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ENTROPY OF CONTROLLED INVARIANT SUBSPACES�
FRITZ COLONIUSz AND UWE HELMKE{
Abstract. For continuous-time linear control systems invariance entropy of controlled invariantsubspaces is introduced. It is shown that it coincides with a variant of topological entropy of linear�ows. Upper bounds in terms of eigenvalues are derived.
August 25, 2011Key words. invariance entropy, topological entropy, (A,B)-invariant subspaces
AMS subject classi�cations. 94A17, 37B40, 93C15
1. Introduction. Control of systems through digital communication channelsand estimation of the required data rates have been considered since more than twentyyears. The main motivation for this circle of ideas comes from the increasing needsof controlling systems with communication constraints, i.e. for systems where thestate passes through a communication channel and may thus be delayed, corruptedby noise or even lost, and thus in any case may not be fully available to the controller.Early contributions to the �eld are, among others, due to Wong and Brockett [30],[31]. More recently, problems in networked control system have been treated, see, e.g.,Gupta, Dana, Hespanha, Murray, Hassibi [13], or Carli and Bullo [29] for quantizedcoordination algorithms for robots. In [21], Nair, Evans, Mareels and Moran intro-duced ideas from topological dynamics into this �eld (see Katok and Hasselblatt [14]for an authoritative presentation of the mathematical theory of dynamical systems.)They introduced and studied the notions of topological feedback entropy and symboliccontrollers for the problem of stabilizing a discrete-time system with communicationconstraints at an equilibrium.
The theoretical analysis of linear systems, controlled over communication chan-nels, pro�ts from the combination of concepts from linear systems theory and infor-mation theory alike. In particular, the information theoretic notion of entropy playsa central role here as it leads to e¤ective quantitative bounds on data rates necessaryfor feedback stabilization. Up to now, such information-based investigations have notbeen connected with the constructions and the insights provided by classical geometriccontrol theory which started with the early work of Basile and Marro [2] and Wonham[28]. We also refer to the textbook Basile and Marro [3] and to Trentelman, Stoorvogeland Hautus [24] as well as the recent tutorial Marro et al. [19]. Controlled invariantsubspaces form a cornerstone of geometric control theory and play a crucial role inunderstanding controller design problems such as disturbance decoupling, �ltering,robust observer design, and high gain state feedback. We therefore expect a simi-lar impact on the analysis of linear systems that are controlled over communicationchannels.
In this paper, we begin an investigation of how geometric control design viacontrolled invariant subspaces is a¤ected by entropy estimates and associated data rate
�Supported by DFG grants Co 124/17-1 and HE 1858/12-1 within DFG Priority Program 1305�Control of Digitally Connected Dynamical Systems�.
zInstitut für Mathematik, Universität Augsburg, Universitätsstraße, 86135 Augsburg/Germany,[email protected]
{Institut für Mathematik, Universität Würzburg, Am Hubland, 97074 Würzburg/Germany,[email protected]
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constraints. As a starting point for such an investigation, we associate to any (A;B)�invariant subspace V of a linear control system a state space similarity invariant,called the invariance entropy of V , that measures how di¢ cult it is, using open loopcontrols, to keep the system in V . It is de�ned by the exponential growth rate of thenumber of controls necessary to keep the system in an arbitrarily small "-neighborhoodof V . We show that the invariance entropy is �nite for any (A;B)-invariant subspaceand derive upper bounds in terms of the sum of the eigenvalues of A with positivereal part. Sharper upper bounds are derived for speci�c classes of linear systems.In turned out that all constructions can be naturally extended for the larger classof almost (A;B)-invariant subspaces, hence the latter are considered throughout thispaper (cf. Remark 4.2.) Our approach follows and partially extends that by Coloniusand Kawan [7], where an entropy-like notion was proposed for controlled invarianceof compact subsets of the state space of general control systems. Further results aregiven in Kawan [15, 16, 17] and Colonius and Kawan [7]. This entropy notion may beregarded as a lower bound for the minimum data rate (take the logarithm with base2 instead of the natural logarithm used, for convenience, in the present paper.) Moreexplicit relations to data rates and to the topological feedback entropy introduced in[21] are also given in Kawan�s PhD thesis [18], see also Colonius [5, 6] for a discussionof the required bit rates. We defer applications of the estimates given here to a laterpaper.
The contents of this paper are as follows: Section 2 recalls basic facts on (A;B)-invariant and on almost (A;B)-invariant subspaces. Section 3 introduces a subspaceentropy for homogeneous autonomous linear di¤erential equations. Section 4 presentsthe main concept of the paper, invariance entropy of almost (A;B)-invariant sub-spaces, and shows that it coincides with the subspace entropy of the uncontrolledsystem. Final Section 5 provides an upper bound for invariance entropy in terms ofeigenvalues. In special cases, sharper upper bounds can be shown.
Notation. The distance of a point x in a normed vector space to a closed subsetM is de�ned by dist(x;M) := infy2M kx� yk.
2. Preliminaries on controlled-invariant subspaces. The purpose of thissection is to summarize some well-known de�nitions and facts from geometric controltheory, i.e., controlled and almost controlled invariant subspaces. The notion of con-trolled invariant or (A;B)�invariant subspaces was introduced by Basile and Marro[2], and Wonham [28], while almost controlled invariant subspaces were �rst intro-duced by J.C. Willems [27]; we also refer to the PhD Thesis by J. Trumpf [25] for auseful summary of basic de�nitions and facts.
