve T 7→ D (T) is reeled on a limit cycle.

The curve f can be given by an explicit formula.

Deﬁne function

F (T) = F (T,ξ) = sup

t∈(−∞,T]

∗

Φ(T,t)

∗

ξ), (4)

where the argument t of periodic functions B and U

is omitted. Due to the stability condition, F (T) is a

continuous periodic function of T. The periodicity of

F follows from the equality

Φ(T + 1,t + 1) = Φ(T,t)

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132

for the fundamental matrix of a 1-periodic system.

Furthermore, for each T, function ξ 7→ F (T, ξ) is ho-

mogeneous and convex, and therefore it is the support

function of a body f(T). Thus, the curve f is deﬁned.

On the other hand, from (3), (4) it follows that

f(T)

(ξ) = H

D (T)

(ξ) for large enough T, (5)

since Φ(T,t) = O



−β(T−t)



, where β > 0 owing to

the assumed stability property. The asymptotic equal-

ity D (T) ∼ f(T) follows from (5).

Unstable Case. Assume that the system (1) is strictly

unstable, i.e.

Φ(T, s) = o(1) as T − s → −∞,

where o(1) is uniformly small. System (1) is strictly

unstable iff the spectrum of the monodromy matrix

M = Φ(1, 0) is contained in the complement of the

closed unit disk of the complex plane.

Deﬁne the matrix multiplier C(T) = Φ(0,T) and

consider the set

D (T)

def

= C(T)D (T).

It is easy to see that

D (T)

(ξ) = sup

t∈[0,T]

∗

Φ(0,t)

∗

ξ),

and from the instability criterion it followsthat Φ(0,t)

decreases exponentially fast as t → ∞. Therefore, the

values sup

t∈[0,T ]

∗

Φ(0,t)

∗

ξ) converge as T → ∞,

and the convergence of bodies

D (T) → D

∞

, takes

place, where

∞

(ξ) = sup

t∈[0,∞]

∗

Φ(0,t)

∗

ξ).

Thus, we have the asymptotical equality:

D (T) ∼ Φ(T,0)D

∞

In the view of the behavior of shapes of the reach-

able sets, there is an essential difference between sta-

ble and unstable cases. In the unstable case, shapes

ShD (T) convergeas T → ∞, while, for stable system,

the curve T → ShD (T) is reeled on a limit cycle.

Note that the aboveconsiderations admit an exten-

sion to almost periodic systems.

Yet another choiceC(T) = Φ({T} , T), where {T}

is the fractional part of T, of a normalizing matrix fac-

tor seems also reasonable. It is easy to see at that rate,

that the limit normalized body D

∞

= D

∞

(T) depends

on time periodically. This choice of the matrix factor

ﬁts better the general case, when stable, unstable and

neutral components are present.

Neutral Case. Assume that system (1) is neutral,

meaning that the spectrum of the monodromy matrix

M = Φ(1,0) rests on the unit circle. Consider the Jor-

dan decomposition

M = UD = e

N+D

where D is a diagonalizable matrix of the same spec-

trum that the matrix M has, D is such a diagonalizable

real matrix that D = e

, N is a nilpotent matrix, and

ND = DN. As is well known, there exists a matrix

F = F(N,T) with the following properties:

FNF

−1

= T

−1

N, FD = DF,

and F

∞

= F

∞

(N) = lim

T→+∞

F(N,T) is deﬁned.

Put

N(T) = Φ(T, 0)NΦ(0,T), D(T) = Φ(T, 0)DΦ(0,T).

It is easy to see that N(T) and D(T) are periodic func-

tions of T, since matrices N and D commute with

M = Φ(1,0). Matrix function

∞

(N(T)) = Φ(T,0)F

∞

(N)Φ(0,T)

is also continuous and periodic. Function φ given by

formula

φ(T,t) = e

(t−T)[N(T)+D(T)]

Φ(T,t), (6)

is periodic in T and t. Deﬁne the matrix factor

C(T) = F(N(T),T)e

T[N(T )+D(T)]

and consider the normalized set

D (T)

def

= C(T)D (T).

