LOCAL AND GLOBAL NULL CONTROLLABILITY OF
TIME VARYING LINEAR CONTROL SYSTEMS
F. COLONIUS AND R. JOHNSON
Abstract. For linear control systems with coecients determined by a dynamical system null controllability is discussed. If uniform local null controllability holds, and if the Lyapounov exponents of the homo- geneous equation are all non-positive, then the system is globally null controllable for almost all paths of the dynamical system. Even if some Lyapounov exponents are positive, an irreducibility assumption implies that, for a dense set of paths, the system is globally null controllable.
1. Introduction
The purpose of this paper is to study the local and global null controlla- bility of the family of linear control systems
x
0=A(Tt(!))x+B(Tt(!))u
x2Rn u2U Rm U compact convex (1.1) where the coe cients A(Tt(!)) 2 Rnn and B(Tt(!)) 2 Rnm are deter- mined by a dynamical system Tt : ! (t 2 R) on a compact metric space . We assume that an ergodic measure on is given and that the results of interest may only hold on a set f!g with -probability 1. The variables x and u represent, respectively, the state and control of the sys- tem. The control function u() will be assumed to be a measurable map of Rinto U. The linear control system is driven by the dynamical system
fTtg which is not inuenced by the control system. It can be interpreted in a variety of ways. For example, if the coe cients are almost periodic functions oft, a classical construction in the theory of dierential equations yields the system fTtg as the shift on the closure of the set of translates of the coe cient functions in the space of continuous functions.
We devote special attention to control problems where only the ampli- tudes of time varying perturbations in the coe cients are known:
x
0=A(w(t))x+B(w(t))u
x2Rn u2U Rm w2W Rk:
(1.2) Here U is a compact convex set containing the origin. In regard to the values of the coe cients Aand B, we assume thatf(A(w)B(w))jw2Wg is bounded inRnnRmn. Systems of the form (1.2) can be reformulated as
F. Colonius, Universitat Augsburg, Germany. Research supported by the Deutsche Forschungsgemeinschaft.
R. Johnson, Universita di Firenze, Italy. Research supported by the Ministero delle Universita e delle Ricerche Scientiche e Tecnologiche.
Received by the journal August 23, 1996. Accepted for publication July 22, 1997.
c Societe de Mathematiques Appliquees et Industrielles. Typeset by LATEX.
systems of type (1.1) see Section 2 below. The quantitywcan be interpreted as a background noise which inuences a reference control system
x
0=A(w0)x+B(w0)u
via the perturbation terms A(w);A(w0) and B(w);B(w0). We can re- gard w() as a typical path of a stationary ergodic process fwtg with val- ues in W. The stability properties of the homogeneous equations x0 =
A(w(t))x w()2W, have been studied in 3], see also the survey 2]. Here the sets of Lyapounov exponents and the corresponding initial points have been characterized.
Our goal in the present paper is to prove local controllability results which are uniform in the pathw(), then to prove global controllability statements which hold for almost all paths w() (seex 2 for the denitions of local and global null controllability). Such results have been proved previously by Johnson and Nerurkar 7, 8] when the stationary ergodic process satises a uniform recurrence assumption. The main point here is to relax this assumption. We will instead impose the hypothesis that the hull (see x 2) of the stationary ergodic process fwtg is the topological support of an ergodic measure . This hypothesis is very natural in the present context:
it leads to no restriction on the class of systems (1.2) which we can study, and avoids the uniform recurrence assumption.
Under this hypothesis we will prove the following results.
(i) Uniform local null controllability holds over if it holds for at least one point in each minimal subset of .
(ii) If uniform local null controllability holds, and if the Lyapounov expo- nents of the homogeneous equationx0=A(w(t))xare all non-positive, then (1.2) is globally null controllable for almost all pathsw().
(iii) Even if some Lyapounov exponents are positive, an \irreducibility"
assumption implies that, for a dense set of paths w(), the process (1.2) is globally null controllable.
The paper is organized as follows. Inx2 we repeat some basic denitions, including those of local and global null controllability. We also review the
\randomization" procedure by which the stationary ergodic process fw(t)g is identied with a topological dynamical system. This construction permits the application to (1.2) of various techniques of topological dynamics. In fact such tools will be applied in x 3, where we study (uniform) local null controllability, and in x4, where global null controllability is treated.