Consider linear control systems in state space form
_x(t) = Ax(t) +Bu(t) (2.1)
with matrices A 2 Rn�n and B 2 Rn�m. The solutions '(t; x; u); t 2 R, of (2.1) withinitial condition '(0; x; u) = x are given by the variations-of-constants formula:
'(t; x; u) = eAtx+
Z t
0
eA(t�s)Bu(s)ds:
Recall that a subspace V is called (A;B)-invariant, if for all x 2 V there is u 2 Rmwith Ax + Bu 2 V . Equivalently, there is a matrix F 2 Rm�n, a so-called friend ofV , such that for AF := A+BF
AFV � V:2
This can be seen by choosing for a basis x1; :::; xk of V control values u1; :::; uk 2 Rmwith Axi+Bui 2 V . Then extend this to a basis of Rn and de�ne a linear map F by
Fxi = ui; for i = 1; :::; k ; and F arbitrary outside V:
This also shows that V is (A;B)-invariant if and only if it is controlled invariant, i.e.,for every x 2 V there is an open loop continuous control function u : [0;1) ! Rmwith '(t; x; u) 2 V for all t � 0. In fact, di¤erentiating the solution one �nds
V 3 d
dt'(0; x; u) = Ax+Bu(0):
Conversely, de�ne for x 2 V a control by u(t) = Fe(A+BF )tx; t � 0:More generally, a linear subspace V is called almost (A;B)-invariant, if for any
x 2 V and any " > 0 there exists a control function u(�) such that for all t � 0
dist('(t; x; u); V ) = infy2V
k'(t; x; u)� yk < ":
Almost (A;B)-invariant subspaces are of interest to study subspaces invariant underhigh gain state feedback. Thus, in general, almost (A;B)-invariant subspaces cannotbe made invariant under state feedback, so there is no friend, but they can be madealmost-invariant in the sense that for every x 2 V and any " > 0 there exists afeedback F such that for all t � 0
dist(eAF tx; V ) < ":
In order to derive explicit estimates for the entropy of controlled invariant subspacesit is useful to have explicit parametrizations of the class of all controlled invariantsubspaces. This is a di¢ cult task and we refer to e.g. [11], [25] for further information.Special types of subspaces are of special interest here. A controlled invariant subspaceV is called coasting, if V \ ImB = f0g. Equivalently, f0g is the largest controllabilitysubspace contained in V . Any controlled invariant subspace of a controllable single-input system is coasting. For an (A;B)�invariant subspace V and a friend F 2 Rm�n,the subspace V is AF -invariant. The restriction ( �A; �B) and co-restriction ( ~A; ~B),respectively, then are de�ned as
( �A; �B) = (AF jV;BjB�1V );( ~A; ~B) = (AF : Rn=V ! Rn=V; � �B : Rm=B�1V ! Rn=V ):
Note, that the co-restriction ( ~A; ~B) is controllable, whenever (A;B) is controllable,while the restriction is controllable only for a controllability subspace. Note also, that�B and ~B are both full column rank if B has full column rank. Of course, the co-restriction may well depend upon the choice of a friend F , so there are in fact manypossible co-restrictions and not just one. However, the controllability indices of theco-restrictions are all the same. It is thus a remarkable but simple fact, that for anycoasting subspace V , the co-restriction is uniquely de�ned and independent of F .
If V is an (A;B)-invariant subspace that is coasting, then there is some a-prioriinformation about the dimensions of the bounding subspaces hA jV i and ker(A;V ).Here, ker(A;V ) is de�ned as the largest invariant subspace that is contained in V ,while hA jV i denotes the smallest invariant subspace containing V . Generically, oneexpects hA jV i = Rn and ker(A;V ) = f0g, but one can be more speci�c. For simplic-ity, we focus on the single input case, i.e. m = 1.
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Lemma 2.1. Let (A; b) be controllable and let V be any (A; b)-invariant subspacethat is not A-invariant. Then every A-invariant subspace W � V satis�es W = Rn.Any generic (A; b)-invariant subspace satis�es ker(A;V ) = f0g and hA jV i = Rn.
Proof. Note, that in the single input case only, every (A; b)-invariant subspace isautomatically coasting. By duality, it su¢ ces to show for single-output systems (c; A)that hA jV i = Rn, for any tight (c; A)-invariant subspace V that is not A-invariant.In order to show this we apply the theory of polynomial models; see Fuhrmann [9],Fuhrmann and Willems [10]. Let q(z) = det(zI � A) denote the characteristic poly-nomial and Xq denote the associated polynomial model. Thus, Xq denotes the set ofpolynomials of degree < n with the module structure given by multiplication mod-ulo q. The (tight) conditioned invariant subspaces of codimension d then uniquelycorrespond to the intersection
V = Xq \ t(z)R[z]; (2.2)
�)via a unique monic polynomial t of degree d; see Fuhrmann and Helmke [11]. LetV� (V �) denote the largest (smallest) shift invariant subspace of Xd contained in V(containing V ). In this polynomial framework, the A invariant subspaces are of theform q1Xq2 � Xq, for any factorization q = q1q2. Thus q1Xq2 � Xq \ t(z)R[z] if andonly if t divides q1. In particular, t must then divide q, i.e. V must be an invariantsubspace. Now assume that V is not an invariant subspace (for the shift), i.e. t doesnot divide q. Then V does not contain any nontrivial invariant subspace. Applyingduality, this implies the lemma. But more can be said. Factor t = q1a with q1 apolynomial of degree r dividing q, q = q1q2, and a a polynomial that is coprime to q.Then q1Xq2 is the smallest invariant subspace containing V . The codimension of thissubspace is thus deg gcd fq; tg. In particular, if q and t are coprime, then V� = f0gand V � = Xq.
