MULTISCALE SINGULARLY PERTURBED CONTROL SYSTEMS: LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING ∗ VLADIMIR GAITSGORY † AND MINH-TUAN NGUYEN †‡ SIAM J. CONTROL OPTIM. c 2002 Society for Industrial and Applied Mathematics Vol. 41, No. 3, pp. 954–974 Abstract. An averaging technique for nonlinear multiscale singularly perturbed control systems is developed. Issues concerning the existence and structure of limit occupational measures sets generated by such systems are discussed. General results are illustrated with special cases. Key words. multiscale singularly perturbed control systems, occupational measures, averaging method, limit occupational measures sets, nonlinear control, approximation of slow motions AMS subject classifications. 34E15, 34C29, 34A60, 93C70, 34A4 PII. S0363012901393055 1. Introduction. In this paper we consider a singularly perturbed control sys- tem containing several small parameters 1 ,..., m (m ≥ 1). The parameters are introduced in such a way that the state variables of the system are decomposed into a group of “slow” variables which change their values with the rates of the order O(1) and m groups of “fast” variables which change their values with the rates of the orders O(−1 1 ),O(−1 1 −1 2 ),...,O(−1 1 −1 2 ... −1 m ), respectively. The main contribution of the paper is the description of the structure of the limit control system, the solutions of which allow us to approximate the slow variables when the parameters i , i =1,...,m, tend to zero. The role of controls in the limit system is played by probability measures defined on the product of the original control set and a subset of the state space containing all the fast trajectories (both are assumed to be compact). These probability measures are chosen from a limit set of occupational measures generated by the admissible controls and trajectories of an associated system which describes the dynamics of the fast variables if the slow ones are “frozen” (see exact definitions below). The existence of such a set (called limit occupational measures set (LOMS)) and its structure are the central issues discussed in the paper. Singularly perturbed control systems (SPCS) with one small parameter (m = 1) have been intensively studied in the literature, the most common approaches being related either to Tikhonov-type theorems justifying the equating of the small param- eter to zero with further application of the boundary layer method (see [24], [30]) to asymptotically describe the fast dynamics (see, e.g., [13], [21], [22], [25], [28], [31]) or to different types of averaging techniques (see [1], [2], [3], [4], [5], [8], [11], [14], [15], [16], [17], [18], [19], [20], [27], [32]) which allow us to deal with the situation when the equating of the parameter to zero may not lead to a right approximation. The literature on multiscale SPCS (m> 1) is much less intensive. Most available references concern linear control systems (see, e.g., [12], [26], and references therein). ∗ Received by the editors July 30, 2001; accepted for publication (in revised form) February 25, 2002; published electronically September 19, 2002. This work was supported by Australian Research Council grant A49906132. http://www.siam.org/journals/sicon/41-3/39305.html † University of South Australia, School of Mathematics, The Mawson Lakes Campus, Mawson Lakes SA 5095, Australia ([email protected]). ‡ Present address: Joint Systems Branch, DSTO, P.O. Box 1500, Edinburgh SA 5111, Australia ([email protected]). 954 Downloaded 12/21/13 to 134.99.128.41. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php
Transcript
MULTISCALE SINGULARLY PERTURBED CONTROL SYSTEMS:LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING∗
Abstract. An averaging technique for nonlinear multiscale singularly perturbed control systemsis developed. Issues concerning the existence and structure of limit occupational measures setsgenerated by such systems are discussed. General results are illustrated with special cases.
Key words. multiscale singularly perturbed control systems, occupational measures, averagingmethod, limit occupational measures sets, nonlinear control, approximation of slow motions
1. Introduction. In this paper we consider a singularly perturbed control sys-tem containing several small parameters ε1, . . . , εm (m ≥ 1). The parameters areintroduced in such a way that the state variables of the system are decomposed intoa group of “slow” variables which change their values with the rates of the order O(1)andm groups of “fast” variables which change their values with the rates of the ordersO(ε−1
1 ), O(ε−11 ε−1
2 ), . . . , O(ε−11 ε−1
2 . . . ε−1m ), respectively.
The main contribution of the paper is the description of the structure of the limitcontrol system, the solutions of which allow us to approximate the slow variableswhen the parameters εi, i = 1, . . . ,m, tend to zero. The role of controls in thelimit system is played by probability measures defined on the product of the originalcontrol set and a subset of the state space containing all the fast trajectories (bothare assumed to be compact). These probability measures are chosen from a limit setof occupational measures generated by the admissible controls and trajectories of anassociated system which describes the dynamics of the fast variables if the slow onesare “frozen” (see exact definitions below). The existence of such a set (called limitoccupational measures set (LOMS)) and its structure are the central issues discussedin the paper.
Singularly perturbed control systems (SPCS) with one small parameter (m = 1)have been intensively studied in the literature, the most common approaches beingrelated either to Tikhonov-type theorems justifying the equating of the small param-eter to zero with further application of the boundary layer method (see [24], [30]) toasymptotically describe the fast dynamics (see, e.g., [13], [21], [22], [25], [28], [31]) orto different types of averaging techniques (see [1], [2], [3], [4], [5], [8], [11], [14], [15],[16], [17], [18], [19], [20], [27], [32]) which allow us to deal with the situation when theequating of the parameter to zero may not lead to a right approximation.
The literature on multiscale SPCS (m > 1) is much less intensive. Most availablereferences concern linear control systems (see, e.g., [12], [26], and references therein).
∗Received by the editors July 30, 2001; accepted for publication (in revised form) February 25,2002; published electronically September 19, 2002. This work was supported by Australian ResearchCouncil grant A49906132.
http://www.siam.org/journals/sicon/41-3/39305.html†University of South Australia, School of Mathematics, The Mawson Lakes Campus, Mawson
Lakes SA 5095, Australia ([email protected]).‡Present address: Joint Systems Branch, DSTO, P.O. Box 1500, Edinburgh SA 5111, Australia
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 955
A technique of averaging type applicable to nonlinear control systems having a trian-gular structure (weakly coupled) was proposed in [20].
In [18] an averaging technique allowing us to deal with a general form of SPCS con-taining two small parameters (m = 2) was developed. The extension of the techniqueto the case m > 2 is, however, hardly possible since it involves a multiple averagingover time and leads to really complex expressions which are difficult to comprehend.In this paper, an averaging over time is replaced by averaging over measures fromthe LOMS. It resembles approaches used in dealing with stochastic SPCS (see, e.g.,[9], [23], [34]) and makes the transition from the case m = k to the case m = k + 1(∀k = 1, 2, . . .) very natural.
Different issues related to averaging over occupational measures in SPCS withone small parameter were discussed in [2], [3], [4], [5], [17], [32]. In [17], in particular,LOMS for control systems without small parameters were considered. In this paper,we introduce and study such sets for singularly perturbed control systems (as is theassociated system if the original system is multiscale).
The paper is organized as follows. Section 1 is this introduction. In section 2statements about approximation of the slow motions by the solutions of the averagedsystem are formulated under the assumption that the LOMS of the associated systemexists. An application of these results to problems of optimal control is demonstratedand a special case concerning systems linear in fast variables and controls is consid-ered. In section 3 issues of existence and structure of the LOMS are addressed and amultistage averaging procedure for the construction of the LOMS is presented. Theprocedure is then illustrated with a special case of control systems which have a tri-angular structure (similar to those studied in [20]). Proofs of most of the statementsare provided in section 4.
2. Averaging of multiscale SPCS.
2.1. Preliminaries. Given a compact metric space W , B(W ) will stand for theσ-algebra of its Borel subsets and P(W ) will denote the set of probability measuresdefined on B(W ). The set P(W ) will always be treated as a compact metric spacewith a metric ρ, which is consistent with its weak convergence topology. That is, asequence γk ∈ P(W ), k = 1, 2, . . . , converges to γ ∈ P(W ) in this metric if and onlyif
limk→∞
∫W
φ(w)γk(dw) =
∫W
φ(w)γ(dw)
for any continuous φ(w) : W → R1.