We wish to note that Baranova 1] has published a proof of our Theo- rem 4.5 regarding the global null controllability of (1.2) when the Lyapounov exponents of the homogeneous equation are non-positive.
2. Preliminaries
We begin by considering the control process (1.2) for a xed measurable function w : R ! W, so that the coe cients A(w(t)) and B(w(t)) are bounded measurable functions of t.
Definition 2.1. (a) A point x0 2Rncan be steered to zero in timeT >0 by the process (1.2) if there is a measurable control function u: 0T]!U
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such that the solution of the initial value problem
x
0 = A(w(t))x+B(w(t))u
x(0) = x0 satises x(T) = 0.
(b) The process (1.2) is said to be locally null controllable if there exists a neighborhood V of zero in Rnsuch that eachx02V can be steered to zero in some nite time T >0.
Remark 2.2. In Denition 2.1 (b) the timeT may a priori be a function of
x
0
2V : T =T(x0). However the convexity ofU and the fact that 02U can be used to show that, if 2.1 (b) holds, then there is a neighborhood V1 of 0 in Rnand a xed timeT1>0 such that eachx0 2V1 can be steered to zero in some nite time T1. See, e.g., 7, Corollary 2.7].
Next we briey discuss the randomization process see e.g. 6] for more details. For each path w() of the stationary ergodic process, consider the pair
!= (A(w())B(w()))2L1(RRnn)L1(RRnm): Dene = clsf! j w() is a path of fWtgg
where the closure is taken with respect to the product of the weak-* topolo- gies on L1(RRnn) resp. L1(RRnm). It follows from our assumption thatfA(w)B(w) j w2Wgis bounded inRnnRnmthat is compact.
Moreover is invariant under shift ow dened by
(Tt!)(s) =!(t+s) (!2 ts2R):
The pair (fTtg) denes a topological ow, or continuous dynamical sys- tem, because the map T : R!: (!t)!Tt! is continuous.
Next let W be the path space of the stationary ergodic process fWtg, and let be the corresponding probability measure on W. Let i:W !:
w()!! be the natural map, and letbe the image measure on . Then is a Radon measure on which is ergodic with respect to the ow (fTtg) 6].Convention 2.3. We redene to be the topological support of the mea- sure .
This convention clearly entails no loss of generality if one is interested in properties of the control process (1.2) which are valid for almost all paths
w().
We now consider the family of control processes
x
0=A(Tt(!))x+B(Tt(!))u (2:1)!
where ! ranges over . Here we have abused notation and written
A(Tt(!))A(w(t)) B(Tt(!))B(w(t)):
Of course one can write down bounded Borel functions A : ! Rnn,
B : ! Rnm such that, for each ! 2 , equation (2.1)! coincides with (1.1) with path w(). However for present purposes we can simply identify
t!A(Tt(!)) resp. t!B(Tt(!)) with the rst resp. the second component of !2L1(RRnn)L1(RRnm).
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We emphasize that, in what follows, the only hypotheses on which we will need are: (i) that it is weak-compact and translation invariant in
L
1(RRnn)L1(RRnm) (ii) that it is the topological support of the ergodic measure .
Definition 2.4. The family of control processesf(2.1)! j!2gis said to be uniformly locally null controllable if there is aT >0 and a neighborhood
V of 02Rnsuch that eachx02V can be steered to zero in time T by the process (2.1)! (!2).
In x 3 we will study the concept of uniform local null controllability. We will use a theorem of 7] which we now recall.
Definition 2.5. The ow (fTtg) is called minimal or uniformly recurrent if every orbit fTt(!)jt2Rgis dense in (! 2).
An equivalent denition is that the only nonempty compact invariant subset of is itself. See 5] for a detailed discussion of the theory of minimal sets. It is easy to see that, since is the topological support of the ergodic measure , the orbitfTt(!)jt2Rgis dense in for -a.e. ! 2. However minimality is a much more restrictive condition.
The result of 7] which we will use is the following (Theorem 2.10 of 7]).
Theorem 2.6. Suppose that the ow(fTtg) is minimal. Suppose that the process (2.1)!0 is locally null controllable for a single point !0 2 . Then the family f(2.1)!j! 2g is uniformly locally null controllable.
3. Local Null Controllability
We study the family of control processes (2.1)!where is the topological support of an ergodic measure . Our goal is to generalize Theorem 2.6 to this situation.
