Controllability properties and invariance pressure for linear discrete-time systems

For linear control systems in discrete time controllability properties are characterized. In particular, a unique control set with nonvoid interior exists and it is bounded in the hyperbolic case. Then a formula for the invariance pressure of this control set is proved.


Introduction
Invariance pressure for subsets of the state space generalizes invariance entropy of deterministic control systems by adding potentials on the control range. We consider control systems in discrete time of the form where F : M × U → M is smooth for a smooth manifold M and a compact control range U ⊂ R m . The invariance entropy h inv (K, Q) determines the average data rate needed to keep the system in Q (forward in time) when it starts in K ⊂ Q. Basic references for invariance entropy are Nair, Evans, Mareels, and Moran [12] and the monograph Kawan [10], where also the relation to minimal data rates is explained. With some analogy to classical constructions for dynamical systems, invariance pressure adds continuous functions f : U → R called potentials giving a weight to the control values.
For continuous-time systems, invariance entropy of hyperbolic control sets has been analyzed in Kawan [9] and Kawan and Da Silva [6]. Kawan and Da Silva [11] and [7] analyze invariance entropy of partially hyperbolic controlled invariant sets and chain control sets. Huang and Zhong [8] show dimension-like characterizations of invariance entropy. Measure-theoretic versions of invariance entropy have been considered in Colonius [4] and Wang, Huang, and Sun [15]. Invariance pressure has been analyzed in Colonius, Cossich, and Santana [1,2,3]. In Zhong and Huang [18] it is shown that several generalized notions of invariance pressure fit into the dimension-theoretic framework due to Pesin.
The main results of the present paper are given for linear control systems x k+1 = Ax k + Bu k with an invertible matrix A and control values u k in a compact neighborhood U of the origin in R m . It is shown that a unique control set D with nonvoid interior exists if and only if the system without control constraints is controllable (i.e., the pair (A, B) is controllable), and D is bounded if and only if A is hyperbolic. In this case a formula for the invariance pressure of compact subsets K in D is presented.
The contents of this paper are as follows: Section 2 collects general properties of control sets for nonlinear discrete-time systems. Section 3 characterizes controllability properties of linear discrete-time systems with control constraints and Section 4 shows that here a unique control set with nonvoid interior exists and that it is bounded if and only if the uncontrolled system is hyperbolic. Section 5 introduces invariance entropy and as a generalization total invariance pressure where potentials on the product of the state space and the control range are allowed. For linear systems, Section 6 first derives an upper bound for the total invariance pressure and a lower bound for the invariance pressure. Combined they yield a formula for the invariance pressure in the hyperbolic case.

Control sets for nonlinear systems
In this section we introduce some notation and prove several properties of control sets with nonvoid interior for nonlinear discrete-time systems. They are analogous to properties of systems in continuous time, however, the statements are a bit more involved, since one has to consider in addition to the interior of control sets their transitivity sets. A discussion of various slightly differing versions in the literature is contained in Colonius [4,Section 5].
We consider control systems of the form on a C ∞ -manifold M of dimension d endowed with a corresponding metric. For an initial value x 0 ∈ M at time k = 0 and control u = (u k ) k≥0 ∈ U := U N0 we denote the solutions by ϕ(k, x 0 , u), k ∈ N 0 . Assume that the set of control values U ⊂ R m is nonvoid and satisfies U ⊂ intU . LetŨ be an open set containing U and suppose that the map F : M ×Ũ → M is a C ∞ -map.
Definition 1 For x ∈ M and k ∈ N the reachable set R k (x) and the controllable set C k (x) are R k (x) := {y ∈ M |∃u ∈ U : y = ϕ(k, x, u) }, C k (x) := {y ∈ M |∃u ∈ U : ϕ(k, y, u) = x }, resp., and R(x) and C(x) are the respective unions over all k ∈ N. The system is called accessible in x if Accessibility in x certainly holds if intF (x, U ) = ∅ and int{y ∈ M |x ∈ F (y, U ) } = ∅.
Next we specify maximal subsets of complete approximate controllability.
We define for k ≥ 1 a C ∞ -map Following Wirth [17] we say that a pair (x, u) ∈ M × intU k is regular if rank ∂G k ∂u (x, u) = d (clearly, this implies mk ≥ d). For x ∈ M and k ∈ N the regular reachable set and the regular controllable set at time k arê resp., and the regular reachable setR(x) and controllable setĈ(x) are given by the respective union over all k ∈ N. It is clear thatR(x) andĈ(x) are open for every x.
Accessibility condition (2) implies that there is k 0 ∈ N such that for all k ≥ k 0 one has intR k (x) = ∅ and By Sard's Theorem the set of points ϕ(k, x, u) ∈ R k (x) such that (x, u) is not regular has Lebesgue measure zero.
Proposition 3 Assume that accessibility condition (2) holds for all x ∈ M . Then for every control set D with nonvoid interior the transitivity set D 0 is nonvoid and dense in intD.
Proof. For x ∈ intD there is k 0 ∈ N such that the reachable set R k (x) at time k has nonvoid interior for all k ≥ k 0 . There is k ≥ k 0 with R k (x) ∩ intD = ∅, hence we may assume that there is y := ϕ(k, x, u) ∈ intR k (x) ∩ intD. Then, by Sard's Theorem, it follows that there is a point y = ϕ(k, x, u) ∈ intD with some regular (x, u), i.e., y ∈ intD ∩R k (x). Then x ∈ intC(y). Let V ⊂ intC(y) be a neighborhood of x. Since x ∈ intD and D ⊂ R(y), there is z ∈ V ∩R(y) ⊂ D and thus y ∈ C(z). By construction, the point z ∈ D satisfies z ∈ intC(y) ⊂ intC(z), hence it is in the transitivity set of D and D 0 is dense in intD.