3. Subspace entropy of linear �ows. In this section we de�ne for a linearautonomous di¤erential equation the entropy of a linear subspace V � Rn. It is asuitable adaptation of the well-known topological entropy of the �ow associated withan autonomous linear di¤erential equation, see e.g. Walters [26, §8.4]. Later we willuse it for the uncontrolled system _x = Ax and relate it to the entropy of controlledinvariant subspaces. We would like to emphasize that the open loop control system(2.1) does not de�ne a �ow, since the control functions u(�) are time-dependent, andhence it is not covered by this de�nition.
Let V be a linear subspace of Rn. For a linear map A : Rn ! Rn, let � : R�Rn !Rn;�(t; x) := eAtx; t 2 R; x 2 Rn be the induced �ow (actually, throughout thispaper, only the semi�ow de�ned for t � 0 will be relevant.). For any compact subsetK � V and for given T; " > 0 we call R � K a (T; ";K; V ; �)-spanning set, if for allx 2 K there exists y 2 R with
max0�t�T
dist(etA(x� y); V ) < ": (3.1)
Let r(T; ";K; V ; �) denote the minimal cardinality of a (T; ";K; V ; �)-spanning set. Ifno �nite (T; ";K; V ; �)-spanning set exists, we set r(T; ";K; V ; �) =1. If there existssome (T; ";K; V ; �)-spanning set, then one �nds a �nite (T; ";K; V ; �)-spanning setusing compactness of K and continuous dependence on the initial value. Similarly,we call S � K a (T; ";K; V ; �)-separated set, if for all x 6= y in S
max0�t�T
dist(etA(x� y); V ) � ":
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The maximal cardinality of such a set is denoted by s(T; ";K; V ; �). Note that thepoints y in R (and in S) will, in general, not lead to solutions eAty which remain forall t � 0 in the "-neighborhood of V .
Definition 3.1. Let A be a linear map on Rn with associated �ow � and considera subspace V of Rn. For a compact subset K � V; we consider the exponential growthrate of r(T; ";K; V ; �) and set
hspan(";K; V ; �) := lim supT!1
1
Tln r(T; ";K; V ; �);
hspan(K;V ; �) := lim"&0
hspan(";K; V ; �);
and de�ne the entropy of V with respect to � by
h(V ; �) := supKhspan(K;V ; �);
where the supremum is taken over all compact subsets K � V .Analogously, an entropy of V can be de�ned via maximal separated sets.As usual in the context of topological entropy, one sees that, by monotonicity, the
limit for " & 0 exists (it might be in�nite.). Since all norms on a �nite dimensionalvector space are equivalent, the entropy does not depend on the norm used in (3.1).One easily sees that the subspace entropy h(V ; �) is invariant under state spacesimilarity, i.e., h(SV ;S�S�1) = h(V ; �) for S in the set GL(n;R) of invertible n�n-matrices; here S�(t; �)S�1 = SeAtS�1 = eSAS�1t; t � 0.
Remark 3.2. If we choose V = f0g condition (3.1) is trivial, since only K = f0gis allowed; furthermore, if we choose V = Rn, the distance in (3.1) is always equalto zero. In particular, the subspace entropy does not recover the usual de�nition oftopological entropy for the linear �ow �(t; x) = eAtx, where, for a compact subsetK � Rn, spanning sets are de�ned as minimal sets R such that for all x 2 K thereexists y 2 R with
max0�t�T
d(etAx; eAty) = max0�t�T
etA(x� y) < ":Equivalently, on can use the number stop(T; ";K; �) of elements in maximal separatedsets. For the topological entropy of linear �ows, Bowen [4] could show
htop(�) := supKhtop(K; �) =
nXi=1
max(0;Re�i);
where �1; :::; �n denote the eigenvalues of A; see also Walters [26, Theorem 8.14] andMatveev and Savkin [20, Theorem 2.4.2] for proofs.
The de�nitions via separated and spanning sets coincide, which easily follows fromthe next proposition (cf. Robinson [23, Lemma VIII.1.10] for similar arguments).
Proposition 3.3. Let K � V be compact and �x T; " > 0. Then
s(T; 2";K; V ; �) � r(T; ";K; V ; �) � s(T; ";K; V ; �):
Proof. Let S � K be a maximal (T; ";K; V ; �)-separated set and let x 2 K. Bymaximality of S, there is some y 2 S such that
max0�t�T
dist(etA(x� y); V ) < ":
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Therefore S is (T; ";K; V ; �)-spanning showing the second inequality. For the �rstone, consider a maximal (T; 2";K; V ; �)-separated set S and a minimal (T; ";K; V ; �)-spanning set R. We de�ne a map H : S ! R in the following way: For x 2 S there isy := H(x) 2 R with dist(etA(x� y); V ) < " for all t 2 [0; T ]. If H(x1) = H(x2) = y,then
max0�t�T
dist(etA(x1 � x2); V )
� max0�t�T
dist(etA(x1 � y); V ) + max0�t�T
dist(etA(x2 � y); V ) < 2":
Thus x1 = x2 follows. This shows that H is injective, and hence r(T; ";K; V ; �) �s(T; 2";K; V ; �):
Although this will not be used in the sequel, we describe the behavior of thisentropy notion under a semiconjugacy to the induced �ow on a quotient space..