Our starting point is a result of Barmish-Schmitendorf, which also began developments in 7]. For a subsetUofRmthe support functionHU :Rm!R is dened by
HU() = supf<u>ju2Ug
where<>denotes the Euclidean inner product onRm. Let A:R!Rnn and B :R!Rnm be locally integrable matrix functions, and consider the control process
x
0=A(t)x+B(t)u: (3.1) Here the control function u :R! U is any measurable U-valued function.
The set U is assumed to be compact and to contain 02Rm. Theorem 3.1. 13] The following are equivalent.
(a) The process (3.1) is locally null controllable.
(b) There exists >0 such that
Z
1
0
HU(B(s)z(s))ds
whereB is the transpose ofB andz(t) is any solution of the adjoint systemz0=;A(t)z such that kz(0)k= 1.
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Remark 3.2. It follows from the Barmish-Schmitendorf proof of Theorem 3.1 that, if > 0 is a number for which 3.1 (b) holds, then each x0 2 Rn with kx0k<=2 can be steered to zero by the process (3.1).
Now we can state and prove the main result of this section. As in the rst two sections of the paper, U Rm is a compact convex subset containing the origin. We consider the family of control processes
x
0=A(Tt(!))x+B(Tt(!))u (3:2)!
where ! ranges over .
Theorem 3.3. Suppose that each minimal subset M contains a point
!
0 such that the process (3.2)!0 is locally null controllable. Then the family
f(3:2)! j!2g of control processes is uniformly locally null controllable.
Proof. First note that, by Theorem 2.8, the conclusion of Theorem 3.3 holds over every minimal subset M .
Suppose now for contradiction that there exists ! 2 such that (3.2)!
is not locally null controllable . By Theorem 3.1 there exists a sequence
fz
n(0)g of vectors of norm 1 in Rn such that the corresponding solutions
zn(t) of the adjoint equation z0=;A(Tt(!))zsatisfy:
Z
1
0
HU(B(Tt(!))zn(t))dt< 1
n :
Passing to a subsequence and using Fatou's Lemma (note thatHU 0 since 0 2 U), we conclude that there exists z(0) 2 Rn of norm 1 such that, if
z
(t) is the corresponding solution of the adjoint equation:
Z
1
0
HU(B(Tt(!))z(t))dt= 0:
Next let tn! 1 be a sequence such that Ttn(!) converges to a point !e in a minimal subset of . Elementary arguments of topological dynamics 5] show that such a sequence exists. We have
0 =
Z
1
0
HU(B(Tt(!))z(t))dt
=
Z tn
0
HU(B(Tt(!))z(t))dt+
Z
1
tn HU(B(Tt(!))z(t))dt
Z
1
0
HU(B(Tt+tn(!))z(t+tn))dt:
Hence writing !n=Ttn(!):
0 =Z 1
0
HU(B(Tt(!n))ezn(t))dt (3.3) where zen(t) is the solution of the adjoint system z0= ;A(Tt(!n))z which satises ezn(0) = z(tn)
kz (tn)k.
Passing to a subsequence, we can assume that ezn(0) converges to a point
e
z(0) 2 Rn of norm 1. Let ze(t) be the solution of the adjoint equation
z
0 = ;A(Tt(!e))z with initial condition ze(0). We cannot apply Fubini's theorem directly to (3.3) to conclude that
Z
1
0
HU(B(Tt(e!))ez(t))dt= 0
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because it is not clear that HU(B(Tt(!n))ezn(t)) converges pointwise to
HU(B(Tt(!e))ez(t)). However we can apply the theory of measurable selec- tions 4]. Write bn(t) =B(Tt(!n)),eb(t) =B(Tt(e!)), and choose T >0. We have rst of all:
0 =
Z T
0
HU(bn(t)zen(t))dt=
Z T
0
sup2U<b
n(t)zen(t)> dt
usup
2U Z T
0
<u(t)bn(t)ezn(t)> dt
where U is the set of all measurable mappings u: 0T]!U. Hence
Z T
0
<u(t)bn(t)ezn(t)> dt0 (u2U):
Now use the compactness of U, the uniform convergence ofzen(t) toze(t) on 0T], the uniform boundedness offbng, and the weak-* convergence of
b
nto b to obtain 0
Z T
0
<u(t)bn(t)ezn(t)> dt !
