Robust Control Theory and Applications Part 12 pptx

429 Consensuability Conditions of Multi Agent Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics... In the case 2, we consider a node dynamic faster than

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(a) (b)

(c)Fig 1.Procedure of redirectioning of links in a regular network (a) with increasing probability p As p

increases the network moves from regular (a) to random (c), becoming small world (b) for a critical value

Now we would like to point out the case of undirected topology with symmetric adjacency

matrix U If we assume A and BC being symmetric, then A gis symmetric with real eigenvalue.Moreover from the ﬁeld value property (Horn R.A & Johnson C.R., 1995), letσ(A ) = { α j }and

σ(BC ) = { ν j } the eigenvalues set of A and BC, then the eigenvalues of A+μ i BC are in the

interval [minj { α j } + μ iminj { ν j }, maxj { α j } + μ imaxj { ν j }], for every 1≤ i ≤ n , 1 ≤ j ≤ m.

In this way, there is a bound need to be satisﬁed by the topology structure, node dynamic andcoupling matrix for MAS stabilization

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In the literature, the MAS consensuability results have been given in terms of Laplacianmatrix properties Here, differently, we have given bounds as function of the adjacencymatrix features Anyway we can use the results on the Laplacian eigenvalue for recasting

the bounds given on the adjacency matrix To this aim, deﬁned the degree d i of i-th node of

an undirected graph as∑j u ij , the Laplacian matrix is deﬁned as L = D − U with D is the diagonal matrix with the degree of node i-th in position i-th Clearly L is a zero row sums

matrix with non-positive off-diagonal elements It has at least one zero eigenvalue and all

nonzero eigenvalues have nonnegative real parts So U=D − L and being the minimum and

maximum Laplacian eigenvalues respectively bounded by 0 and the highest node degree, wehave:

Lemma 2 Let U the adjacency matrix of undirected and connected graph G= (V, E, U), witheigenvaluesμ1≤ μ2≤ .≤ μ n, then results:

topology (p=0), by increasing the probability p of rewiring the links, it is possible smoothly

to change its topology into a random one (p = 1), with small world typically occurring atsome intermediate value In so doing neither the number of nodes nor the overall number ofedges is changed In Fig 1 it shown the results in the case of MAS of 20 nodes with each one

having k=4 neighbors

Among the simulation results we focus our attention on the maximum and minimum

eigenvalues of the matrixes U (i.e μ n andμ1) and A g (i.e.λ M and λ m) and their boundscomputed by using the results of the previous section In particular, by Lemma 2, we convey

the bounds on U eigenvalues in bounds on A geigenvalues suitable for the case of time varying

topology structure We assume in the simulations the matrices A and BC to be symmetric In this way, if U eigenvalues are in[v1, v2], letσ(A ) = { α i },σ(BC ) = { ν i } , the eigenvalues of Ag

will be in the interval [min

i α i+min

j { v1ν j , v2 ν j }, max

i α i+max

j { v1ν j , v2 ν j } ] for i, j=1, 2, , n.

Notice that, known the interval of variation[v1, v2]of the eigenvalues set of U under switching

topologies, we can recast the conditions (8), (9), (12), (13), (6), (7) and to use it for designpurpose Speciﬁcally, given the interval[v1, v2]associated to the topology possible variations,

we derive conditions on A or BC for MAS consensuability.

We consider a graph of n=400 and k=4 In the evolving network simulations, we started

with k=4 and bounded it to the order of O(log(n))for setting a sparse graph In Tab 1 aredrawn the node dynamic and coupling matrices considered in the ﬁrst set of simulations

429

Consensuability Conditions of Multi Agent

Systems with Varying Interconnection Topology and Different Kinds of Node Dynamics

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4 5 6 7 8 9 10 11

eigenvalue of A g , (b) Maximum eigenvalue of U, (c) Minimum eigenvalue of A g, (d) Minimum

eigenvalue of U

0 5 10 15 20

t

Fig 3.Case 1: State dynamic evolution in the time

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4 5 6 7 8 9 10

eigenvalue of U

A B C

Case 1: -4.1 1 1Case 2: -12 1 1Case 3: -6 1 1Case 4: -6 2 1Table 1.Node system matrices (A,B,C)