It is easy to see that

D (T)

(ξ) = supH



(t)



= supH



(t)



= supH (g

(t)) + o(1),

(7)

where sup is taken over t ∈ [0, T], H = H

and

(t) = B

∗

φ(T,t)

∗

t(N

∗

(T)+D

∗

(T))

F(N(T),T)

∗

ξ,

(t) = B

∗

φ(T,t)

∗

F(N(T),T)

∗

(T)+tD

∗

(T)

ξ,

(t) = B

∗

φ(T,t)

∗

∞

(N(T))

∗

(T)+tD

∗

(T)

ξ.

The last term o(1) in (7) appears on account of the

difference between F(T) and F

∞

(N(T)). Since other

matrices involved in g

(t) are uniformly bounded, the

difference of the arguments comes to o(1).

Consider the following function of two arguments

T and L:

I(L,T) = sup

t∈[0,L]

(t,t,τ) =

= sup

t∈[0,L]



∗

φ(T,t)

∗

∞

(N(T))

∗

(T)+tD

∗

(T)



ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR PERIODIC IMPULSIVE CONTROL SYSTEMS

133

Function f

(t,t,τ), where t = e

tD(T)

, τ = t/L, is pe-

riodic in t and depends on the parameter T periodi-

cally as well. By using the Hermann Weyl averaging

method (Weyl, 1938; Weyl, 1939; Arnold, 1989), like

in (Goncharova and Ovseevich, 2007), we obtain the

asymptotic representation

I(L,T) = sup

T ×J

(t,t,τ) + o(1) (8)

as L → ∞, where o(1) is small uniformly in T. In

formula (8), the interval J = [0,1] and a torus T =

T (T) are involved. The torus is the closure of the

one-parameter subgroup {(e

2πit

tD(T)

)} in the group

× GL(V). Notice that the torus

T (T) = Φ(T,0)T (0)Φ(0,T)

depends on T continuously and periodically. The

torus can be naturally represented as a ﬁbre bundle

over the circle, at that the ﬁbre over e

2πiT

∈ S

the closure of the cyclic group generated by matrix

D(T)

∈ GL(V).

It is clear that

I(T) = sup

T ×J

(t,t,τ)

is a periodic function of parameter T. Hence, for large

I(L,T) = I(T) + o(1)

is a periodic function of T up to o(1). In particular,

this is true for L = T and large T. From this we con-

clude that the curve T 7→ ShD (T) is periodic up to

o(1), i.e. it is reeled on a limit cycle.

Stable-neutral Case. Suppose that system (1) is

stable-neutral. This means that fundamental matrix

admits a polynomial estimate:

|Φ(T,t)| = O(1+ |T − t|

) as T − t → +∞.

It is not difﬁcult to obtain the criterion of stable-

neutrality: system (1) is stable-neutral iff the spec-

trum of the monodromy matrix M = Φ(1,0) is con-

tained in the closed unit disk of the complex plane.

Consider the canonical decomposition of the mon-

odromy matrix M = Φ(1,0)

M = M

⊕ M

−

into the stable and neutral components (in accordance

with the relations |λ| < 1, |λ| = 1 for eigenvalues),

and the corresponding decomposition of phase space:

V = V

⊕ V

−

For an arbitrary time moment T, the monodromy ma-

trix

M(T) = Φ(T + 1,T) = Φ(T,0)MΦ(0, T)

depends on T periodically. The corresponding de-

composition

V(T) = V

(T) ⊕ V

−

(T).

is also periodic in T.

The scaling matric factor C(T) can be taken in the

block-diagonal form

C(T) = C

(T) ⊕C

−

(T),

where C

(T) : V

(T) → V

(T) are given by formulas

(T) = F(N(T),T), C

−

(T) = I.

The support function H

D (T)

(ξ) of the normalized

body

D (T)

def

= C(T)D (T) is as follows

D (T)

(ξ) = sup

t∈[0,T ]

(t) = sup

t∈[0,T]

(t)), (9)

(t) = B

∗

Φ(T,t)

∗

−

∗

φ(T,t)

∗

(T−t)(N

∗

(T)+D

∗

(T))

F(N(T),T)

∗

= ξ

(T) ∈ V

(T)

∗

, i ∈ {−, 0} are componentsof the

canonical decomposition of a vector ξ ∈ V

∗

, and the

periodic in both arguments function φ(T,t) is deﬁned

in (6). Notice that in the generic case, matrix F is the

identity one, and therefore, the formula for the sup-

port function is just as simple as (3):

D (T)

(ξ) = sup

t∈[0,T]

∗

Φ(T,t)

∗

ξ).