Proposition 3.4. Let W be an A-invariant subspace for a linear map A on Rn.Then, for a subspace V of Rn the entropies of the induced �ows �(t; x) = eAtx on Rnand ��(t; �x) on the quotient space Rn=W , respectively, satisfy
h(V ; �) � h(V=W ; ��):
Proof. Let K � V be compact and for T; " > 0 consider a (T; ";K; V ; �)-spanningset R � K. Denote the projection of Rn to Rn=W by �, hence �V = V=W . Then theset �R is a (T; "; �K; �V ; ��)-spanning set. In fact, let R = fx1; :::; x`g and consider�x 2 �K for some element x 2 K. Then there exists xj 2 R with
max0�t�T
dist(etA(x� xj); V ) < ":
Denoting the map induced by A on Rn=W by �A one �nds for all t 2 [0; T ]
dist(et�A(�x� �xj); �V ) = inf
z2V
et �A(�x� �xj)� �z = inf
z2V;w2W
etA(x� xj)� z � w � dist(etA(x� xj); V )< ":
It follows that the minimal cardinality of a (T; ";K; V ; �)-spanning set is greater thanor equal to the minimal cardinality of a (T; "; �K; �V ; ��)-spanning set. Then take thelimit superior for T !1 and let " tend to 0. Finally, observe that for every compactset K1 � V=W there is a compact set K � V with �K = K1. Hence taking thesupremum over all compact K1 � V=W one obtains the assertion.
4. Entropy for controlled invariant subspaces. We now introduce the cen-tral notion of this paper, invariance entropy for almost (A;B)-invariant subspaces oflinear control system (2.1) on Rn and relate it to the subspace entropy de�ned in theprevious section.
In the following, we consider a �xed almost (A;B)-invariant subspace V of Rnwith dimV = d. Furthermore, we admit arbitrary controls in the space C([0;1);Rm)of continuous functions u : [0;1)! Rm.
Definition 4.1. For a compact subset K � V and for given T; " > 0 we calla set R � C([0;1);Rm) of control functions (T; ";K; V )-spanning if for all x0 2 Kthere is u 2 R with
dist('(t; x0; u); V ) < " for all t 2 [0; T ]:6
By rinv(T; ";K; V ) we denote the minimal cardinality of a (T; ";K; V )-spanning set.If no �nite (T; ";K; V )-spanning set exists, we set rinv(T; ";K; V ) =1.
In other words: we require for a (T; ";K; V )-spanning set R that, for every initialvalue in K, there is a control in R such that up to time T the trajectory remainsin the "-neighborhood of V . Note that, in contrast to the de�nition of the subspaceentropy for �ows, De�nition 3.1, here a number of control functions is counted, not anumber of initial values. Hence this is a notion which is intrinsic to control systems.We also note that the de�nition above di¤ers from earlier ones used for invarianceentropy (cf. [7, 8]) by the fact, that the set V whose invariance is studied here, is notcompact and that the controls are unrestricted.
The following observation shows that there are always �nite (T; ";K; V )-spanningsets of control functions.
Remark 4.2. Let K � V be compact and "; T > 0. By almost (A;B)-invarianceof V there is for every x 2 K � V a control function u with dist('(t; x; u); V ) < "for all t � 0. Hence, using continuous dependence on initial values and compactnessof K, one �nds �nitely many controls u1; :::; ur such that for every x 2 K there is ujwith dist('(t; x; uj); V ) < " for all t 2 [0; T ]. Hence rinv(T; ";K; V ) <1. This is thereason why we consider in the following almost (A;B)-invariant subspaces, not just(A;B)-invariant subspaces. It also seems, that the class of almost (A;B)-invariantsubspaces is the largest class of subspaces such that for every compact subset K andevery T; " > 0 this number is �nite.
Now we consider the exponential growth rate of rinv(T; ";K; V ) for T ! 1 andlet "! 0. The resulting invariance entropy is the main subject of the present paper.
Definition 4.3. Let V be an almost (A;B)-invariant subspace. Then, for acompact subset K � V , the invariance entropy hinv(K;V ) is de�ned by
hinv(";K; V ) := lim supT!1
1
Tln rinv(T; ";K; V ); hinv(K;V ) := lim
"&0hinv(";K; V ):
Finally, the invariance entropy of V is de�ned by
hinv(V ;A;B) := supKhinv(K;V );
where the supremum is taken over all compact subsets K � V .In the sequel, we will always use for a given underlying system (A;B) the short-
hand notation hinv(V ) for hinv(V ;A;B). Note that hinv("1;K; V ) � hinv("2;K; V )for "2 � "1. Hence the limit for " ! 0 exists (it might be in�nite.) Since all normson �nite dimensional vector spaces are equivalent, the invariance entropy of V is in-dependent of the chosen norm. We will show later that every almost (A;B)-invariantsubspace has �nite invariance entropy. It is clear by inspection, that, as the subspaceentropy h(V ; �), also the invariance entropy hinv(V ) is invariant under state spacesimilarity; i.e. hinv(SV ;SAS�1; SB) = hinv(V ;A;B) for S 2 GL(n;R).
We are interested in the problem to keep the system in the subspace V for allt � 0. Then the exponential growth rate of the required number of control functionswill give information on the di¢ culty of this task. Since, also on �nite intervals, thenumber of controls for achieving invariance need not be �nite, an appropriate math-ematical formulation requires a slightly relaxed notion, which allows for arbitrarilysmall deviations from the subspace V . The motivation to consider open-loop controlsin this context comes, in particular, from model predictive control (see, e.g., the col-lection Allgöwer and Zheng [1] and Grüne and Pannek [12]), where optimal open-loopcontrols are computed and applied on short time intervals.