Remark 4
In the general context of semigroups of continuous maps (and with slightly different notation), Patrão and San Martin [13,Propositions 4.8 and 4.10] show that the transitivity set D 0 is dense in a control set D with nonvoid interior provided that D 0 = ∅.
We note the following further results for control sets.
Proposition 5 Assume that D is a control set for a control system which is accessible for all x ∈ M . Then its transitivity set D 0 satisfies D 0 ⊂ R(x) for all x ∈ D.
Proof. Let x ∈ D and x 0 ∈ D 0 . By approximate controllability of D and Proposition 6 Assume that D is a control set with nonvoid interior of a control system, which is accessible for all x ∈ M . Then the transitivity set D 0 of D is nonvoid and in particular, the set D is measurable.
Proof. By Proposition 3 the transitivity set D 0 is nonvoid. Let x 0 ∈ D 0 . Note that D ⊂ R(x 0 ) by the definition of control sets. For every x ∈ D, Proposition 5 shows that . It is not difficult to see that the set D ′ is a set of approximate controllability with nonvoid interior. It follows that D ′ is contained in a maximal set D ′′ of approximate controllability with nonvoid interior, which by Kawan [10, Proposition 1.20] is a control set. By the maximality property of control sets and D ⊂ D ′′ , it follows that D = D ′ = D ′′ , which concludes the proof.
The following proposition shows that a trajectory starting in the interior of a control set D and remaining in it up to a positive time must actually remain in the interior of D.
Proposition 7 Assume that the maps F (·, u) are local diffeomorphisms on M for all u ∈ U . Let x be in the interior of a control set D and suppose that for some τ ∈ N and u ∈ U one has ϕ(k, x, u) ∈ D, k ∈ {1, . . . , τ }. Then ϕ(k, x, u) ∈ intD, k ∈ {1, . . . , τ }.
Proof. Suppose that y := ϕ(k, x, u) ∈ D ∩ ∂D for some k ∈ {1, . . . , τ }. By the assumption on the maps F (·, u) and x ∈ intD, there is a neighborhood N 0 (y) of y with N 0 (y) = ϕ(k, N (x), u) for a neighborhood N (x) ⊂ D of x. Since y ∈ D, there are a control v ∈ U and k 0 ∈ N with ϕ(k 0 , y, v) ∈ intD. Then there is a neighborhood N 1 (y) with ϕ(k 0 , N 1 (y), u) ⊂ intD. By the maximality property of control sets it follows that the neighborhood N 0 (y) ∩ N 1 (y) of y is contained in D, contradicting y ∈ ∂D.