431

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4 5 6 7 8 9 10

Fig 5.Case 3 Dashed line: bound on the eigenvalue; continuous line: eigenvalue: (a) Maximum

eigenvalue of U

0 2 4 6 8 10

t

Fig 6.Case 3: state dynamic evolution in the time

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4 5 6 7 8 9 10

Fig 7.Case 4 Dashed line: bound on the eigenvalues; continuous line: eigenvalues: (a) Maximum

eigenvalue of U

In the case 1 (Fig 2), we note as although we start from a stable MAS network, the topologyvariation leads the network instability condition (namelyλ Mbecomes positive) In Fig 3 it isshown the time state evolution of the ﬁrsts 10 nodes, under the switching frequency of 1 Hz

We note as the MAS converges to the consensus state till it is stable, then goes in instabilitycondition

In the case 2, we consider a node dynamic faster than the maximum network degree d Mofall evolving network topologies from compete to random graph Notice that although thisassures MAS consensuability as drawn in Fig 4, it can be much conservative

In the case 3 (Fig 5), we consider a slower node dynamic than the cases 2 The MAS is robuststable under topology variations In Fig 6 the state dynamic evolution is convergent and thesettling time is about 4.6/| λ M(A g )|

Then we have varied the value for BC by doubling the B matrix value leaving unchanged the

node dynamic matrix As appears in Fig 7, the MAS goes in instability condition pointing out

433

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4 6 8 10

p

μ 1

(d)

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3 4 5 6 7 8 9

p

μ 1

(d)

eigenvalue of U

On the other side, a reduction on BC increases the MAS stability margin So we can tune the BC value in order to guarantee stability or desired robust stability MAS margin under a speciﬁed node dynamic and topology network variations Indeed if BC has eigenvalues above

1, its effect is to amplify the eigenvalues of U and we need a faster node dynamic for assessing MAS stability If BC has eigenvalues less of 1, its effect is of attenuation and the node dynamic

can be slower without affecting the network stability

Now we consider SISO system of second order at the node as shown in Tab.2 In this case the

matrix BC has one zero eigenvalue being the rows linearly dependent.

In the case 5 the eigenvalues of A are α1 = −4.76 andα2 = −13.23, the eigenvalues of the

coupling matrix BC are ν1=1 andν2=0 In this case the node dynamic is sufﬁciently fast forguaranteeing MAS consensuability (Fig 8) In the case 6, we reduce the node dynamic matrix

A to α1 = −1.15 eα2 = −7.85 Fig 9 shows instability condition for the MAS network We

435

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3 4 5 6 7 8 9

p

μ 1

(d)

Fig 10.Case 7 Dashed line: bound on the eigenvalue; continuous line: eigenvalue (a) Maximum

eigenvalue of A g , (b) Maximum eigenvalue U, (c) Minimum eigenvalue of A g, (d) Minimum eigenvalue

of U

can lead the MAS in stability condition by designing the coupling matrix BC as appear by the

case 7 and the associate Fig 10

4.1 Robustness to node fault

Now we deal with the case of node fault We can state the following Theorem

Theorem 2 Let A and BC symmetric matrix and G(V, E, U)an undirected graph If the MAS

system described by A gis stable, it is stable also in the presence of node faults Moreover theMAS dynamic becomes faster after the node fault

Proof Being the graph undirected and A and BC symmetric then A gis symmetric Let ˜A gtheMAS dynamic matrix associated to the network after a node fault ˜A g is obtained from A gbyeliminating the rows and columns corresponding to the nodes went down So ˜A gis a minor of

A gand for the interlacing theorem (Horn R.A & Johnson C.R., 1995) it has eigenvalues inside

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4 4.2 4.4 4.6 4.8 5

Fig 11.Eigenvalues in the case l=1 Dashed line: eigenvalue in the case of complete topology with

n=100; continuous line: eigenvalue in the case of node fault: (a) Maximum eigenvalue of A g, (b)

Maximum eigenvalue of U, (c) Minimum eigenvalue of A g , (d) Minimum eigenvalue of U

the real interval with extremes the minimum and maximum A g eigenvalues Hence if A gisstable, ˜A g is stable too Moreover, the maximum eigenvalue of ˜A g is less than one of A g So

the slowest dynamic of the system ˙x(t) =A˜g x(t)is faster than the system ˙x(t) =A g x(t)