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The following theorem shows that the entropy of a controlled invariant subspaceV can be characterized by the entropy of V for the corresponding uncontrolled system_x = Ax. This result will be useful in order to compute entropy bounds.
Theorem 4.4. Let V be an almost (A;B)-invariant subspace for system (2.1)and consider the invariance entropy hinv(V ) of control system (2.1) and the subspaceentropy h(V ; �) of V of the uncontrolled system �(t; x) = eAtx. Then
hinv(V ) = h(V ; �):
Proof. (i) Let K � V be compact, and �x T; " > 0. Consider a (T; ";K; V )-spanning set R = fu1; :::; u`g of controls with minimal cardinality rinv(T; ";K; V ).This means that for every x 2 K there is uj with dist('(t; x; uj); V ) < " for all t 2[0; T ]. By minimality, we can, for every uj , pick xj 2 K with dist('(t; xj ; uj); V ) < "for all t 2 [0; T ]. Then, using linearity, one �nds for all x 2 K a control uj and apoint xj 2 K such that for all t 2 [0; T ]
dist(eAtx� eAtxj ; V ) = dist('(t; x; uj)� '(t; xj ; uj); V ) < 2":
This shows that the points xj form a (T; 2";K; V ; �)-spanning set, and hence
rinv(T; ";K; V ) � r(T; 2";K; V ; �):
Letting T tend to in�nity, then " ! 0 and, �nally, taking the supremum over allcompact subsets K � V , one obtains hinv(V ) � h(V ; �).
(ii) For the converse inequality, let K be a compact subset of V and T; " > 0.Let E � K a maximal (T; ";K; V ; �)-separated set with respect to the �ow �, sayE = fy1; : : : ; ysg with s = s(T; ";K; �). Then E is also (T; ";K; V ; �)-spanning whichmeans that for all x 2 K there is j 2 f1; : : : ; sg with
maxt2[0;T ]
dist(eAtx� eAtyj ; V ) = maxt2[0;T ]
infz2V
eAtx� eAtyj � z < ":Since V is almost (A;B)-invariant, we can assign to each yj ; j 2 f1; : : : ; sg, a controlfunction uj 2 C([0;1);Rm) such that dist('(t; yj ; uj); V ) < " for all t � 0. Let R :=fu1; : : : ; usg � C([0;1);Rm). By linearity one has '(t; x; u)�'(t; y; u) = eAtx�eAtyfor all t � 0, x; y 2 Rn and u 2 C([0;1);Rm). We obtain that for every x 2 K thereis j such that
maxt2[0;T ]
dist('(t; x; uj)� '(t; yj ; uj); V ) = maxt2[0;T ]
dist(eAtx� eAtyj ; V ) < ":
Since dist('(t; yj ; uj); V ) < " for t 2 [0; T ], there is z1 2 V with k'(t; yj ; uj)� z1k < "and hence
dist('(t; x; uj); V ) = infz2V
k'(t; x; uj)� zk
� infz2V
k'(t; x; uj)� '(t; yj ; uj) + z1 � zk+ k'(t; yj ; uj)� z1k
< infz2V
k'(t; x; uj)� '(t; yj ; uj)� zk+ "
< 2":
This implies that for all x 2 K there is uj 2 R such that
maxt2[0;T ]
dist('(t; x; uj); V ) < 2":
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Hence R is (T; 2";K; V )-spanning and it follows that
rinv(T; 2";K; V ) � s(T; ";K; V ; �) for all T; " > 0;
and consequently hinv(K;V ) � hsep(K;V ; �) � h(V ; �).
5. Entropy Bounds. This section provides a general upper bound for entropy ofalmost (A;B)-invariant subspaces of control system (2.1). Using the characterizationvia the subspace entropy of the associated autonomous �ow, sharper upper boundsare obtained in special cases.
Theorem 5.1. (i) Let V be an almost (A;B)-invariant subspace for system (2.1)and consider the smallest A-invariant subspace hA jV i containing V and the largestA-invariant subspace ker(A;V ) contained in V . Then the subspace entropy of V withrespect to the �ow �(t; x) = eAtx is �nite and satis�es the inequality
h(V ; �) �X
max (0;Re�i) ; (5.1)
where summation is over the eigenvalues �i of the map induced by A on the quotientspace hA jV i = ker(A;V ).
(ii) The invariance entropy of any almost (A;B)-invariant subspace V for system(2.1) is �nite and satis�es the bound (5.1).
Proof. Assertion (ii) follows from (i) and Theorem 4.4. In order to prove (i)note �rst, that by A-invariance the restriction AjhAjV i as well as the induced mapon hA jV i = ker(A;V ) are well-de�ned. In order to show inequality (5.1), it su¢ ces toshow that it is satis�ed for hspan(K;V ; �), where K is an arbitrary compact subsetof V . Our proof then depends on Bowen�s formula for the topological entropy linear�ows, see Remark 3.2.
Consider the linear �ow �(t; x) = eAtx, � : R+0 � Rn ! Rn. By Bowen�s result
htop(�) =
nXi=1
max (0;Re�i) :
Analogously, the restriction of � to hA jV i has topological entropy given by thesum of the positive real parts of eigenvalues of �jhAjV i and the induced �ow �̂ onhA jV i = ker(A;V ) has topological entropy given by the right hand side of (5.1). Henceit su¢ ces to prove that h(V ; �) � htop(�̂).