Using the metric ρ, one can define the Hausdorff metric ρH on the set of subsetsof P(W ):
ρH(Γ1,Γ2)def
= maxsupγ∈Γ1
ρ(γ,Γ2), supγ∈Γ2
ρ(γ,Γ1)
∀Γ1,Γ2 ∈ P(W ),(2.1)
where ρ(γ,Γi)def
= infγ′∈Γi
ρ(γ, γ′), i = 1, 2.
We will deal with the convergence in the Hausdorff metric of sets in P(W ) defined asunions of occupational measures. Given a measurable function w(t) : [0, T ] → W , theoccupational measure pw(·) ∈ P(W ) generated by this function is defined by taking
pw(·)(Q)def
=1
Tmeas
t∣∣ w(t) ∈ Q
∀Q ∈ B(W ),(2.2)
where meas · stands for the Lebesgue measure on [0, T ].
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
956 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
2.2. Setting. Consider the SPCS
ε1ε2 . . . εm−1εmy1(t) = f1
(u(t), y1(t), . . . , ym(t), z(t)
),
......
......
......
εm−1εmym−1(t) = fm−1
(u(t), y1(t), . . . , ym(t), z(t)
),(2.3)
εmym(t) = fm(u(t), y1(t), . . . , ym(t), z(t)
),
z(t) = g(u(t), y1(t), . . . , ym(t), z(t)
),
where εdef
= (ε1, ε2, . . . , εm) is a vector of small positive parameters, t ∈ [0, T ], and thefunctions fi : U×R
M1 ×· · ·×RMm ×R
N → RMi , i = 1, . . . ,m, and g : U×R
M1 ×· · ·×RMm × R
N → RN are continuous and satisfy Lipschitz conditions in (y1, . . . , ym, z).
Admissible controls are Lebesgue measurable functions u(t) : [0, T ] → U , where U isa compact metric space.
Consider also the system
ε1ε2 . . . εm−1y1(τ) = f1
(u(τ), y1(τ), . . . , ym(τ), z
),
......
......
......
εm−1ym−1(τ) = fm−1
(u(τ), y1(τ), . . . , ym(τ), z
),(2.4)
ym(τ) = fm(u(τ), y1(τ), . . . , ym(τ), z
),
z = constant,
in which z is fixed and τ ∈ [0, S]. This system will be referred to as an associatedsystem with respect to SPCS (2.3). It is formally obtained from the “fast” subsystemof (2.3) via the replacement of the time scale τ = tε−1
m . Admissible controls for theassociated system (2.4) are Lebesgue measurable functions u(τ) : [0, S] → U . Thesolutions of (2.3) and (2.4) which are obtained with admissible controls are calledadmissible trajectories.
Assumption 2.1. (i) There exist compact sets Y ′′i ⊆ Y ′
i ⊂ RMi , i = 1, . . . ,m,
and Z ′′ ⊆ Z ′ ⊂ RN such that the admissible trajectories of SPCS (2.3) which satisfy
the initial conditions(y1(0), . . . , ym(0), z(0)
) ∈ Y ′′1 × · · · × Y ′′
m × Z ′′(2.5)
do not leave the set Y ′1 × · · · × Y ′
m × Z ′ on the interval [0, T ].(ii) There exist compact sets Yi (Y
′i ⊆ Yi), i = 1, . . . ,m, and Z (Z ′ ∈ intZ) such
that for any z from Z, the admissible trajectories of system (2.4) which satisfy theinitial conditions (
y1(0), . . . , ym(0)) ∈ Y ′
1 × · · · × Y ′m(2.6)
do not leave the set Y1 × · · · × Ym on the interval [0,∞).Note that to verify this assumption, one can use results from viability theory (see
Chapter 5 in [6] and also [29] for further references).Let us introduce the following notation:
y(τ)def
=(y1(τ), . . . , ym(τ)
), Y
def
= Y1 × · · · × Ym,
and also Y ′ def
= Y ′1 × · · · × Y ′
m, Y′′ def
= Y ′′1 × · · · × Y ′′
m.
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 957
Let u(τ) be an admissible control defined on the interval [0, S] and let y(τ) bethe solution of the associated system (2.4) obtained with this control and the initialconditions (2.6). Let p(u(·),y(·)) ∈ P(U × Y ) be the occupational measure generatedby the pair
(u(τ), y(τ)
): [0, S] → U × Y and let
Γ(z, ε1, . . . , εm−1, S, y(0)
) def
=⋃
(u(·),y(·))
p(u(·),y(·))
,(2.7)
where the union is taken over all admissible controls and the corresponding solutions of(2.4). Notice that the dependence on (z, ε1, . . . , εm−1) in (2.7) is due to the dependenceof the solutions of (2.4) on these parameters.
Assumption 2.2. For any z ∈ Z , there exists a convex and compact set Γ(z) ⊂P(U × Y ) such that
ρH
(Γ(z, ε1, . . . , εm−1, S, y(0)
),Γ(z)
)≤ ν(ε1, . . . , εm−1, S) ∀y(0) ∈ Y ′,(2.8)
where lim(ε1,...,εm−1,S−1)→0 ν(ε1, . . . , εm−1, S) = 0.The set Γ(z) introduced in Assumption 2.2 will be referred to as the limit oc-
cupational measures set (LOMS). Some sufficient conditions for the existence of theLOMS are considered in section 3.
Assumption 2.3. For any S > 0, any absolutely continuous function z(τ) :[0, S] → Z, and any admissible control u(τ) : [0, S] → U ,
where yz(τ) is the solution of (2.4) obtained with a given z ∈ Z and y(τ) is thesolution of the same system obtained with the replacement of z by the function z(τ).Initial conditions for yz(τ) and y(τ) are the same: yz(0) = y(0) ∈ Y ′ and the functionκ(ε1, . . . , εm−1) is either zero (for m = 1) or tends to zero as (ε1, . . . , εm−1) tends tozero (for m > 1).
Lemma 2.4. Let Assumptions 2.1–2.3 be satisfied. Then for any vector functionh(u, y, z) : U ×Y ×Z → R
j, j = 1, 2, . . . , which is continuous in (u, y, z) and satisfiesLipschitz conditions in (y, z), there exists a constant ch such that
dH
(Vh(z
′), Vh(z′′)) ≤ ch‖z′ − z′′‖ ∀z′, z′′ ∈ Z,(2.10)
where
Vh(z)def
=⋃
p∈Γ(z)
∫U×Y
h(u, y, z)p (du, dy).(2.11)
Note that dH(·, ·) in (2.10) stands for the Hausdorff metric in a finite-dimensionalspace. That is, for arbitrary bounded subsets V1, V2 of R
j (j = 1, 2, . . .),
dH(V1, V2)def
= maxsupv∈V1
d(v, V2), supv∈V2
d(v, V1), d(v, Vi)
def
= infv′∈Vi
‖v − v′‖,(2.12)
where ‖·‖ is a norm in Rj .
The proof of Lemma 2.4 is in section 4.1.Note that Assumption 2.3 is satisfied automatically if the functions f1, . . . , fm−1
defining the right-hand side of the associated systems (2.4) do not depend on z. In a
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
958 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
general case, Assumption 2.3 can be verified to be valid if the associated system (2.4)satisfies stability conditions similar to that introduced in [16] (see [16, Assumption 4.1,Lemma 4.1]), the latter being implied by the existence of a Lyapunov-like function(as in [17, p. 467]). For the case m = 1 (one singular perturbation parameter),Assumption 2.3 can be replaced by the assumption that the statement of Lemma 2.4is valid (see [17]). A slightly different assumption which can replace Assumption 2.3for m > 1 is discussed in Remark 4.1.
2.3. Approximation of the slow trajectories. Let the function g(γ, z) :P(U × Y )× R
N → RN be defined as follows:
g(γ, z)def
=
∫U×Y
g(u, y, z) γ(du, dy).(2.13)
We will assume that the metric ρ of P(U×Y ) is chosen in such a way that the functiong(γ, z) satisfies the Lipschitz conditions:
where b is a positive constant. Let us consider the system
z(t) = g(γ(t), z(t)
),(2.15)
which will be referred to as the averaged system. The role of controls in the averagedsystem is played by functions γ(t) satisfying the inclusion
γ(t) ∈ Γ(z(t)
).(2.16)
Note that the fact that the functions γ(t) are measure valued underlines the similarityof our description with classical relaxed control setting (see [33]).