In the follows we will show the eigenvalues of MAS dynamic in the presence of node fault

We consider MAS network with n= 100 We compare for each evolving network topology

at each time simulation step, the maximum and minimum eigenvalues of A gthan those ones

resulting with the fault of randomly chosen l nodes Figures 11 and 12 show the eigenvalues

of system dynamic for the cases l=1 and l=50

Notice that as the eigenvalues of U and A g of fault network are inside the real interval

containing the eigenvalues of U and A gof the complete graph In Fig 13 are shown the timeevolutions of state of the complete and faulted graphs Notice that the fault network is faster

than the initial network as stated by the analysis of the spectra of A gand ˜A g

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3 3.5 4 4.5 5

Fig 12.Eigenvalues in the case of l=50 Dashed line: eigenvalue in the case of complete topology with

n=100; continuous line: eigenvalue in the case of node fault: (a) Maximum eigenvalue of A g, (b)

Maximum eigenvalue of U, (c) Minimum eigenvalue of A g , (d) Minimum eigenvalue of U

5 Conclusions

In this book chapter we have investigated the consensuability of the MASs under both thedynamic agent structure and communication topology variations Speciﬁcally, it has givenconsensusability conditions of linear MASs as function of the agent dynamic structure,communication topology and coupling strength parameters The theoretical results are given

by transferring the consensusability problem to the stability analysis of LTI-MASs Moreover,

it is shown that the interplay among consensusability, node dynamic and topology must

be taken into account for MASs stabilization: consensuability of MASs is assessed forall topologies, dynamic and coupling strength satisfying a pre-speciﬁed bound From thepractical point of view the consensuability conditions can be used for both the analysisand planning of MASs protocols to guarantee robust stability for a wide range of possibleinterconnection topologies, coupling strength and node dynamics Also, the consensuability

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J.K Hedrick, D.H McMahon, V.K Narendran, and D Swaroop (1990) Longitudinal vehical

controller design for IVHS systems Proceedings of the American Control Conference,

pages 3107-3112

P Kundur (1994) Power System Stability and Control McGraw-Hill.

D Limebeer and Y.S Hung (1983) Robust stability of interconnected systems IEEE Trans.

Automatic Control, pages 710-716

A Michel and R Miller (1977) Qualitative analysis of large scale dynamical systems Academic

Press

P.J Moylan and D.J Hill (1978) Stability criteria for large scale systems IEEE Trans Automatic

Control, pages 143-149

F Paganini, J Doyle, and S Low (2001) Scalable laws for stable network congestion control.

Proceedings of the IEEE Conference on Decision and Control, pages 185-190, 2001

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M Vidyasagar (1977) L2 stability of interconnected systems using a reformulation of the passivity

theorem IEEE Transactions on Circuits and Systems, 24, 637-645.

D ˚uSiljak (1978) Large-Scale Dynamic Systems Elsevier North-Holland.

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Saber R.O., Murray R.M (2004) Consensus Problems in Networks of Agents with Switching

Topology and Time-Delays, IEEE Transactions on Automatic Control, Vol 49, 9.

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agents., Transactions on Automatic Control, 49, vol 4, pages: 622 ˝U-629

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coordination, consensus, and ﬂocking, 44th IEEE Conference on Decision and Control

and European Control Conference, pages 2996 ˝U-3000

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VicsekŠs model., Transactions on Automatic Control, Vol 49, 6, pages: 981- ˝U982

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Transactions on Automatic Control Vol 54, 6, pages: 1330–1335

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http://arxiv.org/abs/0801.3390

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Automatica, Volume 45, Issue 11, Pages 2557-2562

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changing interaction topologies, IEEE Trans Automatic Control, vol 50, no 5, pp.

655-661

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with stochastic switching topology., Automatica, Vol 45, 5, 2009, Pages 1195–1201.