Let K be a compact subset of V and � denote the projection of hA jV i to thequotient space hA jV i = ker(A;V ). Thus, the set �K is compact. Let T; " > 0 be givenand denote by E � �K a maximal (T; "; �K)-separated set with respect to the �ow �̂on hA jV i = ker(A;V ), say E = f�y1; : : : ; �y`g with yj 2 K and ` = stop(T; "; �K; �̂).Recall that, as noted in Remark 3.2, the topological entropy of the �ow �̂ can becharacterized by the exponential growth rate of stop(T; "; �K; �̂). Then E is also(T; "; �K)-spanning which means that for all x 2 K there is j 2 f1; : : : ; `g with
maxt2[0;T ]
dist(eAtx� eAtyj ; ker(A;V )) = maxt2[0;T ]
infz2ker(A;V )
eAtx� eAtyj � z < ":Since V is almost (A;B)-invariant, we can assign to each yj ; j 2 f1; : : : ; `g, a controlfunction uj 2 C([0;1);Rm) such that dist('(t; yj ; uj); V ) < " for all t � 0. Let R :=fu1; : : : ; u`g � C([0;1);Rm). By linearity one has '(t; x; u)�'(t; y; u) = eAtx�eAty
9
for all t � 0, x; y 2 Rn and u 2 C([0;1);Rm). We obtain that for every x 2 K thereis j such that
maxt2[0;T ]
dist('(t; x; uj)� '(t; yj ; uj); V )
� maxt2[0;T ]
dist('(t; x; uj)� '(t; yj ; uj); ker(A;V ))
= maxt2[0;T ]
dist(eAtx� eAtyj ; ker(A;V ))
< ":
Since dist('(t; yj ; uj); V ) < " for t 2 [0; T ], there is z1 2 V with k'(t; yj ; uj)� z1k < "and hence, using that V is a linear subspace, one �nds
dist('(t; x; uj); V ) = infz2V
k'(t; x; uj)� zk
� infz2V
k'(t; x; uj)� '(t; yj ; uj) + z1 � zk+ k'(t; yj ; uj)� z1k
< infz2V
k'(t; x; uj)� '(t; yj ; uj)� zk+ "
< 2":
This shows that for all x 2 K there is uj 2 R such that dist('(t; x; uj); V ) < 2" forall t 2 [0; T ]. Hence R is (T; 2";K; V )-spanning and it follows that
rinv(T; 2";K; V ) � stop(T; "; �K; �̂) for all T; " > 0:
Finally, taking the logarithm, dividing by T and letting T tend to in�nity, one obtainshspan(K;V ; �̂) � htop(�K; �̂) � htop(�̂).
The above bound is rather conservative and can be improved in several cases.We therefore turn to the computation of sharper bounds for h(V ; �) under suitablegenericity conditions on the almost controlled invariant subspace V . Here one mayexpect that starting in a neighborhood of the origin in V , only d := dimV eigenvaluesdetermine the behavior. This will be made precise below.
We begin with a few lemmas. In the sequel, e1; � � � ; en denotes the standard basisvectors of Rn. For a real diagonalizable matrix A, order the eigenvalues of A suchthat
�1 � �2 � ::: � �n: (5.2)
Lemma 5.2. Let A 2 Rn�n be diagonalizable and consider a d-dimensional sub-space V � Rn which satis�es V \W = f0g for any (n� d)-dimensional A-invariantsubspace W � Rn. We can write the Jordan representation J of A as
J = diag(�1; :::; �n);
and we abbreviate
�1 := diag(�1; :::; �d); and �2 := diag(�d+1; :::; �n); : (5.3)
Then there exist S 2 GLn(R) and G 2 R(n�d)�d with V = fSe1; � � � ; Sedg, and
S�1AS =
��1 0G �2
�: (5.4)
10
Proof. Let w1; :::; wn be a corresponding basis of eigenvectors and denote
V1 := hw1; :::; wdi and W = hwd+1; :::; wni :
By assumption on V , we have V \W = f0g and therefore the canonical projection map� : Rn ! V1 along W maps V isomorphically onto V1. Choose any basis v1; � � � ; vd ofV and extend it to a basis S1 = (v1; � � � ; vd; wd+1; � � � ; wn). Then
S�11 AS1 =
��1 0A2 �2
�;
and therefore �1 has the same eigenvalues as �1. Finally, we can transform �1 toJordan normal form by a matrix S2. Then conjugation with the matrix
S1 ��S2 00 I
�leads to (5.4).
