Outer invariance entropy for discrete-time linear systems on Lie groups

We introduce discrete-time linear control systems on connected Lie groups and present an upper bound for the outer invariance entropy of admissible pairs ( K; Q ) . If the stable subgroup of the uncontrolled system is closed and K has positive measure for a left invariant Haar measure, the upper bound coincides with the outer invariance entropy.


Introduction
In 2009, Colonius and Kawan [6] introduced the theory of invariance entropy for control systems. This concept is closely related to feedback entropy introduced by Nair, Evans, Mareels and Moran [14] in an engineering context. Speci…cally, for invariance entropy a pair (K; Q) of nonempty subsets of the state space is called admissible if K is a compact subset of Q and for each x 2 K there exists a control v such that the trajectory '(R + ; x; v) Q. For T > 0 denote by r inv (T; K; Q) the minimal number of controls u such that for every initial point x 2 K there is u with trajectory '([0; T ]; x; u) Q. Then the invariance entropy is the exponential growth rate of these numbers as T tends to in…nity, From this paper the theory was developed culminating in Kawan's book [13] that compiled all the theory achieved until 2013. In particular, this book also developed the theory of invariance entropy for discrete time. Most of the results lead to more explicit expressions when studied in the case of linear systems. For linear discrete-time systems on Euclidean spaces the present authors in [5] described the invariance pressure and, in particular, the invariance entropy for subsets K of the control set D, which is unique and bounded under hyperbolicity assumptions. For the hyperbolic theory of invariance entropy for continuous-time nonlinear control systems, cf. Da Silva and Kawan [9] and also [10]. Invariance entropy for continuous-time linear systems on Lie groups has been analyzed by Ayala, Da Silva, Jouan, and Zsigmond [2], cf. also the references therein for the theory of continuous-time linear control systems on Lie groups. For discrete time, the present paper introduces linear control systems on connected Lie groups and starts the investigation of invariance entropy. We de…ne discrete-time linear systems on connected Lie groups G as systems of the form : g k+1 = f (g k ; u k ); u k 2 U; where f u ( ) := f ( ; u) : G ! G is an automorphism for u = 0 and otherwise a di¤eomorphism such that f u (g) = f u (e) f 0 (g). We refer to g k+1 = f (g k ; 0) = f 0 (g k ) as the uncontrolled system. Outer invariance entropy weakens the requirements on the trajectories: the trajectories are allowed to go out to "-neighborhoods N " (Q) of Q, and Our main result establishes that an upper bound for outer invariance entropy of an admissible pair (K; Q) of discrete-time linear systems is given by the sum of the logarithms of the modulus of the eigenvalues of the di¤erential (df 0 ) e with j j > 1, where e is the identity of G. The upper bound coincides with the outer invariance entropy, if the stable subgroup of the uncontrolled system (cf. the de…nition after formula (2)) is closed and K has positive measure for a left invariant Haar measure on G. This bears some similarity to the characterization of outer invariance entropy in the continuous-time case, cf. Da Silva [7]. Section 2 presents the main concepts for discrete-time linear control system on Lie groups and some examples. In Section 3 we introduce discrete-time linear control system induced on homogeneous spaces and Section 4 proves the main result of the paper, the characterization of outer invariance entropy for discretetime linear control systems on Lie groups. Finally, Section 5 derives a su¢ cient condition for closedness of the stable subgroup of the uncontrolled system.