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Conference on Decision and Control, Atlantis, Paradise Island, Bahamas, Dicembre2004

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Publishers Ltd, Vol 393, Giugno 1998

Horn R.A and Johnson C.R., (1995) Topics in Matrix Analysis Cambridge University Press

1995

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Bogdan Sasu*and Adina Lumini¸ta Sasu

Department of Mathematics, Faculty of Mathematics and Computer Science, West

University of Timi¸soara, V Pârvan Blvd No 4 300223 Timi¸soara

Romania

1 Introduction

The aim of this chapter is to present several interesting connections between the input-outputstability properties and the stabilizability and detectability of variational control systems,proposing a new perspective concerning the interference of the interpolation methods incontrol theory and extending the applicability area of the input-output methods in the stabilitytheory

Indeed, let X be a Banach space, let(Θ, d)be a locally compact metric space and letE = X ×Θ

We denote byB( X)the Banach algebra of all bounded linear operators on X If Y, U are two

Banach spaces, we denote byB( U, Y)the space of all bounded linear operators from U into Y

and byC s(Θ,B( U, Y))the space of all continuous bounded mappings H :Θ→ B( U, Y) Withrespect to the norm||| H |||:=sup

θ∈Θ || H(θ )||,C s(Θ,B( U, Y))is a Banach space

If H ∈ C s(Θ,B( U, Y))and Q ∈ C s(Θ,B( Y, Z))we denote by QH the mappingΘ θ →

Q(θ)H(θ) It is obvious that QH ∈ C s(Θ,B( U, Z))

Deﬁnition 1.1. Let J ∈ {R+,R} A continuous mappingσ : Θ × J → Θ is called a ﬂow on Θ

ifσ(θ, 0) =θ and σ(θ, s+t) =σ(σ(θ, s), t), for all(θ, s, t ) ∈Θ× J2

Deﬁnition 1.2. A pairπ = (Φ, σ)is called a linear skew-product ﬂow on E = X × Θ if σ is a

ﬂow onΘ and Φ : Θ×R+→ B( X)satisﬁes the following conditions:

(i)Φ(θ, 0) =I d , the identity operator on X, for all θ ∈Θ;

(ii)Φ(θ, t+s) =Φ(σ(θ, t), s)Φ(θ, t), for all(θ, t, s ) ∈Θ×R2

+(the cocycle identity);

(iii)(θ, t ) →Φ(θ, t)x is continuous, for every x ∈ X;

(iv) there are M ≥1 andω >0 such that||Φ(θ, t )|| ≤ Me ωt, for all(θ, t ) ∈Θ×R+

The mappingΦ is called the cocycle associated to the linear skew-product ﬂow π= (Φ, σ)

Let L1

loc(R+, X)denote the linear space of all locally Bochner integrable functions u :R+→ X.

Letπ = (Φ, σ)be a linear skew-product ﬂow onE = X ×Θ We consider the variationalintegral system

(S ) x θ(t; x0, u) =Φ(θ, t)x0+t

0 Φ(σ(θ, s), t − s)u(s)ds, t ≥0,θ ∈Θ

* The work is supported by The National Research Council CNCSIS-UEFISCSU, PN II Research Grant

ID 1081 code 550.

On Stabilizability and Detectability of

Variational Control Systems

19

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with u ∈ L1loc(R+, X)and x0∈ X.

Deﬁnition 1.3. The system(S )is said to be uniformly exponentially stable if there are N, ν >0such that

|| x θ(t; x0, 0)|| ≤ Ne −νt || x0||, ∀( θ, t ) ∈Θ×R+,∀ x0∈ X.

Remark 1.4. It is easily seen that the system(S )is uniformly exponentially stable if and only

if there are N, ν >0 such that||Φ(θ, t )|| ≤ Ne −νt, for all(θ, t ) ∈Θ×R+

Ifπ = (Φ, σ)is a linear skew-product flow onE = X × Θ and P ∈ C s(Θ,B( X)), then thereexists a unique linear skew-product flow denoted π P = (ΦP,σ) on X ×Θ such that thissatisfies the variation of constants formula:

ΦP(θ, t)x=Φ(θ, t)x+t

0 Φ(σ(θ, s), t − s)P(σ(θ, s))ΦP(θ, s)x ds (1.1)and respectively

ΦP(θ, t)x=Φ(θ, t)x+t

0 ΦP(σ(θ, s), t − s)P(σ(θ, s))Φ(θ, s)x ds (1.2)for all(x, θ, t ) ∈ E ×R+ Moreover, if M,ω are the exponential growth constants given by