Definition 2.5. A pair(γ(t), z(t)
): [0, T ] → P(U×Y )×R
N is called admissiblefor the averaged system if γ(t) is Lebesgue measurable, z(t) is absolutely continuous,and (2.15)–(2.16) are satisfied for almost all t ∈ [0, T ].
Theorem 2.6. Let Assumptions 2.1–2.3 be satisfied and let h(u, y, z) : U × Y ×Z → R
j, j = 1, 2, . . . , be an arbitrary Lipschitz continuous vector function. Thereexist µ(ε, T ) and µh(ε, T ),
limε→0
µ(ε, T ) = 0, limε→0
µh(ε, T ) = 0,(2.17)
such that the following two statements are valid:(i) Let u(t) be an admissible control and let
(y(t), z(t)
)be the corresponding tra-
jectory of SPCS (2.3) which satisfies initial condition (2.5). There exists an ad-missible pair
(γa(t), za(t)
)of the averaged system (2.15) with the initial conditions
za(0) = z(0) such that
maxt∈[0,T ]
∥∥z(t)− za(t)∥∥ ≤ µ(ε, T ),(2.18)
and also ∥∥∥∥∫ T
0
h(u(t), y(t), z(t)
)dt −
∫ T
0
h(γa(t), za(t)
)dt
∥∥∥∥ ≤ µh(ε, T ),(2.19)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 959
where
h(γ, z)def
=
∫U×Y
h(u, y, z) γ(du, dy).(2.20)
(ii) Conversely, let(γa(t), za(t)
)be an admissible pair of the averaged system
(2.15), which satisfies initial conditions za(0) ∈ Z ′′. One can construct an admissiblecontrol u(t) such that the trajectory
(y(t), z(t)
)of SPCS (2.3) obtained with this
control and initial conditions (2.5)(z(0) = za(0)
)will satisfy (2.18)–(2.19).
The proof of the theorem is in section 4.1. Estimates (2.18)–(2.19) of Theorem2.6 are not uniform with respect to the length T of the time interval. Additionalassumptions are needed to make them uniform. The assumption we use in this paperis as follows.
Assumption 2.7. There exist positive definite matrices C, D and a constant asuch that corresponding to any z′, z′′ from Z and any γ′ ∈ Γ(z′) there exists γ′′ ∈ Γ(z′′)such that (
g(γ′, z′)− g(γ′′, z′′))T
C (z′ − z′′) ≤ −‖z′ − z′′‖2D(2.21)
and
ρ(γ′, γ′′) ≤ a‖z′ − z′′‖,(2.22)
where ‖x‖2D in (2.21) (and in what follows) stands for xTDx.
Note that Assumption 2.7 is satisfied if the inequality (2.21) is valid for anyγ′ = γ′′ and the LOMS Γ(z) is independent of z (that is, the associated system doesnot depend on z).
Theorem 2.8. Let Assumptions 2.1–2.3 and 2.7 be satisfied. Assume also thatall the admissible trajectories of averaged system (2.15) which start in Z ′′ do not leaveZ ′ and those which start in Z ′ do not leave intZ on the infinite time horizon. Thenthere exist µ(ε) and µh(ε),
limε→0
µ(ε) = 0, limε→0
µh(ε) = 0,
such that statements (i) and (ii) of Theorem 2.6 remain valid with
supt>0
∥∥z(t)− za(t)∥∥ ≤ µ(ε)(2.23)
replacing (2.18) and
supT>T0
∥∥∥∥T−1
∫ T
0
h(u(t), y(t), z(t)
)dt − T−1
∫ T
0
h(γa(t), za(t)
)dt
∥∥∥∥(2.24) ≤ µh(ε), T0 = const
replacing (2.19) for any Lipschitz continuous vector function h(u, y, z) : U ×Y ×Z →Rj, j = 1, 2, . . . , such that the corresponding h(γ, z) defined by (2.20) satisfies the
where ah is some positive constant.The proof of the theorem is in section 4.1.
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
960 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
2.4. Application to optimal control. Let h(u, y, z) : U × Y × Z → R1 be
continuous and satisfy the Lipschitz conditions in (y, z). Consider the optimal controlproblem
inf(u(·),y(·),z(·))
∫ T
0
h(u(t), y(t), z(t))dt
,(2.26)
where inf is sought over all admissible controls and trajectories of (2.3). Under the as-sumptions of Theorem 2.6, the optimal value of this problem converges to the optimalvalue of the problem
inf(γ(·),z(·))
∫ T
0
h(γ(t), z(t))dt
,(2.27)
where h(γ, z) is defined according to (2.20) and inf is over the admissible pairs of theaveraged system (2.15). Near optimal controls of (2.26) can also be constructed on thebasis of the solution of (2.27). These will be the controls which provide the validity of(2.18)–(2.19) for the admissible pair (γa(t), za(t)) which delivers the optimal (or nearoptimal) value to (2.27) (see statement (ii) of Theorem 2.6). If the assumptions ofTheorem 2.8 are satisfied, then a similar approximation of a problem on the infinitetime horizon with a time average criterion is possible.
In some cases the “limit” problem (2.27) can be significantly simplified with thehelp of the following proposition.
Proposition 2.9. Let φ(yi) : Yi → R1 be continuously differentiable. Then∫
U×Y(φ′(yi))T fi(u, y, z)γ(du, dy) = 0 ∀γ ∈ Γ(z),(2.28)
and, in particular, ∫U×Y
fi(u, y, z)γ(du, dy) = 0 ∀γ ∈ Γ(z),(2.29)
where fi(u, y, z), i = 1, . . . ,m, are the functions defining the right-hand side of (2.4).The proof of the proposition is in section 4.1. To illustrate how this proposition
can be applied let us consider the following special case. Assume that the set U isconvex and the functions fi(u, y, z), g(z, y, u) are linear in fast variables and controls.That is,
fi(u, y, z) =
m∑j=1
Ai,j(z)yj +Ai,m+1(z)u+Ai,m+2(z), i = 1, . . . ,m,(2.30)
g(u, y, z) =
m∑j=1
A0,j(z)yj +A0,m+1(z)u+A0,m+2(z),(2.31)
where Ai,j are matrix functions of the corresponding dimensions. By (2.31), theaveraged system is equivalent to
where y = ψ(u, z) is the root of the system of equations fi(u, y, z) = 0, i = 1, . . . ,m.This is a so-called reduced system and can be obtained from (2.3) via formally equatingε to zero. If, in addition, the function h(u, y, z) used in (2.26) is convex in (u, y), thenlimit problem (2.27) becomes equivalent to
inf(u(·),z(·))
∫ T
0
h(u(t), ψ(u(t), z(t)), z(t))dt
,(2.35)
where inf is over the admissible controls and corresponding trajectories of (2.34).Notice that the reasoning above is valid if Assumptions 2.1–2.3 are satisfied. It canbe shown (although it is quite technical and we do not prove it in this paper) that these
assumptions are satisfied if the eigenvalues of the matrices A(l−1)l,l (z), l = 1, . . . ,m,
defined below have negative real parts for all z from a sufficiently large domain. Thematrices are defined recursively for l = 1, . . . ,m by the equations
A(l)i,j(z) = A
(l−1)i,j (z)−A
(l−1)i,l (z)(A
(l−1)l,l (z))−1A
(l−1)l,j (z)(2.36)
(i = l + 1, . . . ,m, j = l + 1, . . . ,m + 2), with A(0)i,j (z)
def
= Ai,j(z) (i = 1, . . . ,m,j = 1, . . . ,m+2). Note that the condition that the matrices (2.36) have negative realparts is similar to that used in [12] to asymptotically describe the reachability set ofa multiscale linear SPCS.