In the situation as above, by invariance of the problem under similarity, we canassume without loss of generality, that
A =
��1 0G �2
�; V = Rd � f0g: (5.5)
Note that for any vector z =�xy
�2 Rn = V � Rn�d we have dist(z; V ) = kyk. For
x =
�x10
�; y =
�y10
�2 V we compute
etA(x� y) =�et�1 0M(t) et�2
��x1 � y10
�=
�et�1(x1 � y1)M(t)(x1 � y1)
�(5.6)
and
dist(etA(x� y); V ) = kM(t)(x1 � y1)k :
Di¤erentiating equation (5.6), one �nds that the (n� d)� d-matrix function M(t) isthe unique solution to the linear di¤erential equation _M = �2M +Get�1 with initialcondition M(0) = 0 and therefore
M(t) = et�2Z t
0
e�s�2Ges�1ds:
The formula for M(t) shows that for diagonal �2 and �1 one �nds, with ej = jthstandard basis vector and gj = Gej 2 Rn�d; j = 1; :::; d, for the jth column of M(t)
M(t)ej = et�2
Z t
0
e�s�2Ges�1ds ej = et�2
Z t
0
e�s�2Ges�jejds
= et�2Z t
0
es(�jIn�d��2)ds gj :
11
In addition to (5.2), assume that �d > �d+1 and let for k = 1; :::; n� d; j = 1; :::; d
�kj(t) :=1
�j � �d+k
h1� et(�d+k��j)
i; t � 0:
They satisfy for t > 0 the inequalities
0 < �kj(t) � (�j � �d+k)�1 � (�j � �d+1)�1:
One computes that for j = 1; :::; d
M(t)ej = et�2diag[�1j(t); :::; �n�d;j(t)] gj (5.7)
Proposition 5.3. Let A 2 Rn�n be diagonalizable and consider a d-dimensionalsubspace V � Rn which satis�es V \W = f0g for any (n�d)-dimensional A-invariantsubspace W � Rn. Assume that the eigenvalues of A satisfy
�1 � ::: � �d > �d+1 � ::: � �n:
Then the entropy h(V ; �) of V with respect to the linear �ow �(t; x) = eAtx isbounded above by the topological entropy of the �ow �1(t; x) = e�1tx where �1 =diag[�1; :::; �d]. Thus it satis�es the upper bound
h(V ; �) �dXi=1
max (0; �i) : (5.8)
Proof. Let K � V be compact. We show that for T; " > 0 any (T; ";K; �1)-spanning set R for the topological entropy of the �ow �1 is (T; c";K; V ; �)-spanningfor �, with
c := (�d � �d+1)�1 maxj=1;:::;d
kgjk :
Recall that the entropies do not depend on the norm, hence, here and in the following,we may use the 1-norm. For every x 2 K there is y 2 R such that, by formula (5.7),
kM(t)(x� y)k1 =
dXj=1
M(t)ej(xj � yj)
1
=
dXj=1
et�2diag[�1j(t); :::; �n�d;j(t)] gj(xj � yj)
1
�dXj=1
jxj � yj j diag[et�d+1�1j(t); :::; et�n�n�d;j(t)] gj 1
�dXj=1
jxj � yj j et�d+1(�j � �d+1)�1 kgjk1
� et�1(x� y)
1(�d � �d+1)�1 max
j=1;:::;dkgjk1
� c maxt2[0;T ]
et�1(x� y) 1
< c ":
12
This implies inequality (5.8).Proposition 5.3 yields the following estimate for the invariance entropy of almost
(A;B)-invariant subspaces. This estimate is sharper than the one provided in Theo-rem 5.1 (in particular, for low dimensional spaces V .)
Theorem 5.4. Consider an almost (A;B)-invariant subspace V � Rn withdimension d and denote the largest A-invariant subspace contained in V by ker(A;V )and its dimension by `. Suppose that the map A induced by A on the quotient spaceRn= ker(A;V ) is diagonalizable and that the eigenvalues satisfy
�1 � ::: � �d�` > �d+1�` � ::: � �n�`:
Assume further that V= ker(A;V ) intersects trivially any A-invariant subspace W �Rn= ker(A;V ) of codimension d � `. Then the invariance entropy of V satis�es theinequality
hinv(V ) �d�X̀i=1
max (0; �i) : (5.9)
Proof. We argue similarly as in the proof of Theorem 5.1, using now the subspaceentropy with respect to V instead of the topological entropy.
Let K be a compact subset of V . Then, for the projection � of Rn to the quotientspace Rn= ker(A;V ), the set �K is compact. Let T; " > 0 be given and denote byE � �K a maximal (T; "; �K; �V ; �̂)-separated set with respect to the �ow �̂ onRn= ker(A;V ), say E = f�y1; : : : ; �ysg with yj 2 K and s = s(T; "; �K; �V ; �̂).Then E is also (T; "; �K; �V ; �̂)-spanning which means that for all x 2 K there isj 2 f1; : : : ; sg with
maxt2[0;T ]
dist(eAtx� eAtyj ; V + ker(A;V )) = maxt2[0;T ]
infz2V
eAtx� eAtyj � z < ":Since V is almost (A;B)-invariant, we can assign to each yj ; j 2 f1; : : : ; sg, a controlfunction uj 2 C([0;1);Rm) such that dist('(t; yj ; uj); V ) < " for all t � 0. Let R :=fu1; : : : ; usg � C([0;1);Rm). By linearity one has '(t; x; u)�'(t; y; u) = eAtx�eAtyfor all t � 0, x; y 2 Rn and u 2 C([0;1);Rm). We obtain that for every x 2 K thereis j such that
maxt2[0;T ]
dist('(t; x; uj)� '(t; yj ; uj); V ) = maxt2[0;T ]
dist(eAtx� eAtyj ; V ) < ":
Since dist('(t; yj ; uj); V ) < " for t 2 [0; T ], there is z1 2 V with
k'(t; yj ; uj)� z1k < "
and hence, using that V is a linear subspace, one �nds
dist('(t; x; uj); V ) = infz2V
k'(t; x; uj)� zk
� infz2V
k'(t; x; uj)� '(t; yj ; uj) + z1 � zk+ k'(t; yj ; uj)� z1k
< infz2V
k'(t; x; uj)� '(t; yj ; uj)� zk+ "
< 2":
13
This implies that for all x 2 K there is uj 2 R such that
maxt2[0;T ]
dist('(t; x; uj); V ) < 2":
Hence R is (T; 2";K; V )-spanning and it follows that
rinv(T; 2";K; V ) � s(T; "; �K; �V ; �̂) for all T; " > 0;
and consequently
hinv(K;V ) � hsep(�K; �V ; �̂) = h(�K; �V ; �̂) � h(�V ; �̂):
Since, by assumption, �V = V= ker(A;V ) intersects trivially any A-invariant subspaceW � Rn= ker(A;V ) of codimension d�`, it does not contain any nontrivial �A-invariantsubspace, and we can apply Proposition 5.3 in order to prove the assertion.