To study H

D (T)

(ξ) as T → ∞ let us apply the decom-

position method like in the autonomous case (Gon-

charova and Ovseevich, 2007). Owing to the basic

commutativity relations for matrices F, N, and D, we

have

(T−t)(N

∗

(T)+D

∗

(T))

F(N(T),T)

∗

= F(N(T),T)

∗

T−t

∗

(T)+(T−t)D

∗

(T)

and, thus, we have the uniform asymptotic equality

Φ(T,t)

∗

F(N(T),T)

∗

φ(T,t)

∗

∞

(N(T))

∗

T−t

∗

(T)+(T−t)D

∗

(T)

+ o(1)

as T → ∞. Like in (Goncharova and Ovseevich,

2007), we divide the time interval I = [0,T] into the

two subintervals

I = I

∪ I

−

= [0,(1− ε)T] ∪ [(1− ε)T,T],

where ε = ε(T) is such that ε(T) = o(1), while

ε(T)T → ∞ as T → ∞. By adopting arguments from

(Goncharova and Ovseevich, 2007), we obtain the

asymptotic equality

D (T)

(ξ) = max{H

−0

(T, ξ),H

(T, ξ)} + o(1), (10)

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134

where in accordance with notations (9)

−0

= sup

t∈I

−

(t) + o(1) = sup

t∈(−∞,T]

−0T

(t)),

where g

−0T

(t) stands for

∗

Φ(T,t)

∗

−

+ B

∗

φ(T,t)

∗

∞

(N(T))

∗

(T−t)D

∗

(T)

while

= sup

t∈I

(t) + o(1) = sup

t∈R,τ∈[0,1]

where g

(t,τ) stands for

∗

φ(T,t)

∗

∞

(N(T))

∗

τN

∗

(T)+(T−t)D

∗

(T)

Functions H

−0

(T, ξ) and H

(T, ξ) are periodic in T,

and convex, homogeneous in ξ. From this it fol-

lows that H

(T, ξ), i = −0,0, are the support func-

tions of some convex compact sets Ω

−0

(T) ⊂ V(T)

and Ω

(T) ⊂ V

(T), which periodically depend on

T. Furthermore, the convex compact

Ω(T) = Ω

−0

(T) ∗ Ω

(T) ⊂ V(T),

and Ω

(T)) is a body with the support function

max{H

−0

(T, ξ),H

(T, ξ)},

and also periodically depends on T. Here, we use the

join notation:

Ω = Ω

′

∗ Ω

′′

meaning that Ω is the convex hull of the union

Ω

′

Ω

′′

, or what is the same H

Ω

= max{H

Ω

′

Ω

′′

Thus, the curve T 7→

D (T) is reeled on the limit cycle

T 7→ Ω(T).

Unstable-neutral Case. Similarly to stable-neutral

case, fundamental matrix admits a polynomial esti-

mate

|Φ(T,t)| = O(1+ |T −t|

) as T − t → −∞.

System (1) is unstable-neutral iff the spectrum of the

monodromy matrix M = Φ(1,0) is contained in the

closed complement of the unit disk of the complex

plane. In this case, the normalizing matrix factor

C(T) can be taken in the block-diagonal form:

C(T) = C

(T) ⊕C

(T),

whereC

(T) : V

(T) → V

(T) are deﬁned by formulas

(T) = F(N(T),T ), C

(T) = Φ({T},T).

We note in passing that if we put C

(T) = Φ(0,T),

then the block-diagonal structure of the normalizing

matrix would be lost, since, in general, Φ(0,T) do

not map V

(T) into itself.

The support function H

D (T)

(ξ) of the normalized

body

D (T)

def

= C(T)D (T) is similar to (9):

sup

t∈[0,T ]

∗

Φ({T},t)

∗

+ B

∗

φ(T,t)

∗

×e

(T−t)(N

∗

(T)+D

∗

(T))

F(N(T),T)

∗

and like in (10) we have the asymptotics

D (T)

(ξ) = max{H

(T, ξ),H

(T, ξ)} + o(1),

where H

(T, ξ) = sup

t∈[0,∞)

+0T

(t)), and

+0T

(t) = B

∗

Φ({T},t)

∗

φ(T,t)

∗

∞

(N(T))

∗

N(T)

∗

(T−t)D

∗

(T)

The essential difference, in contrast to stable-neutral

situation, consists of that, on this occasion, func-

tion H

(T, ξ) is not periodic in T, however, it be-

comes periodic by the transformation of variable

7→ e

∗

(T)

. Geometrically, this means that the

asymptotic equality

D (T) ∼ p(t

)Ω

(T) ∗ Ω

(T) as T → ∞,

holds, where t

= (e

2πiT

TD(T)

) is an element of the

torus T (T), matrix p(t

) = e

TD(T)

is the second com-

ponent of t

, Ω

(T) are periodically depending on

time convex compacts in spaces V

(T), α ∈ { +0,0}.