3. Existence of LOMS.
3.1. Approximation of the occupational measures set. Let u(t) be anadmissible control and let
(y(t), z(t)
)be the corresponding admissible trajectory of
SPCS (2.3) which satisfies initial conditions (2.5). Let p(u(·),y(·),z(·)) ∈ P(U×Y ×Z) bethe occupational measure generated by the vector function
(u(·), y(·), z(·)): [0, T ] →
U × Y ′ × Z ′ ⊂ U× Y × Z and let
Γ(ε, T, y(0), z(0)
) def
=⋃
(u(·),y(·),z(·))
p(u(·),y(·),z(·))
,(3.1)
where the union is taken over all admissible controls and the corresponding trajectoriesof SPCS (2.3). In this section, we will describe the asymptotics of this set as the vectorof small parameters ε = (ε1, . . . , εm−1, εm) tends to zero.
Let (γ(t), z(t)) : [0, T ] → P(U × Y ) × Z be an admissible pair of the averagedsystem (2.15) with the initial condition
z(0) ∈ Z ′′.(3.2)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
962 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
Let p(γ(t),z(t)) ∈ P(P(U×Y )×Z)be the occupational measure generated by this pair
and let Γ(T, z(0)) be the union of the occupational measures generated by all suchpairs
Γ(T, z(0)
) def
=⋃
(γ(·),z(·))
p(γ(·),z(·))
.(3.3)
We will use Γ(T, z(0)) to specify the limit of (3.1) as ε tends to zero. To do that letus define a map ψ(p) : p ∈ P(P(U ×Y )×Z
)→ P(U ×Y ×Z) in such a way that forany Q ∈ B(U × Y ) and any F ∈ B(Z),
ψ(p)(Q× F ) =
∫P(U×Y )×Z
γ(Q)χF (z)p(dγ, dz),(3.4)
where χF (·) is the indicator function of F . The integration in (3.4) is legitimate sincethe function
γ(Q)χF (z) : (γ, z) ∈ P(U × Y )× Z → [0, 1](3.5)
is measurable with respect to B(P(U × Y ) × Z) (see [10, Proposition 7.25, p. 133]).Notice that for any p ∈ P(P(U × Y ) × Z
)and any continuous function h(u, y, z) :
U × Y × Z → Rj , j = 1, 2, . . . ,∫
U×Y×Zh(u, y, z)ψ(p)(du, dy, dz) =
∫P(U×Y )×Z
h(γ, z)p(dγ, dz),(3.6)
where h(γ, z) is defined by (2.20). For p = p(γ(·),z(·)) (that is, for p being the occupa-tional measure generated by an admissible pair
(γ(·), z(·)) of (2.15))
∫U×Y×Z
h(u, y, z)ψ(p(γ(·),z(·))
)(du, dy, dz) =
1
T
∫ T
0
h(γ(t), z(t)
)dt.(3.7)
Let us now define the set Γ(T, z(0)) ⊂ P(U × Y × Z) as follows:
Γ(T, z(0))def
=⋃
p∈Γ(T,z(0))
ψ(p)
=
⋃(γ(·),z(·))
ψ(p(γ(·),z(·))
),(3.8)
where the second union is taken over all admissible pairs of (2.15) satisfying initialconditions (3.2). (The second equality follows from the definition (3.3) of the setΓ(T, z(0)
).)
Theorem 3.1. (i) Let the assumptions of Theorem 2.6 be satisfied. Then thereexists ν(ε, T ), limε→0 ν(ε, T ) = 0, such that
ρH
(Γ(ε, T, y(0), z(0)
),Γ(T, z(0)
)) ≤ ν(ε, T ) ∀(y(0), z(0)) ∈ Y ′′ × Z ′′.(3.9)
(ii) Let the assumptions of Theorem 2.8 be satisfied and let there be a sequenceqk(u, y, z) : U ×Y ×Z → R
1, k = 1, 2, . . . , of Lipschitz continuous functions such thatit is dense in C(U × Y × Z) and for any
h(z, y, u)def
= (q1(u, y, z), . . . , qj(u, y, z)), j = 1, 2, . . . ,(3.10)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 963
the corresponding h(γ, z) defined by (2.20) satisfies Lipschitz condition (2.25). Thenestimate (3.9) becomes uniform with respect to T ≥ T0. That is, there exists ν(ε),limε→0 ν(ε) = 0, such that ∀T ≥ T0,
ρH
(Γ(ε, T, y(0), z(0)
),Γ(T, z(0)
)) ≤ ν(ε) ∀(y(0), z(0)) ∈ Y ′′ × Z ′′.(3.11)
The proof of the theorem is in section 3.4.
3.2. LOMS of the averaged system and LOMS of the multiscale SPCS.Proposition 3.2. Let the uniform estimate (3.11) be valid and let the LOMS
of the averaged system (2.15) exist. That is, there exists the convex and compact setΓ ⊂ P(P(U × Y )× Z
)such that
ρH
(Γ(T, z(0)
), Γ)≤ µ(T ) ∀z(0) ∈ Z ′′,(3.12)
where limT→∞ µ(T ) = 0. Then the set
Γdef
=⋃p∈Γ
ψ(p)
⊂ P(U × Y × Z
)(3.13)
is convex and compact, and the following estimate is valid:
ρH
(Γ(T, z(0)
),Γ)≤ µ(T ) ∀z(0) ∈ Z ′′,(3.14)
where limT→∞ µ(T ) = 0. Also,
ρH
(Γ(ε, T, y(0), z(0)
),Γ)≤ µ(T ) + ν(ε) ∀(y(0), z(0)) ∈ Y ′′ × Z ′′,(3.15)
where µ(T ) and ν(ε) are as in (3.14) and (3.11), respectively. Thus, Γ is the LOMSof SPCS (2.3).
Proof. The validity of (3.14) is implied by (3.12) and by the fact that the map ψ(p)defined by (3.4) is continuous (see Lemma 4.3 in section 4.2). This continuity impliesalso the fact that the set Γ is compact. The convexity of Γ follows from the linearityof ψ(p). Estimate (3.15) follows from (3.14), (3.11), and the triangle inequality.
Theorem 3.3. Let the assumptions of Theorem 3.1(ii) be satisfied. Then(i) the LOMS Γ of the averaged system (2.15) exists and the estimate (3.12) is
valid;(ii) the LOMS Γ of the SPCS system (2.3) exists and the estimate (3.15) is valid;
Γ is presented in the form (3.13).Proof. The statements included in (ii) follow from Theorem 3.1(ii), Proposi-
tion 3.2, and Theorem 3.3(i). The proof of Theorem 3.3(i) is in section 4.2.
3.3. LOMS via multistage averaging. System (2.4), which was introduced asassociated with respect to (2.3), is singularly perturbed itself. One can thus considera system which would be associated with respect to (2.4):
ε1ε2 . . . εm−2y1(τ) = f1
(u(τ), y1(τ), . . . , ym−1(τ), ym, z
),
......
......
......
εm−2ym−2(τ) = fm−2
(u(τ), y1(τ), . . . , ym−1(τ), ym, z
),(3.16)
ym−1(τ) = fm−1
(u(τ), y1(τ), . . . , ym−1(τ), ym, z
),
(ym, z) = constant,
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
964 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
in which both ym and z are fixed. For the sake of convenience, in this section we willrefer to (2.4) and (3.16) as to ym- and ym−1-associated systems, respectively (by thename of the group of variables changing their values with rates of the order O(1)).One can also consider ym−2-, . . . , y2- and y1-associated systems, the latter two beingof the form
ε1y1(τ) = f1
(u(τ), y1(τ), y2(τ), y3, . . . , ym, z
),
y2(τ) = f2
(u(τ), y1(τ), y2(τ), y3, . . . , ym, z
),(3.17)
(y3, . . . , ym, z) = constant
and
y1(τ) = f1
(u(τ), y1(τ), y2, y3, . . . , ym, z
),(3.18)
(y2, y3, . . . , ym, z) = constant.