Discrete-time linear control systems on Lie groups
In this section we present our de…nition of discrete-time linear control systems on Lie groups in analogy to the de…nition of the continuous-time linear control systems on Lie groups, derive some properties, and provide several examples.
Recall (cf. e.g. Sontag [16]) that a discrete-time control system on a topological space M is given by di¤erence equations where k 2 N 0 , the control range U is a nonempty set and f : M U ! M is a map such that f u ( ) := f ( ; u) : M ! M is continuous for each u 2 U . For initial value x 0 := x 2 M and control u = (u i ) i2N0 2 U = U N0 the solution of this system is given by Where convenient, we also write ' k;u := '(k; ; u). Now we present our de…nition of discrete-time linear control systems on Lie groups.
Here " " denotes the product of G, it will be omitted when it is clear by the context. Thus f u (g) is given by left translation L fu(e) of f 0 (g) by f u (e), and f 0 (g) 1 = f 0 (g 1 ) implies that for each u 2 U the map f u : G ! G is a di¤eomorphism with inverse 1 g = (f 0 ) 1 L (fu(e)) 1 (g); g 2 G: Example 2 Consider on the additive Lie group G = R d the control system given by In this case, the solutions are given by The next result shows that for linear systems the solution starting in a point g is a translation of the solution starting in the identity e.
Proposition 3 Consider a discrete-time linear control system g k+1 = f (g k ; u k ), u k 2 U , on a Lie group G. Then it follows for all g 2 G and u = (u i ) 2 U that '(k; g; u) = '(k; e; u)f k 0 (g) for all k 2 N: Proof. The proof will follow by induction over k 2 N. Note initially that for g 2 G and u 2 U we have Now, suppose that the equality holds for k 2 N, that is, '(k; g; u) = '(k; e; u)f k 0 (g). With the k-shift to the right given by k (u i ) i2N = (u i+k ) i2N , this implies Remark 4 For a continuous-time linear control system on a connected Lie group the solution is given by '(t; g; u) = '(t; e; u)'(t; g; 0); cf. Ayala, Da Silva, and Zsigmond [3, formula (7)]. The solution formula (1) for discrete-time linear systems is analogous. This group can be seen as R + R provided with the product is a discrete-time linear control system on A¤(2; R) 0 .
For u = (u i ) i2N0 2 U, denote by S k (u) the sum P k i=0 u i . Proposition 3 and induction over k imply that for all k 2 N 0 , (x; y) 2 A¤(2; R) 0 and u 2 U, A : be the Heisenberg group. Note that G is di¤ eomorphic to R 3 with the product Consider the automorphism f 0 : G ! G given by and for each u 2 U , de…ne the di¤ eomorphism f u : G ! G by It is not di¢ cult to see that for each g 2 G, f u (g) = f u (e)f 0 (g). Hence, the system on G is linear. By Proposition 3, the solutions are given by Remark 7 For a discrete-time linear control system g k+1 = f (g k ; u k ), u k 2 U , on a Lie group G, we have that (df 0 ) e : g ! g is a Lie algebra isomorphism, because f 0 is an automorphism of G. Also, we can see that (df n 0 ) e = [(df 0 ) e ] n for each n 2 Z and hence for each X 2 g Given the automorphism f 0 , we can de…ne several Lie subalgebras that are intrinsically associated with its dynamics. Consider an eigenvalue of (df 0 ) e and its generalized eigenspace ) n X = 0; for some n 1g: The following proposition, whose proof can be found in Ayala and Da Silva [1, Proposition 2.1], will be useful in order to de…ne dynamical subalgebras of g related to the di¤erential of f 0 at e 2 G.
Proposition 8 If and are eigenvalues of (df 0 ) e , then is not an eigenvalue of (df 0 ) e .
Due to Proposition 8, we de…ne the unstable, center, and unstable Lie subalgebras (cf. [1]) by Proposition 8 implies that g + and g are nilpotent Lie subalgebras of g. Since (df 0 ) e is a Lie algebra isomorphism, the decomposition g = g + g 0 g holds. The center-unstable and center-stable Lie subalgebras are g +;0 = g + g 0 and g ;0 = g g 0 : We will also need the connected Lie groups G ;0 , G +;0 , G and G + corresponding to g ;0 , g +;0 , g and g + , respectively. In particular, we refer to G and G + as the stable and the unstable subgroup of the uncontrolled system, resp.

Remark 9
The restrictions of (df 0 ) e to the Lie subalgebras g + , g 0 , and g satisfy: j[(df 0 ) e ] n Xj c n jXj; for any X 2 g + ; n 2 N; for some c 1 and 2 (0; 1) and, for all a > 0 and Z 2 g 0 it holds that j[(df 0 ) e ] n Zj ajnj ! 0 as n ! 1:

Linear systems induced on homogeneous spaces
In this section, we de…ne a induced (discrete-time) linear system on a homogeneous space G=H from a discrete-time linear system on a Lie group G. This construction will be important to get a formula for the outer invariance entropy presented in the Theorem 17.
Consider the discrete-time linear control system on a Lie group G and let H a Lie subgroup of G which is f 0 -invariant. If H is closed, this induces a control system on the homogeneous space G=H in the following way: de…ne f : on G=H with solutions denoted by '. Since for each k 2 N, f k 0 = f k 0 , we have that ('(k; g; u)) = '(k; (g); u); for all k 2 N 0 , g 2 G and u 2 U. In other words, ( ; id U ) is a semi-conjugacy between the systems (3) and (4) This shows that the solutions of (4) satisfy properties similar to equality (1) for a linear system on G, hence we call it the induced linear system on the homogeneous space G=H.
Proposition 10 Let G a Lie group with Lie algebra g. Assume that g decomposes as g = h l, where h and l are (df 0 ) e -invariant Lie subalgebras of g.
Consider the connected Lie subgroup H of G with Lie algebra h. If H is closed, then (d f k 0 ) eH = (df 0 ) k e j l , for all k 2 N 0 .
Proof. The (df 0 ) e -invariance of h and l allows us to consider the well de…ned linear isomorphism (df 0 ) e : g=h ! g=h given by If is the natural projection of G onto G=H, we can see that (df 0 ) e satis…es (df 0 ) e (d ) e = (d ) e (df 0 ) e . Hence, for all X 2 g and k 2 N 0 we have By invariance, we can identify g=h with l and, therefore, (df 0 ) e with (df 0 ) k e j l and the desired equality holds.

De…nition 11 A measure on a homogeneous space G=H is a G-invariant
Borel measure if (L g (A)) = (A), for all g 2 G and all Borel set A G=H.
The following proposition is an adaptation of Da Silva [7,Proposition 4.6] in our context of discrete-time linear systems.

Proposition 12
If the connected subgroup G corresponding to g de…ned in (2) is closed, the homogeneous space G=G admits a unique (up to a scalar) G-invariant Borel measure.
Proof. Denote by G and G the modular functions of G and G , respectively, cf. Knapp [11,Chapter VIII,Section 2]. The result will follow from [11,Theorem 8.36], if we can show that ( G )j G = G .
Since G is nilpotent, G (h) = 1 for all h 2 G . On the other hand, for any two eigenvalues ; of (df 0 ) e we have ad(g ) n g g n ; 8 n 2 N; which implies that for each X 2 g , ad(X) : g ! g is a nilpotent linear map.
Writing for X 2 g X = X j j<1 a X ; a 2 R; where the sum is taken over all eigenvalues of (df 0 ) e with j j < 1, we get tr(ad(X)) = X j j<1 a tr(ad(X )) = 0: Now, consider g 2 G . Then g can be written as for some X 1 ; : : : ; X r 2 g , because G is connected. Therefore, = e tr(ad(X1)) e tr(ad(Xr)) = 1: This shows that ( G )j G = G and, by [11,Theorem 8.36], the result follows.
Remark 13 According to [11,Theorem 8.36], if G and H are the left invariant Haar measures on G and H, respectively, the G-invariant Borel measure on G=H can be normalized so that Z for all continuous function : G ! R with compact support. Note that if K G is compact, then G (K) > 0 implies ( (K)) > 0. This follows from the fact that K (gH) = R H K (gh) d H (h) is a bounded positive function and K > 0 if and only if (K) > 0.

Outer invariance entropy
In this section we prove our main result that provides an upper bound for outer invariance entropy of discrete-time linear control systems on Lie groups. We begin by recalling the de…nition of invariance entropy and outer invariance entropy as given in Kawan [13, De…nitions 2.2 and 2.3].
Consider a discrete-time control system with solutions '(k; x; u); k 2 N 0 . A pair (K; Q) of nonempty subsets of M is called admissible, if K is compact and for each x 2 K, there exists u 2 U such that '(k; x; u) 2 Q for all k 2 N 0 . Given an admissible pair (K; Q) and n 2 N, we say that a set S U is (n; K; Q)-spanning if 8 x 2 K 9 u 2 S 8 j 2 f1; : : : ; ng : '(j; x; u) 2 Q: Denote by r inv (n; K; Q) the minimal number of elements such a set can have (if there is no …nite set with this property we set r inv (n; K; Q) = 1).
The existence of (n; K; Q)-spanning sets is guaranteed, since U is (n; K; Q)spanning for every n 2 N.