Thus, the limit behavior of the normalized reach-

able sets

D (T) is the same as the behavior of the curve

T 7→ t

. The closure of the curve might have an ar-

bitrary large dimension so that the curve is reeled on

a multidimensional manifold. Still, the shapes of the

reachable sets ShD (T) have a simpler behavior, since

ShD (T) ∼ Sh(Ω

(T) ∗ Ω

(T)), and, therefore, the

curve T 7→ ShD (T) is reeled on a limit cycle of di-

mension not greater than 1.

General Case. The general result can be obtained

by using the decomposition method (see (Goncharova

and Ovseevich, 2007)) and the considered above

cases. Consider the canonical decomposition of the

monodromy matrix M = Φ(1,0)

M = M

⊕ M

−

into the unstable, neutral, and stable components ( in

accordance with the relations |λ| < 1, |λ| > 1, |λ| = 1

for eigenvalues), and the corresponding decomposi-

tion of phase space

V = V

⊕ V

−

For an arbitrary time moment T, the monodromy ma-

trix

M(T) = Φ(T + 1,T) = Φ(T,0)MΦ(0,T)

ASYMPTOTIC THEORY OF THE REACHABLE SETS TO LINEAR PERIODIC IMPULSIVE CONTROL SYSTEMS

135

depends on T periodically, so does the corresponding

decomposition of phase space

V = V

(T) ⊕ V

−

(T).

In the general case, the scaling matric factor C(T) can

be chosen in the block-diagonal form

C(T) = C

(T) ⊕C

−

(T),

where C

(T) : V

(T) → V

(T) are given by formulas

(T) = Φ({T},T),

(T) = F(N(T),T), C

−

(T) = I.

The normalized body

D (T) = C(T)D (T) has the fol-

lowing asymptotics

D (T) ∼ p(t)Ω

(T) ∗ Ω

−0

(T) (11)

as T → ∞, where t = (e

2πiT

TD(T)

) is an element

of the torus T (T), matrix p(t) = e

TD(T)

is the sec-

ond component of t, Ω

(T) are periodically depend-

ing on time convex compacts in V

(T), α ∈ {+0, 0}.

Asymptotic equality (11) can be naturally interpreted

in the terms of attractors in space S of shapes of con-

vex bodies. Deﬁne a ﬁbre bundle over circle P → S

as follows:

P = {(e

2πiT

TD(T)

z) ∈ S

× GL(V) : z ∈ Z (T)},

2πiT

TD(T)

z) 7→ e

2πiT

where Z (T) is the closure in GL(V) of a cyclic group

generated by matrix e

D(T)

. Then, the relation (11) as-

serts that the totality A of all the limit shapes of the

reachable sets (attractor) is parameterized by the set

P : there is a continuous map σ from P onto A . At

that, in the limit, the curve T 7→ ShD (T) is parame-

terized by the curve T 7→ t

= (e

2πiT

TD(T)

) in P in

the sense that

ShD (T) ∼ σ(t

)

as T → ∞.

This asymptotic equality is an incarnation of (11)

and the main result of the paper. It says that the limit

dynamics of the shapes ShD (T) can be described via

the “straight winding” T 7→ t

in the toric bundle P .

4 CONCLUSIONS

In this paper we determined completely the asymp-

totic behavior of reachable sets to periodic linear dy-

namic systems with impulsive control. This is just a

single step in the long road directed to understanding

the limit behavior of reachable sets for general linear

systems. Still, our results for the periodic case suggest

a reasonable conjectural description of this behavior.

In fact, it is possible to state a precise conjecture per-

taining to the quasi-periodic case.

ACKNOWLEDGEMENTS

The work was partially supported by RFBR (projects

08-01-00156, 08-08-00292).

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