Deﬁnition 1.2 (iv) forπ, then

Φ0(θ, t)x=Φ(θ, t)x and Φn(θ, t)x=t

0 Φ(σ(θ, s), t − s)P(σ(θ, s))Φn−1(θ, s)x ds, n ≥1for every(x, θ ) ∈ E and t ≥0

Let U, Y be two Banach spaces, let B ∈ C s(Θ,B( U, X))and C ∈ C s(Θ,B( X, Y)) We considerthe variational control system(π, B, C)described by the following integral model

by stabilizability and detectability as follows:

Deﬁnition 1.5. The system(π, B, C)is said to be:

(i) stabilizable if there exists a mapping F ∈ C s(Θ,B( X, U))such that the system(S BF)isuniformly exponentially stable;

(ii) detectable if there exists a mapping K ∈ C s(Θ,B( Y, X))such that the system (S KC)isuniformly exponentially stable

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Remark 1.6. (i) The system(π, B, C)is stabilizable if and only if there exists a mapping F ∈

C s(Θ,B( X, U))and two constants N, ν >0 such that the perturbed linear skew-product ﬂow

π BF= (ΦBF,σ)has the property

||ΦBF(θ, t )|| ≤ Ne −νt, ∀( θ, t ) ∈Θ×R+;(ii) The system(π, B, C)is detectable if and only if there exists a mapping K ∈ C s(Θ,B( Y, X))

and two constants N, ν >0 such that the perturbed linear skew-product ﬂowπ KC= (ΦKC,σ)has the property

A special application of our main results will be the study of the connections betweenthe exponential stability and the stabilizability and detectability of nonautonomous controlsystems in inﬁnite dimensional spaces The nonautonomous case treated in this chapter willinclude as consequences many interesting situations among which we mention the resultsobtained by Clark, Latushkin, Montgomery-Smith and Randolph (see (Clark et al., 2000))and the authors (see (Sasu & Sasu, 2004)) concerning the connections between stabilizability,detectability and exponential stability

2 Preliminaries on Banach function spaces and auxiliary results

In what follows we recall several fundamental properties of Banach function spaces and weintroduce the main tools of our investigation Indeed, letM(R+,R)be the linear space of all

Lebesgue measurable functions u :R+→R, identifying the functions equal a.e.

Deﬁnition 2.1. A linear subspace B of M(R+,R)is called a normed function space, if there is a

mapping| · | B : B →R+such that:

(i ) | u | B=0 if and only if u=0 a.e.;

(ii ) | αu | B = | α | | u | B, for all(α, u ) ∈R× B;

(iii ) | u+v | B ≤ | u | B + | v | B , for all u, v ∈ B;

(iv)if| u(t )| ≤ | v(t )| a.e t ∈R+and v ∈ B, then u ∈ B and | u | B ≤ | v | B

If(B, | · | B)is complete, then B is called a Banach function space.

Remark 2.2. If(B, | · | B)is a Banach function space and u ∈ B then | u (· )| ∈ B.

A remarkable class of Banach function spaces is represented by the translations invariantspaces These spaces have a special role in the study of the asymptotic properties of thedynamical systems using control type techniques (see Sasu (2008), Sasu & Sasu (2004))

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On Stabilizability and Detectability of Variational Control Systems

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Deﬁnition 2.3. A Banach function space(B, | · | B)is said to be invariant to translations if for every u :R+→ R and every t > 0, u ∈ B if and only if the function

Let C c(R+,R)denote the linear space of all continuous functions v :R+ →R with compact

support contained inR+ and let L1loc(R+,R)denote the linear space of all locally integrable

functions u :R+→R.

We denote by T (R+) the class of all Banach function spaces B which are invariant to

translations and satisfy the following properties:

(i) C c(R+,R) ⊂ B ⊂ L1loc(R+,R);

(ii) if B \ L1(R+,R ∅ then there is a continuous function δ ∈ B \ L1(R+,R)

For every A ⊂R+we denote byχ A the characteristic function of the set A.