Assume that the LOMS Γ1(y2, y3, . . . , ym, z) ⊂ P(U × Y1
)of system (3.18) exists
(sufficient conditions for the existence of LOMS of systems which, like (3.18), do notinvolve small parameters were discussed in [17]) and that Theorem 2.6 is applicable tosystem (3.17). Then y2-components of the trajectories of this system are approximatedby the trajectories of the averaged system
y2(τ) = f2
(γ1(τ), y2(τ), y3, . . . , ym, z
), γ1(τ) ∈ Γ1(y2(τ), y3, . . . , ym, z),(3.19)
where (y3, . . . , ym, z) are fixed and
f2
(γ1, y2, y3, . . . , ym, z
) def
=
∫U×Y1
f2
(u, y1, y2, y3, . . . , ym, z
)γ1(du, dy1).(3.20)
Suppose that the LOMS Γ2(y3, . . . , ym, z) ⊂ P(P(U×Y1)×Y2
)of system (3.19) exists
and that the other assumptions of Proposition 3.2 or Theorem 3.3 are satisfied. Onethen can come to the conclusion that the LOMS Γ2(y3, . . . , ym, z) ⊂ P(U × Y1 × Y2
)of system (3.17) exists and is presented in the form
Γ2(y3, . . . , ym, z) =⋃
p∈Γ2(y3,...,ym,z)
ψ1(p)
,(3.21)
where the map ψ1(p) : p ∈ P(P(U × Y1) × Y2
) → P(U × Y1 × Y2) is such (comparewith (3.4) above) that for any Q ∈ B(U × Y1) and any F ∈ B(Y2),
ψ1(p)(Q× F ) =
∫P(U×Y1)×Y2
γ1(Q)χF (y2)p(dγ1, dy2),(3.22)
χF (·) being the indicator function of F . Assuming further that Proposition 3.2 orTheorem 3.3 can be applied step by step to y3-, . . . , ym-associated systems, one can
establish the existence of the LOMS Γ(z)def
= Γm(z) of system (2.4), which is presentedin the form
Γm(z) =⋃
p∈Γm(z)
ψm−1(p)
,(3.23)
with the corresponding definition of ψm−1(p) and Γm(z) being the LOMS of theaveraged system
ym(τ) = fm(γm−1(τ), ym(τ), z
), γm−1(τ) ∈ Γm−1(ym(τ), z),(3.24)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 965
where z = const, Γm−1(ym, z) is the LOMS of the ym−1-associated system, and
fm(γm−1, ym, z
) def
=
∫U×Y1×···×Ym−1
fm(u, y1, . . . , ym−1, ym, z
)γm−1(du, dy1, . . . , dym−1).
The applicability of Theorem 3.3 to each of the above systems is easy to verify, forexample, if
fi(u, y, z)def
= fi(u, y1, . . . , yi), i = 1, . . . ,m.(3.25)
That is, the dynamics of yi-components in (2.3) is not influenced by the dynamics ofyi+1-, . . . , ym- and z-components. Assuming that this is the case, let us also introducethe following assumption about the functions fi(·).
Assumption 3.4. There exist positive definite matrices Ci, Di (i = 1, . . . ,m)such that for any u ∈ U and any y1, . . . , yi−1, y
By (3.25), the y1-associated system (3.18) does not depend on (y2, . . . , ym, z) and,by (3.26) with i = 1, the LOMS Γ1 of this system exists (see Proposition 3.3 in [17]).Again, by (3.25), the dependence on (y3, . . . , ym, z) in the function (3.20) defining theright-hand side of (3.19) disappears and, by (3.26) with i = 2, this function satisfiesthe inequality
(f2(γ1, y
′2)− f2(γ1, y
′′2 ))T
C2 (y′2 − y′′2 ) ≤ −‖y′2 − y′′2‖2D2
∀γ1 ∈ P(U × Y1),
∀y′2, y′′2 ∈ RM2 and ∀γ1 ∈ P(U × Y1). This implies the applicability of Theorem 3.3
according to which the LOMS Γ2 of averaged system (3.19) and the LOMS Γ2 of they2-associated system both exist and the representation (3.21) is valid. Continuing ina similar way, one can finally verify that the LOMS Γm of averaged system (3.24) andthe LOMS Γm of ym-associated system (2.4) exist and that the representation (3.23)is valid. The applicability of Theorem 3.3 at this final stage can be verified by usingthe fact that the function fm
(γm−1, ym
)defining the right-hand side of the averaged
system (3.24) (which, by (3.25), does not involve the dependence on z) satisfies theinequality
Note that a different multistage averaging procedure for SPCS with fi(·) havingthe form (3.25) and satisfying an assumption similar to Assumption 3.4 (with Ci, Di
being identity matrices) was suggested in [20].
3.4. Basic lemma and the proof of Theorem 3.1. The proofs of Theorems3.1 and 3.3 are based on the lemma and its corollaries presented below. Let W be acompact metric space and qk(w) : W → R
1, k = 1, 2, . . . , be a sequence of Lipschitzcontinuous functions which is dense in C(W ).
Lemma 3.5. Let Γi(α, β) ⊂ P(W ), i = 1, 2, where α and β take values in somemetric spaces A and B. Assume that corresponding to any vector function
h(w) =(q1(w), . . . , qj(w)
), j = 1, 2, . . . ,(3.27)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
966 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
there exists a function
νh(α) : A → R1, lim
α→α0
νh(α) = 0,(3.28)
such that
supv∈V 1
h(α,β)
d(v, V 2
h (α, β)) ≤ νh(α),(3.29)
where
V ih(α, β)
def
=⋃
γ∈Γi(α,β)
∫W
h(w)γ(dw)
, i = 1, 2, . . . .(3.30)
Then there also exists another function
ν(α) : A → R1, lim
α→α0
ν(α) = 0,(3.31)
such that
supγ∈Γ1(α,β)
ρ(γ,Γ2(α, β)
) ≤ ν(α).(3.32)
Corollary 3.6. If for any h(w) : W → Rj as in (3.27) there exists a func-
tion (3.28) such that
dH
(V 1h (α, β), V
2h (α, β)
) ≤ νh(α),(3.33)
then there also exists a function (3.31) such that
ρH
(Γ1(α, β),Γ2(α, β)
) ≤ ν(α).(3.34)
Corollary 3.7. Let Γ(α, β) ⊂ P(W ) for (α, β) ∈ A × B, and for any h(w) :W → R
j as in (3.27) there exists a convex and compact set Vh ⊂ Rj and a func-
tion (3.28) such that
dH
(Vh(α, β), Vh
) ≤ νh(α),(3.35)
where
Vh(α, β) =⋃
γ∈Γ(α,β)
∫W
h(w)γ(dw)
.(3.36)
Then there exists a function (3.31) such that
ρH
(Γ(α, β),Γ
) ≤ ν(α),(3.37)
where Γ is a convex and compact subset of P(W ) defined by
Γdef
=
γ∣∣∣ γ ∈ P(W ),
∫W
h(w)γ(dw) ∈ Vh ∀h(w): W→ Rj as in (3.27)
.(3.38)
The proof of Lemma 3.5 is in section 4.2. Corollary 3.6 is implied by Lemma 3.5in an obvious way. The proof of Corollary 3.7 is similar to the proof of Theorem 3.1(i)in [17].
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 967
Proof of Theorem 3.1. Let h(u, y, z) : U × Y × Z → Rj , j = 1, 2, . . . , be an
arbitrary Lipschitz continuous vector function. Let u(t) be an admissible control andlet(y(t), z(t)
)be the corresponding admissible trajectory of SPCS (2.3) which satisfies
initial conditions (2.5). Let Vh(ε, T, y(0), z(0)
)be the set of time averages
Vh(ε, T, y(0), z(0)
) def
=⋃
(u(·),y(·),z(·))
1
T
∫ T
0
h(u(t), y(t), z(t)
)dt
,(3.39)
where the union is taken over all admissible controls and the corresponding trajectoriesof (2.3). Notice that by definition (3.1) of Γ
(ε, T, y(0), z(0)
), the set (3.39) also allows
the representation
Vh(ε, T, y(0), z(0)
)=
⋃γ∈Γ(ε,T,y(0),z(0))
∫h(u, y, z)γ(du, dy, dz)
.(3.40)
Let the set Vh(T, z(0)
)be defined as follows:
Vh(T, z(0)
) def
=⋃
γ∈Γ(T,z(0))
∫h(u, y, z)γ(du, dy, dz)
(3.41)
=⋃
(γ(·),z(·))
∫h(u, y, z)ψ
(p(γ(·),z(·)))(du, dy, dz),
where, as in (3.8), the second union is taken over all admissible pairs of (2.15) whichsatisfy the initial conditions (3.2).