Z T
0
<u(t)eb(t)ez(t)> dt for eachu2U. Hence using measurable selection theory 4]:
0 supu
2U Z T
0
<u(t)eb(t)ez(t)> dt=
Z T
0
sup2U<
e
b
(t)ez(t)> dt
=
Z T
0
HU(eb(t)ez(t))dt0 and since this holds for everyT >0 we get
Z
1
0
HU(B(Tt(!e))ez(t))dt= 0:
This contradicts Theorem 3.1 and the Theorem 2.10 of 7] referred to above.
Hence (3.2)! is locally null controllable for each! 2.
We remark that the proof of Theorem 2.10 in 7] tacitly assumes that
B is continuous as a function of !. To generalize the proof to the case of the measurable B considered here requires only the use of the measurable selection theorem as just illustrated.
We nish the proof of Theorem 3.3 by proving the uniform local null controllability. For this we use Remarks 2.2 and 3.2: it su ces to nd >0 such that, for each ! 2 and each solution z(t) of the adjoint system
z
0=;A(Tt(!))zwithkz(0)k= 1, there holds
Z
1
0
HU(B(Tt(!))z(t))dt:
Assume for contradiction that there are sequences fzn(0)g Rn of norm 1 and f!ng such that R01HU(B(Tt(!n))z(t))dt < 1=n. Passing to convergent subsequences and using the measurable selection theorem as above, we can nd !e2 and ze(0)2Rnof norm 1 such that, if ze(t) is the corresponding solution of z0=;A(Tt(!e))z, then
Z
1
0
HU(B(Tt(!e))ez(t))dt= 0:
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But then (3.2)!eis not locally null controllable, and this contradicts the rst part of the proof. So Theorem 3.3 is veried.
4. Global Null Controllability Consider for a moment the initial value problem
x
0 = A(t)x x2Rn
x(0) = x06= 0
where A(t) is a locally-integrable matrix function dened on R. Letting
x(tx0) be the corresponding solution, the Lyapounov exponent is given by
(x0) = limt
!1
1
t
lnkx(tx0)k:
Of course this denition can be applied to each equation in the ergodic family
x
0=A(Tt(!))x (4.1)
where!ranges over . The well-known theorem of Oseledec 10] states that, for -a.e. ! 2, there are nitely many Lyapounov exponents 1:::k
where kn and the exponents do not depend on!. Furthermore there is a measurable decomposition of the product bundle Rn
Rn=V1Vk
into invariant measurable subbundles V1:::Vk where (!x0) 2 Vr if and only if either x0 = 0 or limt!1t;1lnkx(tx0!)k = r. See 10, 9] for details.
Next we review a basic construction which will be useful in developing our theory. Write (!t) for thennmatrix solution of (4.1) which is the
nnidentity at t= 0. We note that, by a convenient smoothing trick due to Ellis 5], there is a change of variables x = P1(Tt(!))y with continuous invertible coe cient function P1 : ! Rnn such that the transformed coe cient matrix
P
;1
1 AP
1
;P
;1
1 dP
1
is a continuous function of!. The introduction of such a change of variablesdt
clearly has no eect on the local or global controllability properties of the processes (3.2)!. So we can and will assume WLOG thatA: !Rnnis a continuous function. (HoweverB cannot be made continuous in this way).
Turning to the promised construction, let g be an element of the orthog- onal group O(n). We can use the Gram-Schmidt procedure to write
(!t) =G(!gt)E(!gt)
whereG(!gt)2O(n) andE(!gt)is a triangular matrix with zeros above the main diagonal and positive diagonal entries. Writez= (!g)2O(n).
Using the \cocycle identity" (!ts) = (Tt(!)s) (!t), one checks that the maps Tbt : z = (!g) ! (Tt(!)G(!gt)) dene a ow on O(n).
Furthermore the continuous matrix function
e(z) = d
dt E(zt)
t=0
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is lower triangular, and E(zt) is the nnmatrix solution of
x
0=e(Tbt(z))x (4.2) which satisesE(z0) =I. See 9, 8].
Next let : O(n) ! : (!g) ! ! be the natural projection. Let
!
0
2 be a point whose orbit fTt(!0) j t 2 Rg is dense in , and let
g
0
2 O(n). Then the orbit closure Z1 = clsfTbt(!0g0) j t 2 Rg has the property that (Z1) = . Let be a Radon probability measure on Z1 which is ergodic with respect to fTbtg and whose image () = such a measure always exists. Dene Z to be the topological support of.