Remark 2.4. (i) If B ∈ T (R+), thenχ [0,t) ∈ B, for all t >0

(ii) Let B ∈ T (R+), u ∈ B and t >0 Then, the function ˜u t:R+→ R, ˜ut(s) =u(s+t)belongs

to B and | u˜t | B ≤ | u | B(see (Sasu, 2008), Lemma 5.4)

Deﬁnition 2.5. (i) Let u, v ∈ M(R+,R) We say that u and v are equimeasurable if for every

t >0 the sets{ s ∈R+:| u(s )| > t }and{ s ∈R+:| v(s )| > t }have the same measure.(ii) A Banach function space(B, | · | B)is rearrangement invariant if for every equimeasurable functions u, v :R+→R+with u ∈ B we have that v ∈ B and | u | B = | v | B

We denote by R(R+) the class of all Banach function spaces B ∈ T (R+) which arerearrangement invariant

A remarkable class of rearrangement invariant function spaces is represented by the so-called

Orlicz spaces which are introduced in the following remark:

Remark 2.6. Let ϕ : R+ → R+ be a non-decreasing left-continuous function, which isnot identically zero on(0,∞) The Young function associated with ϕ is deﬁned by Y ϕ(t) =

t

0ϕ(s) ds For every u ∈ M(R+,R) let M ϕ(u) := 0∞Y ϕ (| u(s )|) ds The set O ϕ of all

u ∈ M(R+,R)with the property that there is k > 0 such that M ϕ(ku ) < ∞, is a linearspace With respect to the norm| u | ϕ := inf{ k > 0 : M ϕ(u/k ) ≤ 1} , O ϕ is a Banach space,

called the Orlicz space associated with ϕ.

The Orlicz spaces are rearrangement invariant (see (Bennet & Sharpley, 1988), Theorem 8.9)

Moreover, it is well known that, for every p ∈ [1,∞], the space L p(R+,R)is a particular case

of Orlicz space

Let now (X, || · ||) be a real or complex Banach space For every B ∈ T (R+) we denote

by B(R+, X), the linear space of all Bochner measurable functions u : R+ → X with the property that the mapping N u:R+→R+, Nu(t ) = || u(t )|| lies in B Endowed with the norm

|| u || B(R+,X):= | N u | B , B(R+, X)is a Banach space

Let(Θ, d)be a metric space and letE = X × Θ Let π= (Φ, σ)be a linear skew-product ﬂow

onE = X ×Θ We consider the variational integral system

(S ) x θ(t; x0, u) =Φ(θ, t)x0+t

0 Φ(σ(θ, s), t − s)u(s)ds, t ≥0,θ ∈Θ

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with u ∈ L1loc(R+, X)and x0∈ X.

An important stability concept related with the asymptotic behavior of dynamical systems isdescribed by the following concept:

Deﬁnition 2.7. Let W ∈ T (R+) The system (S ) is said to be completely (W(R+, X),

W(R+, X))-stable if the following assertions hold:

(i) for every u ∈ W(R+, X)and everyθ ∈ Θ the solution x θ (· ; 0, u ) ∈ W(R+, X);

(ii) there isλ >0 such that|| x θ (· ; 0, u )|| W(R+,X) ≤ λ || u || W(R+,X), for all(u, θ ) ∈ W(R+, X ) ×

Θ

A characterization of uniform exponential stability of variational systems in terms of thecomplete stability of a pair of function spaces has been obtained in (Sasu, 2008) (see Corollary3.19) and this is given by:

Theorem 2.8. Let W ∈ R(R+) The system(S )is uniformly exponentially stable if and only if

(S )is completely(W(R+, X), W(R+, X))-stable.

The problem can be also treated in the setting of the continuous functions Indeed, let

C b(R+,R)be the space of all bounded continuous functions u : R+ → R Let C0(R+,R)

be the space of all continuous functions u :R+→R with lim

t→∞ u(t) =0 and let C00(R+,R):=

{ u ∈ C0(R+,R): u(0) =0}

Deﬁnition 2.9. Let V ∈ { C b(R+,R), C0(R+,R), C00(R+,R)} The system(S )is said to be

completely(V(R+, X), V(R+, X))-stable if the following assertions hold:

(i) for every u ∈ V(R+, X)and everyθ ∈ Θ the solution x θ (· ; 0, u ) ∈ V(R+, X);

(ii) there isλ >0 such that|| x θ (· ; 0, u )|| V(R+,X) ≤ λ || u || V(R+,X), for all(u, θ ) ∈ V(R+, X) ×Θ.For the proof of the next result we refer to Corollary 3.24 in (Sasu, 2008) or, alternatively, toTheorem 5.1 in (Megan et al., 2005)

Theorem 2.10. Let V ∈ { C b(R+,R), C0(R+,R), C00(R+,R)} The system(S ) is uniformly exponentially stable if and only if(S )is completely(V(R+, X), V(R+, X))-stable.