De…nition 14
Given an admissible pair (K; Q) for a discrete-time control system , the invariance entropy of (K; Q) is de…ned by Here, and throughout the paper, log denotes the logarithm with base 2.
The invariance entropy of (K; Q) measures the exponential growth rate of the minimal number of control functions su¢ cient to stay in Q when starting in K as time tends to in…nity. Hence, invariance entropy is a nonnegative (possibly in…nite) quantity which is assigned to an admissible pair (K; Q). For our main result we need the following related quantity. Note that for an admissible pair (K; Q) also every pair (K; N " (Q)); " > 0, is admissible, where N " (Q) = fx 2 M jd(x; Q) < " g denotes the "-neighborhood of Q. Obviously, the inequality h inv (K; Q) h inv;out (K; Q) holds. The next proposition (cf. Kawan [13,Proposition 2.13]) describes the behavior of outer invariance entropy under semi-conjugacy. Recall that a semi-conjugacy of two discrete-time control systems on metric spaces M 1 and M 2 , respectively, given by x k+1 = f 1 (x k ; u k ); u k 2 U , and y k+1 = f 2 (y k ; v k ); V k 2 V; with solutions ' 1 and ' 2 , resp., is a pair ( ; h) of continuous maps : Proposition 16 Consider two discrete-time control systems 1 and 2 and let ( ; h) be a semi-conjugacy from 1 to 2 . Then every admissible pair (K; Q) for 1 with compact Q de…nes an admissible pair ( (K); (Q)) for 2 and h inv;out (K; Q; 1 ) h inv;out ( (K); (Q); 2 ): Now, we will brie ‡y recall the notion of topological entropy as de…ned by Bowen and Dinaburg (cf. e.g. Walters [17]). Let (M; d) be a metric space and : M ! M be a continuous map. Given a compact set K X and n 2 N we say that a set F M is (n; ")-spanning set for K with respect to if, for every y 2 K, there exists x 2 F such that d( j (x); j (y)) < "; for all j 2 f0; 1; : : : ; ng: If we denote by r n ("; K) the minimal cardinality of an (n; ")-spanning set for K with respect to , the topological entropy of over K is de…ned by Alternatively, topological entropy can be de…ned using (n; ")-separated sets E K with respect to which are de…ned as sets satisfying 8 x; y 2 E : x 6 = y ) 9j 2 f0; : : : ; ng : d( j (x); j (y)) > ": Let s n ("; K) denote the largest cardinality of any (n; ")-separated subset of K with respect to . If E is an (n; ")-separated subset of K of maximal cardinality, then E is (n; ")-spanning for K. The topological entropy of over K can also be characterized as the limit Bowen [4,Corollary 16] shows that for an endomorphism of a Lie group G the topological entropy is log j j; where the sum is taken over all eigenvalues of d( ) e with j j > 1. Now we formulate the main result of this paper.
Theorem 17 Let (K; Q) be an admissible pair for the discrete-time linear system : g k+1 = f (g k ; u k ), u k 2 U , on a connected Lie group G. Assume that Q is compact.
i) Then the outer invariance entropy satis…es where the sum is taken over all eigenvalues of (df 0 ) e with j j > 1.
ii) Moreover, suppose that the connected subgroup G corresponding to g de…ned in (2) is closed and that G (K) > 0 for a left invariant Haar measure G on G. Then where the sum is taken over all eigenvalues of (df 0 ) e with j j > 1.
Proof. (i) The characterization of the topological entropy of the automorphism f 0 follows from Bowen's result (7). Since Q is compact, Kawan [13, Proposition 2.5] implies that the invariance entropy does not depend on the metric. Hence we may choose a left invariant Riemannian metric % on G. This yields for g 2 G, %(' k;u (g); ' k;u (h)) = %(' k;u (e)f k 0 (g); ' k;u (e)f k 0 (h)) = %(f k 0 (g); f k 0 (h)): Now, …x " > 0 and n 2 N. Let E K be an (n; ")-separated subset of K with respect to f 0 of maximal cardinality s n ("; K). Since (K; Q) is admissible, for each h 2 E, there exists u h 2 U such that '(k; h; u h ) 2 Q for k = 1; : : : ; n. We claim that S := fu h 2 U; h 2 Eg is an (n; K; N " (Q))-spanning set. In fact, since the set E is also (n; ")-spanning, one …nds for all g 2 K an element h 2 E with %(' k;u h (g); ' k;u h (h)) = %(f k 0 (g); f k 0 (h)) < " for all k 2 f0; : : : ; ng. This shows that ' k;u h (g) 2 N " (Q) and hence proves the claim. It follows that r inv (n; K; N " (Q)) s n ("; K) implying h inv;out (K; N " (Q)) = lim sup n!1 1 n log r inv (n; K; N " (Q)) lim sup n!1 1 n log s n ("; K): Letting " tend to 0, one …nds by formula (7) for the topological entropy where the sum is taken over all eigenvalues of (df 0 ) e with j j > 1.
Denoting C := ( (K)) (N " (Q)) > 0 we get h inv ( (K); N " ( (Q))) lim sup where the product is taken over all eigenvalues of (df 0 ) e with j j 1 and the sum is taken over all eigenvalues of (df 0 ) e with j j > 1. Taking " & 0 the result follows.