We list a few explicit cases in the single-input case in which the hypotheses ofTheorem 5.4 are satis�ed.
Corollary 5.5. Assume that (A; b) 2 Rn�n � Rn is controllable and A isdiagonalizable with n distinct real eigenvalues
�1 > ::: > �d > �d+1 > ::: > �n:
Let �1; � � � ; �d denote any distinct real numbers that are disjoint from �1; � � � ; �n.Then
V = span�(A� �1I)�1b; � � � ; (A� �dI)�1b)
�(5.10)
is an (A; b)-invariant subspace with ker(A;V ) = f0g and < AjV >= Rn. The entropyof V satis�es the inequality
hinv(V ) �dXi=1
max (0; �i) ; (5.11)
Proof. Without loss of generality we can assume that (A; b) is in Jordan canonicalform, i.e. A = diag(�1; � � � ; �n) and b = (1; � � � ; 1)>. Thus V coincides with thecolumn span of the n� d matrix
264 (�1 � �1)�1 � � � (�1 � �d)�1...
...(�n � �1)�1 � � � (�n � �d)�1
375 :For any column v of this matrix, the pair (A; v) is controllable, which implies < AjV >= Rn. LetW denote an arbitrary A-invariant eigenspace of codimension d and assumeW \V 6= f0g. Then there exists nonzero real numbers c1; � � � cd such that the rationalfunction
p(�)
q(�):=
dXi=1
ci�� �i
vanishes at d eigenvalues � 2 f�i1 ; � � � ; �idg. But then p = 0, as degp < d andtherefore c1 = � � � = cd = 0, which is a contradiction. Hence W \ V = f0g for
14
any A-invariant subspace of codimension d. This implies also ker(A;V ) = f0g, asany invariant subspace V0 � V can be extended to an (n � d)-dimensional invariantsubspace W . This shows that V satis�es the assumptions of Theorem 5.4.
We note that the (A; b)-invariant subspaces constructed in Corollary 5.5 are notof the most general form; however for d = 1 they parameterize all one-dimensionalcontrolled invariant subspaces. In the scalar case, it can be shown that the estimateabove is sharp.
Example 5.6. Let d = 1; n = 2. We can suppose that A has the form (5.5)and we use small letters instead of capital letters. Let K � V = R� f0g be compact.Choose a (T; ";K; V ; �)-spanning set R � K. Thus for all x 2 K there is y 2 R suchthat for all t 2 [0; T ]
dist
�etA
��x0
���y0
��; V
�= km(t)(x� y)k < ":
If V is not invariant, one has g 6= 0. For t � 1
" > km(t)(x� y)k = et�2 Z t
0
e�s�2ges�1ds (x� y)
=
� g�1��2 e
t�1 [1� et(�2��1)] kx� yk for �1 > �2et�1t jgj kx� yk for �1 = �2
� cet�1 kx� yk ;
with a constant c > 0 given by
c :=
� jgj�1��2 [1� e
�2��1 ] for �1 > �2jgj for �1 = �2
(recall that �1 � �2.) Hence
et�1 kx� yk � c�1" for t 2 [1; T ]:
It is an easy observation that one also may de�ne topological entropy by requiring thespanning property for all times t 2 [1; T ] only. Then the set R is a spanning set forthe topological entropy of the �ow et�1 ; t � 0; x 2 V . It follows that
r(T; c�1";K; V ) � r(T; ";K; e�1�):
Hence hinv(V ) � htop(e�1�) follows showing that equality holds in (5.11).
Example 5.7. Here we treat the n-dimensional generalization of the above ex-ample, i.e. d = 1 and n � 2. Assume further, that �1 > �2 � � � � � �n andG = (gj) 2 R(n�1)�1 is nonzero. Then
kM(t)(x� y)k = et�1 kx� yk kv(t)k
with (we take the 1-norm)
kv(t)k =nXj=2
jgj j1� et(�j��1)�1 � �j
upper bounded on [0;1) by c :=Pn
j=2 jgj j 1�1��j and lower bounded on [1;1) by
c =Pn
j=2 jgj j 1�e�j��1
�1��j . Proceeding as in the above example, we conclude that theentropy is given by max(0; �1).
15
6. Conclusions. This paper introduces a notion of invariance entropy for con-trolled invariant subspaces. It essentially measures how fast the number of open-loopcontrol functions has to grow over time, if the system is to be kept in such a sub-space (more precisely, within an arbitrarily small neighborhood). If the present stateof the system is available for the controller and hence a state feedback law can beimplemented, this question is, clearly, obsolete. However, if this information is notavailable for the controller, the controls have to be adjusted over time, and the invari-ance entropy gives an indication about the average number of readjustments. Sincethe notion of controlled invariance is of fundamental importance in many feedback de-sign problems, we hope that this will shed light on the amount of information in theseproblems. In particular, combining the invariance entropy discussed in the presentpaper with the entropy for exponential stabilization considered in Colonius [6] appearspromising.
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