Controllability properties of linear systems
Next we consider linear control systems in K d , K = R or K = C, of the form where A ∈ Gl(d, K) and B ∈ K d×m and the control range U is a compact convex neighborhood of 0 ∈ K m with U = intU . For initial value x ∈ K d and control u ∈ U = U N0 the solutions of (3) are given by Where convenient, we also use the notation ϕ k,u := ϕ(k, ·, u) : R d → R d . Note the following observation.
Proof. Convexity follows from the convexity of U . Since U ⊂ K m is compact, there is M > 0 such that u ≤ M , for all u ∈ U . Then, if y = ϕ(k, x, u) ∈ R k (x), u = (u i ) ∈ U k , we get hence R k (x) is bounded. In order to show that R k (x) is closed, consider a sequence y n = ϕ(k, x, u n ) in R k (x) such that y n → y ∈ K d and u n ∈ U k . By compactness of U , we have that U k is compact, hence there is a subsequence converging to some u ∈ U k . Therefore y = ϕ(k, x, u) ∈ R k (x) by continuity.
Proposition 9 For all k, l ∈ N we have Proof. Let x 1 ∈ R k (0) and x 2 ∈ R l (0). Then there are u, v ∈ U such that Then . The converse inclusion follows by reversing these steps. The second assertion follows since the set on left hand side is open. Define the time reversed counterpart of system (3) by The reachable and controllable sets from the origin at time k for this system are denoted by R − k (0) and C − k (0), respectively.

Proposition 10
The reachable and controllable sets for system (3) and the time reversed system (4) satisfy for all k ∈ N R k (0) = C − k (0) and C k (0) = R − k (0).
Proof. Note that x ∈ C k (0) if and only if there is u ∈ U with Hence we conclude that x ∈ C k (0) if and only if there exists a control v ∈ U k such that x = ϕ − (k, 0, v), where ϕ − is the solution of (4). This proves that C k (0) = R − k (0). The other equality follows analogously.
Proof. Since the control range is a neighborhood of 0, controllability implies that there is δ > 0 with B δ (0) ⊂ intR d−1 (0). The second assertion follows since 0 is an equilibrium for u = 0.
Proof. The inclusion intR(0) ⊂ R(0) holds trivially. For the converse we first show that R(y) ⊂ intR(0) for y ∈ intR(0). In fact, let there exists a neighborhood V y of y such that V y ⊂ R(0). Given z ∈ R(y), there are k ∈ N and u ∈ U such that z = ϕ(k, y, u). Since A ∈ Gl(d, R), the map ϕ k,u is a diffeomorphism and we have that ϕ k,u (V y ) is a neighborhood of z and clearly (0) and hence x ∈ intR(0).
We will need the following lemmas.

This implies
The next lemma states a property of convex sets.

Lemma 14 If
C is an open convex subset of K n and Y ⊂ C a subspace, then The following theorem describes the general structure of reachable and controllable sets. It is analogous to a well known property of linear systems in continuous time, cf. Sontag [14, Section 3.6] and Hinrichsen and Pritchard [5, Theorem 6.2.15]; the proof for discrete-time systems, however, is more involved. Recall that the state space K d can be decomposed with respect to A into the direct sum of the stable subspace E s , the center space E c and the unstable subspace E u which are the direct sums of all generalized (real) eigenspaces for the eigenvalues λ of A with |λ| < 1, |λ| = 1 and |λ| > 1, respectively. Furthermore, Theorem 15 Consider the control system given by (3) and suppose that the system without control restriction is controllable.
(i) There exists a compact and convex set K ⊂ E s ⊂ K d with nonvoid interior with respect to E s such that R(0) = K + E uc . Moreover 0 ∈ K and E uc ⊂ intR(0).
(ii) There exists a compact and convex set Proof. We will first prove the result for K = C.
(i) In the first step, we will show that E uc ⊂ intR(0). As R(0) is convex, its interior is convex too. Therefore it suffices to prove that the generalized eigenspaces for eigenvalues with absolute value greater than or equal to 1 are contained in intR(0). Fix an eigenvalue λ of A with |λ| ≥ 1 and let E q (λ) = ker(A − λI) q , q ∈ N 0 . It suffices to show that E q (λ) ⊂ intR(0) for all q.
Note that for all |a| < δ and all n ≥ 1 Using aw ∈ intR d−1 (0) Lemma 11 and Lemma 14 imply for n ≥ 1 We write a = α + ıβ and λ n = x n + ıy n with α, β ∈ R and x n , y n ∈ R d . Claim: There are a sequence (n k ) k∈N with n k → ∞ and a n k ∈ C with |a n k | < δ such that λ n k a n k ∈ R.
In fact, we have if and only if x n β + y n α = 0. Case (a): If x n = 0, one may choose α n := 0 and gets λ n a n = −y n β n ∈ R for β n = δ 2 with |a n | = |β n | = δ 2 .
According to Lemma 13 there are n k ∈ N, arbitrarily large, such that with α n k := δ 2 and β n k : It follows for a n k := α n k + β n k that We have shown that with this choice of a n k we have λ n k a n k ∈ R and the Claim is proved. Furthermore in case (a), by |λ| ≥ 1, |λ n a n | = |λ| n |a n | ≥ |a n | = δ 2 , Recall that all points a n k w ∈ intR d−1 (0). We may assume that We may also assume that