By (3.7), the set Vh(T, z(0)
)can also be represented in the form
Vh(T, z(0)
)=
⋃(γ(·),z(·))
1
T
∫ T
0
h(γ(·), z(·)).(3.42)
Using estimate (2.19) from Theorem 2.6 and comparing (3.39) and (3.42), one obtains
dH
(Vh(ε, T, y(0), z(0)
), Vh(T, z(0)
)) ≤ 1
Tµh(ε, T )(3.43)
∀(y(0), z(0)) ∈ Y ′′ × Z ′′. Having in mind representations (3.40), (3.41) and applyingCorollary 3.6, one proves (3.9). Under the conditions of Theorem 2.8, estimate (3.43)can be rewritten in the uniform with respect to the T ≥ T0 form
dH
(Vh(ε, T, y(0), z(0)
), Vh(T, z(0)
)) ≤ µh(ε) ∀T ≥ T0,
where h(·) is as in (3.10). This, by Corollary 3.6, proves (3.11).
4. Proofs and auxiliary results.
4.1. Proofs for section 2.Proof of Lemma 2.4. Consider the set of the time averages
Vh(z, ε, S, y(0)
) def
=⋃
(u(·),y(·))
1
S
∫ S
0
h(u(τ), y(τ), z
),(4.1)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
968 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
where εdef
= (ε1, . . . , εm−1) and the union is taken over all admissible controls and thecorresponding trajectories of (2.4). By Assumption 2.3,
′′(τ) are solutions of (2.4) obtained with the same control and
initial conditions and with z = z′ and z = z′′, respectively. Hence,
dH
(Vh(z′, ε, S, y(0)
), Vh(z′′, ε, S, y(0)
)) ≤ ch‖z′ − z′′‖+ chκ(ε) ∀z′, z′′ ∈ Z,(4.3)
where ch is a constant which is expressed via the Lipschitz constant of h(·) and cfrom (4.2) in an obvious way.
By definition (2.7) of Γ(z, ε, S, y(0)
), the set Vh
(z, ε, S, y(0)
)defined in (4.1) allows
also the representation
Vh(z, ε, S, y(0)
)=
⋃p∈Γ(z,ε,S,y(0))
∫U×Y
h(u, y)p(du, dy)
.(4.4)
It follows from Assumption 2.2 that there exists a function νh(ε, S) such that
lim(ε,S−1)→0
νh(ε, S) = 0
and
dH
(Vh(z, ε, S, y(0)
), Vh(z)
)≤ νh(ε, S) ∀z ∈ Z, ∀y(0) ∈ Y ′.(4.5)
Passing to the limit as (ε, S−1) tends to zero in (4.3), one obtains (2.10).
Proof of Theorem 2.6. Let g(u, y, z)def
=(g(u, y, z), h(u, y, z)
). Consider the set of
time averages
V(z, ε, S, y(0)
)=
⋃(u(·),y(·))
1
S
∫ T
0
g(u(τ), y(τ), z
)dτ
⊂ R
N+j ,
where, as in (4.1), the union is taken over all admissible controls and correspondingtrajectories of (2.4). From Assumption 2.2 it follows (similarly to (4.5)) that thereexists ν(ε, S), lim(ε,S−1)→0 ν(ε, S) = 0 such that
dH
(V(z, ε, S, y(0)
), V (z)
)≤ ν(ε, S) ∀z ∈ Z, ∀y(0) ∈ Y ′,(4.6)
where
V (z)def
=(v, w)
∣∣ (v, w) = (g(γ, z), h(γ, z)), γ ∈ Γ(z)⊂ R
N+j ,(4.7)
with g and h being defined by (2.13) and (2.20), respectively.Let us augment the averaged system (2.15) with the equation
θ(t) = h(γ(t), z(t)
), θ(0) = 0.(4.8)
The map V (z) : Z → 2RN+j
defined by (4.7) is convex and compact valued. It alsosatisfies the Lipschitz conditions (Lemma 2.4)
dH
(V (z′), V (z′′)
) ≤ c‖z′ − z′′‖ ∀z′, z′′ ∈ Z, c = const.(4.9)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 969
By the Filippov theorem (see, e.g., [7, Theorem 8.2.10, p. 316]), the set of admissible
trajectories(z(t), θ(t)
) def
= z(t) of systems (2.15) and (4.8) coincides with the set ofsolutions of the differential inclusion
˙z(t) ∈ V(z(t)
).(4.10)
Let us augment system (2.3) with the equation
θ(t) = h(u(t), y1(t), . . . , ym(t), z(t)
), θ(0) = 0,(4.11)
and again denote z(τ)def
=(z(τ), θ(τ)
). To prove the theorem it is enough to show
that, corresponding to any admissible trajectory(y(t), z(t)
)of (2.3) and (4.11), there
exists a solution za(t) of (4.10) satisfying the inequality
maxt∈[0,T ]
∥∥z(t)− za(t)∥∥ ≤ µ(ε, T ), lim
ε→0µ(ε, T ) = 0,(4.12)
and, conversely, for any solution za(t) of (4.10), there exists an admissible trajectory(y(τ), z(τ)
)of (2.3) and (4.11) which satisfies (4.12).
The proof of these statements is similar to Lemma 2.1 in [16] or Theorem 3.1in [19].
Remark 4.1. Note that an important step of the proof is an introduction of the
new time scale τdef
= tε−1m and a partition of the interval [0, T ε−1
m ] by the points τl =lS(εm), l = 0, 1, . . . , where S(εm) > 0 is a function of εm such that limεm→0 S(εm) =∞ and limεm→0 εmS(εm) = 0. At the cost of making the proof slightly more involved,one can replace Assumption 2.3 by the assumption that the statement of Lemma 2.4is valid and that the ym−1-associated system (3.16) has a property similar to (2.9),with (ym, z) playing the role of z.