Remark 18
The formula presented in Theorem 17 (ii) holds for the class of solvable, connected and simply connected Lie groups G. In fact, by San Martin [15,Proposion 10.6] all connected subgroups of G are closed, hence G is closed. In Section 5 we show that G is still closed even when G is not solvable (see Theorem 24) and Theorem 17 (ii) can be applied for discrete-time linear systems on a broader class of Lie groups.
Example 19 Let (K; Q) be an admissible pair for the system de…ned in Example 5, where Q is compact and K has positive Haar measure. Note that the matrix of the di¤ erential of f 0 is given by In this case, g = f0g and G = f(1; 0)g, which is closed. By Theorem 17 (ii), h inv;out (K; Q) = 2.
Example 20 Again, consider an admissible pair (K; Q) of the system presented in Example 6. Assume that Q is compact and K has positive Haar measure. The matrix of (df 0 ) e in the canonical basis is and we can see that the unique eigenvalue of (df 0 ) e is 1, hence h inv;out (K; Q) = 0 by Theorem 17.
Remark 21 For linear control systems in Euclidean space, cf. Example 2, natural candidates for admissible pairs (K; Q) such that Q has compact closure, as considered in Theorem 17, can be obtained by Colonius, Cossich and Santana [5, Theorem 32] as follows: If A is hyperbolic, there exists a unique control set D with nonvoid interior (i.e., a maximal set of approximate controllability with intD 6 = ?) and it is bounded. Hence its closure D is compact and for any compact subset K Q := D the pair (K; Q) is admissible with compact Q. Results on control sets for continuous-time linear systems on Lie groups are proved in Ayala, Da Silva, and Zsigmond [3] and Ayala, Da Silva, Philippe, and Zsigmond [2]. For discrete-time linear systems on Lie groups, the control sets have not be studied.

Closedness of the stable subgroup
The formula for the outer invariance entropy in Theorem 17 has been derived under the assumption that the stable subgroup G is closed. In this section we provide a su¢ cient condition for this property. The arguments can also be applied to the unstable subgroup G + . First we show that these subgroups are simply connected.

Proposition 22
The stable subgroup G and the unstable subgroup G + are simply connected.
Proof. We will just show this fact for G , because the proof for G + is analogous. Denote the exponential map exp : g ! G restricted to g by exp , which is the exponential map of G . Since G is nilpotent and connected, exp : g ! G is a covering map, hence it is surjective, continuous and open. Hence, if we show that exp is injective, it will be a homeomorphism. Therefore simple connectedness of g implies simple connectedness of G .
In order to show that exp is injective, let X; Y 2 g such that exp X = exp Y . Consider open neighborhoods V and U of 0 2 g and e 2 G, respectively, such that exp : V ! U is a di¤eomorphism. Remark 9 implies that for all Z 2 g it holds j(df 0 ) n e (Z)j c n jZj for all n 2 N; for some c 1 and 2 (0; 1). Consider n large enough such that (df 0 ) n e (Y ); (df 0 ) n e (Y ) 2 V . Hence, exp ((df 0 ) n e (X)) = f n 0 (exp X) = f n 0 (exp Y ) = exp ((df 0 ) n e (Y )): The injectivity of exp j V implies that exp j V = exp j V \g is injective, and hence (df 0 ) n e (X) = (df 0 ) n e (Y ). Since (df 0 ) e j g is an isomorphism, we obtain X = Y .

Remark 23
The maps exp = exp j G : g ! G are di¤ eomorphisms, because G are connected, simply connected and nilpotent (see Knapp [11,  Both G + and G are irrational ‡ows on T 2 which are dense in T 2 , hence they are not closed.