Remark 2.11. Let W ∈ R(R+) ∪ { C0(R+, X), C00(R+, X), C b(R+, X )} If the system(S )isuniformly exponentially stable then for everyθ ∈Θ the linear operator

P W θ : W(R+, X) → W(R+, X), (P W θ u)(t) =t

0 Φ(σ(θ, s), t − s)u(s)ds

is correctly deﬁned and bounded Moreover, ifλ >0 is given by Deﬁnition 2.7 or respectively

by Deﬁnition 2.9, then we have that supθ∈Θ || P W θ || ≤ λ.

These results have several interesting applications in control theory among we mention thoseconcerning the robustness problems (see (Sasu, 2008)) which lead to an inedit estimation ofthe lower bound of the stability radius, as well as to the study of the connections betweenstability and stabilizability and detectability of associated control systems, as we will see inwhat follows It worth mentioning that these aspects were studied for the very ﬁrst time forthe case of systems associated to evolution operators in (Clark et al., 2000) and were extendedfor linear skew-product ﬂows in (Megan et al., 2002)

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On Stabilizability and Detectability of Variational Control Systems

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3 Stabilizability and detectability of variational control systems

As stated from the very beginning, in this section our attention will focus on the connections

between stabilizability, detectability and the uniform exponential stability Let X be a Banach

space, let(Θ, d)be a metric space and letπ = (Φ, σ)be a linear skew-product ﬂow onE =

X ×Θ We consider the variational integral system

where t ≥0,(x0,θ ) ∈ E and u ∈ L1loc(R+, U)

According to Deﬁnition 1.5 it is obvious that if the system(S )is uniformly exponentiallystable, then the control system(π, B, C)is stabilizable (via the trivial feedback F ≡ 0) and

this is also detectable (via the trivial feedback K ≡0) The natural question arises whether theconverse implication holds

Example 3.1. Let X = R, Θ = R and let σ(θ, t) = θ+t Let(S )be a variational integralsystem such thatΦ(θ, t) = I d (the identity operator on X), for all(θ, t ) ∈ Θ×R+ Let U =

Y=X and let B(θ) =C(θ) = I d, for allθ ∈ Θ Let δ > 0 By considering F(θ ) = − δ I d, for all

θ ∈Θ, from relation (1.1), we obtain that

ΦBF(θ, t)x=x − δt

0 ΦBF(θ, s)x ds, ∀ t ≥0for every(x, θ ) ∈ E This implies thatΦBF(θ, t)x = e −δt x, for all t ≥ 0 and all (x, θ ) ∈ E,

so the perturbed system(S BF)is uniformly exponentially stable This shows that the system(π, B, C)is stabilizable

Similarly, ifδ > 0, for K(θ ) = − δ I d, for all θ ∈ Θ, we deduce that the variational controlsystem(π, B, C)is also detectable

In conclusion, the variational control system(π, B, C)is both stabilizable and detectable, butfor all that, the variational integral system(S )is not uniformly exponentially stable

It follows that the stabilizability or/and the detectability of the control system(π, B, C)arenot sufﬁcient conditions for the uniform exponential stability of the system(S ) Naturally,additional hypotheses are required In what follows we shall prove that certain input-outputconditions assure a complete resolution to this problem The answer will be given employingnew methods based on function spaces techniques

Indeed, for everyθ ∈Θ, we deﬁne

P θ : L1loc(R+, X) → L1loc(R+, X), (P θ w)(t) =t

0 Φ(σ(θ, s), t − s)w(s)ds

Tiêu đề	Robust Control, Theory And Applications
Trường học	Standard University
Chuyên ngành	Control Theory
Thể loại	bài báo
Thành phố	City Name

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