Proof of Theorem 2.8. The proof is based on the following result.Proposition 4.2. Given a solution
(z1(t), θ1(t)
)of the differential inclusion
(4.10) satisfying the initial condition(z1(0), θ1(0)
where b1, b2, β are some positive constants.Proof of Proposition 4.2. As mentioned above, the map V (z) defined in (4.7) is
convex and compact valued and satisfies Lipschitz conditions. Also, from Assump-tion 2.7 (see (2.21)–(2.22)) it follows that it has the following property: for any z′ ∈ Z,(v′, w′) ∈ V (z′) and any z′′ ∈ Z, there exists (v′′, w′′) ∈ V (z′′) such that
(v′ − v′′)TC(z′ − z′′) ≤ −‖z′ − z′′‖2D,(4.15)
‖w′ − w′′‖ ≤ bh‖z′ − z′′‖.(4.16)
The claim of the proposition follows now from Lemma A.2 in [18].To prove Theorem 2.8 let us choose T0 in such a way that
b1e−βT0
def
= δ < 1,(4.17)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
970 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
and let(y(t), z(t), θ(t)
)be an admissible trajectory of the systems (2.3) and (4.11)
which satisfies the initial conditions (y(0), z(0)) ∈ Y ′′×Z ′′, θ(0) = 0. By Theorem 2.6,there exists a solution
(za(t), θa(t)
)of the differential inclusion (4.10) satisfying the
initial condition(za(0), θa(0)
)=(z(0), 0
)such that∥∥z(t)− za(t)
∥∥ ≤ µ(ε, T0),∥∥θ(t)− θa(t)
∥∥ ≤ µh(ε, T0) ∀t ∈ [0, T0].(4.18)
When Theorem 2.6 is applied again, one can establish that there exists a solution(za(t), θa(t)
)of (4.10) on the interval [T0, 2T0] such that it satisfies the initial con-
ditions(za(T0), θ
a(T0))=(z(T0), θ(T0)
)and, also, such that the following estimates
are valid:∥∥z(t)− za(t)∥∥ ≤ µ(ε, T0),
∥∥θ(t)− θa(t)∥∥ ≤ µh(ε, T0) ∀t ∈ [T0, 2T0].(4.19)
It follows from Proposition 4.2 that the solution(za(t), θa(t)
)used in (4.18) can be
extended to the interval [T0, 2T0] in such a way that for any t ∈ [T0, 2T0],∥∥za(t)− za(t)∥∥ ≤ b1e
−β(t−T0)∥∥z(T0)− za(T0)
∥∥,∥∥θa(t)− θa(t)∥∥ ≤ ∥∥θ(T0)− θa(T0)
∥∥+ b2∥∥z(T0)− za(T0)
∥∥.These along with (4.19) allow us to establish that for any t ∈ [T0, 2T0],∥∥z(t)− za(t)
∥∥ ≤ µ(ε, T0) + b1e−β(t−T0)
∥∥z(T0)− za(T0)∥∥,∥∥θ(t)− θa(t)
∥∥ ≤ µh(ε, T0) +∥∥θ(T0)− θa(T0)
∥∥+ b2∥∥z(T0)− za(T0)
∥∥.Continuing in a similar fashion, one can construct a solution of (4.10) such that thefollowing inequalities are satisfied ∀t ∈ [lT0, (l + 1)T0], l = 1, 2, . . . :∥∥z(t)− za(t)
∥∥ ≤ µ(ε, T0) + b1e−β(t−lT0)
∥∥z(lT0)− za(lT0)∥∥,(4.20) ∥∥θ(t)− θa(t)
∥∥ ≤ µh(ε, T0) +∥∥θ(lT0)− θa(lT0)
∥∥+b2∥∥z(lT0)− za(lT0)∥∥.(4.21)
It follows now from (4.17) and (4.20)–(4.21) that∥∥∥z((l + 1)T0
)− za((l + 1)T0
)∥∥∥ ≤ µ(ε, T0) + δ∥∥z(lT0)− za(lT0)
∥∥,∥∥∥θ((l + 1)T0
)− θa((l + 1)T0
)∥∥∥ ≤ µh(ε, T0) +∥∥θ(lT0)− θa(lT0)
∥∥+ b2∥∥z(lT0)− za(lT0)
∥∥,which imply that
∥∥z(lT0)− za(lT0)∥∥ ≤ µ(ε, T0)
1− δ, l = 1, 2, . . . ,
∥∥θ(lT0)− θa(lT0)∥∥ ≤ l
(µh(ε, T0) +
b21− δ
µ(ε, T0)
), l = 1, 2, . . . .
These and (4.20)–(4.21) lead to statement (i) of the theorem (see also the proof ofLemma 3.2 in [18]). The proof of (ii) is similar.
Proof of Proposition 2.9. Let γ ∈ Γ(z). By (2.8), there exist sequences εk, Sk,and γk ∈ Γ(z, εk, Sk, y(0)) such that (εk, (Sk)−1) → 0 and γk → γ as k tends toinfinity. The latter convergence is in the metric consistent with the weak convergencetopology of P(U × Y ) and, hence, it implies in particular that
limk→∞
∫U×Y
(φ′(yi))T fi(u, y, z)γk(du, dy) =
∫U×Y
(φ′(yi))T fi(u, y, z)γ(du, dy).(4.22)
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 971
According to the definition of the set Γ(z, εk, Sk, y(0)) (see (2.7)) and the fact thatγk ∈ Γ(z, εk, Sk, y(0)), there exists an admissible control uk(τ) and the correspondingtrajectory yk(τ) of system (2.4) such that
∫U×Y
(φ′(yi))T fi(u, y, z)γk(du, dy) =1
Sk
∫ Sk
0
(φ′(yki (τ)))T fi(uk(τ), yk(τ), z)dτ.
The second integral is apparently equal toφ(yki (Sk))−φ(yki (0))
Sk , which tends to zero as Sk
tends to infinity (since, by Assumption 2.1, the solutions of (2.4) stay in the boundedarea). This and (4.22) imply the validity of the proposition.
4.2. Proofs for section 3.Lemma 4.3. The map ψ(p) defined by (3.4) is continuous. That is, ψ(pl) con-
verges to ψ(p) in the weak convergence topology of P(U × Y ×Z) if pl converges to pin the weak convergence topology of P(P(U × Y )× Z
).
Proof of Lemma 4.3. Let h(u, y, z) : U × Y × Z → R1 be a continuous function.
Then
limpl→p
∫h(u, y, z)ψ(pl)(du, dy, dz) = lim
pl→p
∫ (∫h(u, y, z)γ(du, dy)
)pl(dγ, dz)
=
∫ (∫h(u, y, z)γ(du, dy)
)p(dγ, dz) =
∫h(u, y, z)ψ(p)(du, dy, dz),
where it is taken into account that, because h(u, y, z) is continuous, it follows thatthe function h(γ, z) defined by (2.20) is continuous as well. Since the last equalitiesare valid for any continuous h(·), it follows that limpl→p ψ(pl) = ψ(p).
Proof of Lemma 3.5. Let the metric ρ of P(W ) be defined as follows:
ρ(γ′, γ′′) =∞∑k=0
1
2k|〈γ′, qk〉 − 〈γ′′, qk〉|
1 + |〈γ′, qk〉 − 〈γ′′, qk〉| ∀γ′, γ′′ ∈ P(W )(4.23)
where qk : W → R1, k = 0, 1, . . . , is a sequence of Lipschitz continuous functions which
is dense in the space of continuous functions C(W ) and 〈γ, qk〉 =∫Wqk(w)γ(dw). Note
that this metric is consistent with the weak convergence topology of P(W ). Define
ν(α)def
= supβ∈B
supγ∈Γ1(α,β)
ρ(γ,Γ2(α, β)
)(4.24)
and show that ν(α) tends to zero as α tends to α0. Assume that it does not. Thenthere exists a number δ > 0 and sequences (αl, βl) ∈ A × B, γl ∈ Γ1(αl, βl), l =1, 2, . . . , such that liml→∞ αl = α0 and ρ(γl, γ) ≥ δ ∀ γ ∈ Γ2(α, β). That is,
K . Assume that the norm of a vector in (2.12)is defined as the sum of the absolute values of its components. Then, by (3.30), onecan rewrite (4.26) in the form
d(vl, v) ≥ δ
2∀v ∈ V 2
h (αl, βl), where vldef
=
∫W
h(w)γl(dw) ∈ V 1h (αl, βl).
Hence, d(vl, V 2
h (αl, βl)) ≥ δ
2 , l = 1, 2, . . . , which contradicts (3.29) and thus provesthe lemma.