We can lift the family (3.2)! to a family of processes on Z by simply dening Ab(z) =A((z)),Bb(z) =B((z)). Dene P :Z !O(n): P(z) =g if z= (!g). Introduce the orthogonal change of variables
x=P(Tb(z))y in the family (3.2)! lifted toZ. The result is
y
0=e(Tbt(z))y+b(Tb(z))u (4.3) wheree() is the function introduced above andb(z) =P;1(z)Bb(z). Clearly the local/global null controllability properties of the process (4.3) are the same as those of the process (3.2)! with !=(z).
We proceed to analyze the family of processes (4.3). The rst step is to state the following ergodic theoretic result, proved by Schneiberg 14].
Theorem 4.1. Letf 2L1(Z ) be a function such thatRZfd= 0 and let
> 0. Then for -a.e. z 2Z, there is a sequence tk !1 (which depends on z) such that
Z tk
0
f(Tbs(z))ds
<:
We will use the following variant of Theorem 4.1.
Proposition 4.2. Let f 2 L1(Z ) be a function such that RZfd 0.
LetZe be the set of z2Z such that given>0,T >0, andk1, there are numbers Qj>T such that, ifS0 = 0 and Sj =Xj
i=1
Qi, then
Z Qj
0
f(Tbs(TbSj;1(z)))ds< (1jk): Then (Ze) = 1.
Proof. The statement of the Proposition follows from the Birkho ergodic theorem if RZfd < 0. If RZfd = 0, we x , T and k, and we use Theorem 4.1 to choose Qj > T such that R0Sjf(Ts(z))ds < =2 for all 1j k here z 2Ze and Sj =Q1++Qj. This implies the statement of Proposition 4.2 for each z2Ze and completes the proof.
Now we return to the family of processes (3.2)! and to the Lyapounov exponents 1:::k of the ergodic family (4.1) recall that these exponents are constant-a.e. We will show that, if these exponents are all non-positive,
Esaim: Cocv, November 1997, Vol. 2, pp. 329{341
and if the family (3.2)! is uniformly locally null controllable, then (3.2)! is globally null controllable for-a.e. ! 2. That is, we will nd a set with() = 1 such that, if!2, then each vectorx0 2Rncan be steered to zero by (3.2)!. (The timeT which it takes to steerx0to zero may depend on ! and x0.)
First let eij (1ijn) be the entries of the matrix function e:Z !
Rnn, thus eij = 0 if i < j. It is known (e.g., 9]) that the Lyapounov exponents 1:::k are given by the mean values of the diagonal elements
eii of ewith respect to therefore
Z
Zeii(z)d(z)0 (1in):
Let Zb be the set ofz2Z for which the conclusion of Proposition 4.2 holds for each 1in, and letZ =\fTbn(Zb)jn= 123:::g. Then(Z) = 1, and therefore = (Z) is -measurable and () = 1. We will show that (3.2)! is globally null controllable for each !2.
We prove a preliminary steering lemma. Let us say that a vectorx0 2Rn can be!-steered to another vectorx1 2Rnin timeT if there is a measurable control u: 0T]! U so that the solutionx(t) of
x
0 = A(Tt(!))x+B(Tt(!))u
x(0) = x1 satises x(T) =x2.
Lemma 4.3. Let ! 2 . If a vector x0 2Rn can be !-steered to a vector
x
1, in time T1, and ifx1 can be Tt1(!)-steered to x2 in timeT2, thenx0 can be !-steered to x2 in time T1+T2.
Proof. Letu1 and u2 be admissible controls which steer x0 tox1 and x1 to
x
2 respectively. Then the control
u(t) =
(
u
1(t) 0t<T1
u
2(t;T1) T1 tT1+T2 will !-steer x0 tox2 in timeT1+T2.
We now prove
Lemma 4.4. Suppose that the family of control processesf(3.2)!j! 2gis uniformly locally null controllable (see x 3). Let z2Z. There exists >0 such that, for each ay2:::yn2Rwithjaj>2, the vector (ay2:::yn)t can be z-steered to a vector(bv2:::vn)t with jbj<jaj;.
Proof. It is clear that the family of control processes (4.3) is uniformly locally null controllable. Choose >0 such that each vectory of length kyk <3 can be z-steered to 0 for each z2Z.
Fix z 2 Z, and let "z(t) be the matrix solution of the homogeneous system (4.2)
y
0=e(Tbt(z))y=
0
B
@ e
11(Tbt(z))
... 0
enn(Tbt(z))
1
C
A y
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