Proof of Theorem 3.3(i). Let h(γ, z) : P(U × Y ) × Z → Rj , j = 1, 2, . . . , be an
arbitrary Lipschitz continuous vector function. That is,
Consider a set-valued map V (z) defined by (4.7) with h(γ, z) as above. Note thatthis map is not necessarily convex valued since h(γ, z) may not be represented as theintegral (2.20). By (2.14), (4.27), and (2.22) (see Assumption 2.7), it satisfies Lipschitzconditions (4.9). Hence, by the relaxation theorem (see, e.g., [7, Theorem 10.4.4,p. 402]), the set of solutions of the differential inclusion (4.10) is dense in the set ofsolutions of the differential inclusion
˙z(t) ∈ coV(z(t)
),(4.28)
where coV (z) is the convex hull of V (z).By Corollary 3.7, to establish the existence of a convex and compact set Γ ⊂
P(P(U×Y )×Z) satisfying (3.12) it is enough to show that for any Lipschitz continuoush(γ, z) : P(U × Y ) × Z → R
j , j = 1, 2, . . . , there exist a convex and compact setVh ⊂ R
j and a function µh(T ) such that
dH
(Vh(T, z(0)
), Vh
)≤ µh(T ) ∀z(0) ∈ Z ′′, lim
T→∞µh(T ) = 0,(4.29)
where
Vh(T, z(0)
)=
⋃p∈Γ(T,z(0))
∫P(U×Y )×Z
h(γ, z)p(dγ, dz)
(4.30)
=⋃
(γ(·),z(·))
1
T
∫ T
0
h(γ(t), z(t)
)dt
,
with the second union being taken over all admissible pairs of averaged system (2.15).The closure of the set (4.30), clVh
(T, z(0)
), allows also the representations
clVh(T, z(0)
)= cl
⋃z(·)
θ(T )
T
=⋃z(·)
θ(T )
T
,(4.31)
where the first union is taken over the solutions of (4.10) and the second over thesolutions of (4.28), which satisfy the initial conditions z(0) = (z(0), 0).
As in the proof of Proposition 4.2, from Assumption 2.7 it follows that for anyz′ ∈ Z, (v′, w′) ∈ V (z′), and z′′ ∈ Z, there exists (v′′, w′′) ∈ V (z′′) such that(4.15)–(4.16) are satisfied. It can be verified that the map coV (z) has a similarproperty. That is, for any z′ ∈ Z, (v′, w′) ∈ coV (z′), and z′′ ∈ Z, there exists
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
LIMIT OCCUPATIONAL MEASURES SETS AND AVERAGING 973
(v′′, w′′) ∈ coV (z′′) such that (4.15)–(4.16) are satisfied. As with Proposition 4.2, thisallows us to establish that, given a solution
(z1(t), θ1(t)
)of the differential inclusion
(4.28) satisfying the initial condition(z1(0), θ1(0)
)= (z1, θ1) ∈ Z ′ × R
j and a vec-
tor (z2, θ2) ∈ Z ′ × Rj , there exists a solution
(z2(t), θ2(t)
)of (4.28) which satisfies
the initial condition(z2(0), θ2(0)
)= (z2, θ2) such that estimates (4.13)–(4.14) will be
valid.It follows from (4.14) that
dH
(clVh(T, z
1), clVh(T, z2)) ≤ b2T
−1 ∀zi ∈ Z ′, i = 1, 2, ∀T ≥ 0.(4.32)
Now applying results from [15] or [19, Proposition 3.2], one can establish the existenceof a convex and compact set Vh and a function µh(T ) = O(T−1/2) which satisfy (4.29).This completes the proof of the theorem.
REFERENCES
[1] O. Alvarez and M. Bardi, Viscosity solutions methods for singular perturbations in deter-ministic and stochastic controls, SIAM J. Control Optim., 40 (2001), pp. 1159–1188.
[2] Z. Artstein, Invariant measures of differential inclusions, J. Differential Equations, 152(1999), pp. 289–307.
[3] Z. Artstein, The chattering limit of singularly perturbed optimal control problems, in Pro-ceedings of CDC-2000, Control and Decision Conference, Sydney, 2000, pp. 564–569
[4] Z. Artstein and V. Gaitsgory, Tracking fast trajectories along a slow dynamics: A singularperturbations approach, SIAM J. Control Optim., 35 (1997), pp. 1487–1507.
[5] Z. Artstein and A. Vigodner, Singularly perturbed ordinary differential equations with dy-namical limits, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), pp. 541–569.
[6] J.-P. Aubin, Viability Theory, Birkhauser, Boston, 1991.[7] J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Birkhauser, Boston, 1990.[8] F. Bagagiolo and M. Bardi, Singular perturbation of a finite horizon problem with state-space
constraints, SIAM J. Control Optim., 36 (1998), pp. 2040–2060.[9] A. Bensoussan, Perturbations Methods in Optimal Control Problems, Wiley, New York, 1984.
[10] D.P. Bertsekas and S.E. Shreve, Stochastic Optimal Control: The Discrete Time Case,Academic Press, New York, 1978.
[11] T. Donchev and I. Slavov Averaging method for one-sided Lipschitz differential inclusionswith generalized solutions, SIAM J. Control Optim., 37 (1999), pp. 1600–1613.
[12] A. Dontchev, Time-scale decomposition of the reachable set of constrained linear systems,Math. Control Signals Systems, 5 (1992), pp. 327–340.
[13] A. Dontchev, T. Donchev, and I. Slavov, A Tichonov-type theorem for singularly perturbeddifferential inclusions, Nonlinear Analysis, 26 (1996), pp. 1547–1554.
[14] O.P. Filatov and M.M. Hapaev, Averaging of Systems of Differential Inclusions, MoscowUniversity Publishing House, Moscow, 1998 (in Russian).
[15] V. Gaitsgory, Use of the averaging method in control problems, Differential Equations, 22(1986), pp. 1290–1299 (translated from Russian).
[16] V. Gaitsgory, Suboptimization of singularly perturbed control systems, SIAM J. Control Op-tim., 30 (1992), pp. 1228–1249.
[17] V. Gaitsgory and A. Leizarowitz, Limit occupational measures set for a control systemand averaging of singularly perturbed control systems, J. Math. Anal. Appl., 233 (1999),pp. 461–475.
[18] V. Gaitsgory and M.-T. Nguyen, Averaging of three time scale singularly perturbed controlsystems, Systems Control Lett., 42 (2001), pp. 395–403.
[19] G. Grammel, Averaging of singularly perturbed systems, Nonlinear Analysis, 28 (1997),pp. 1855–1865.
[20] G. Grammel, Order Reduction for Nonlinear Systems by Re-iterated Averaging, in Proceedingsof the IEEE Singapore International Symposium on Control Theory and Applications,Singapore, 1997, pp. 1855–1865.
[21] P.V. Kokotovic, Applications of singular perturbations techniques to control problems, SIAMRev., 26 (1984), pp. 501–550.
[22] P.V. Kokotovic, H.K. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control:Analysis and Design, Academic Press, London, 1986.
Dow
nloa
ded
12/2
1/13
to 1
34.9
9.12
8.41
. Red
istr
ibut
ion
subj
ect t
o SI
AM
lice
nse
or c
opyr
ight
; see
http
://w
ww
.sia
m.o
rg/jo
urna
ls/o
jsa.
php
974 VLADIMIR GAITSGORY AND MINH-TUAN NGUYEN
[23] H. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control andFiltering Problems, Birkhauser, Boston, 1990.
[24] R.E. O’Malley, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.[25] R.E. O’Malley, Jr., Singular perturbations and optimal control, in Mathematical Control
[26] Z. Pan and T. Basar, Multi-time scale zero-sum differential games with perfect state mea-surements, Dynam. Control, 5 (1995), pp. 7–29.
[27] V.A. Plotnikov, Differential Equations with Multivalued Right-Hand Sides: AsymptoticMethods, AstroPrint, Odessa, 1999 (in Russian).
[28] M. Quincampoix and H. Zhang, Singular perturbations in non-linear optimal control systems,Differential Integral Equations, 8 (1995), pp. 931–944.
[29] M. Quincampoix and M. Veliov, Open-loop viable control under uncertain initial state in-formation, Set-Valued Anal., 7 (1999), pp. 55–87.
[30] A.B. Vasil’eva and A.F. Butuzov, Asymptotic Expansions of Solutions of Singularly Per-turbed Equations, Nauka, Moscow, 1973 (in Russian).
[31] V. Veliov, A generalization of Tikhonov theorem for singularly perturbed differential inclu-sions, J. Dynam. Control Systems, 3 (1997), pp. 291–319.
[32] A. Vigodner, Limits of singularly perturbed control problems with statistical dynamics of fastmotions, SIAM J. Control Optim., 35 (1997), pp. 1–28.
[33] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, NewYork, 1972.
[34] G.G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications. A SingularPerturbations Approach, Springer-Verlag, New York, 1998.