Thus Proposition 9 implies
Proceeding in this way, we finally arrive at ℓ k=1 A n k a n k w ∈ intR n ℓ (0).
For the k with λ n k a n k < 0, replace a n k by −a n k , to get the same conclusion. This shows that w is a convex combination of the points 0 and ℓ k=1 λ n k a n k w in intR(0), thus convexity of this set implies w ∈ intR(0) completing the induction step E q (λ) ⊂ intR(0). Hence we have shown that E uc ⊂ intR(0).
It remains to construct a set K as in the assertion. Define K 0 := intR(0) ∩ E s . Then it follows that For the converse inclusion, let v ∈ intR(0), then v = x + y where x ∈ E s and y ∈ E uc , hence by Lemma 14, which shows that x ∈ K 0 and therefore v ∈ K 0 + E s . This shows that In order to show that K 0 is bounded, consider the projection π : Since E s and E uc are A-invariant, π commutes with A and we have πA n = A n π, for all n ∈ N 0 . For each Since A| E s is a linear contraction, there exist constants a ∈ (0, 1) and c ≥ 1 such that A n x ≤ ca n x for all n ∈ N and x ∈ E s . Since U is compact, there is M > 0 such that πBu ≤ M , for all u ∈ U , so showing that K 0 is bounded. As a consequence, K := K 0 = intR(0) ∩ E s is a compact convex set which has nonvoid interior relative to E s . Moreover, K + E uc is closed, because K is compact. Therefore it follows from Proposition 12 and (6) that (ii) Consider the time reversed system (4).
are the sums of the generalized eigenspaces for the eigenvalues µ of A −1 with |µ| < 1, |µ| = 1 and |µ| > 1, respectively. Now λ is (0) and This completes the proof of the theorem for the case K = C.
It remains to prove the theorem for the case K = R. Note that if A ∈ Gl(d, R), then u − ıv ∈ E s , u, v ∈ R d , implies u + ıv, v + ıu ∈ E s and a similar implication holds for E uc . Hence Let U C := U + ıU and apply the result above for K = C. Clearly (A, B) is controllable, when considered as a system with state space C d and U C is a convex compact neighborhood of 0 ∈ C m with U C ⊂ intU C .
Denote the reachable set from 0 of the real and complex system by R R and R C , respectively. It follows from the complex version of the theorem that the compact convex set K C := int(R C ) ∩ E s has non-empty interior relative to E s and satisfies R C = K C ∩ E uc . Since every u ∈ U C is of the form u = v + ıw, where v, w ∈ U, and ϕ(k, 0, u) = ϕ(k, 0, v) + ıϕ(k, 0, w), k ∈ N, we have It follows that where the interior of R R is relative to R d and the interior of R C is relative to C d . Now, if W, Z ⊂ C d are subsets of the form Applying this equality to W = intR C and Z = E s we obtain from (8) and (7) that Hence K is a compact convex subset of R d , which has a non-empty interior relative to Re E s . Using (8) for the second equality we get This concludes the proof. Next we present a necessary and sufficient condition for controllability in K d . This consequence of Theorem 15 illustrates that controllability only holds under very strong assumptions on the spectrum of the matrix A. In the next section, we will instead consider subsets of the state space where complete controllability holds, i.e., control sets. Recall that the system without control restriction is controllable in R d if and only if (A, B) is controllable.

Control sets for linear systems
Next we analyze linear control systems in R d of the form with A ∈ Gl(d, R) and B ∈ R d×m and suppose that U is a convex compact neighborhood of 0 ∈ R m with U = intU . Recall that the system without control Theorem 18 There exists a unique control set D with nonvoid interior of system (9) if and only if the system without control restriction is controllable in R d . In this case 0 ∈ D 0 ∩ intD.
Proof. The controllability condition for (A, B) is necessary for the existence of D, since it guarantees that accessibility condition (2) holds for all x ∈ R d and, for the system without control constraints, the reachable and the nullcontrollable subspaces coincide with R d . Since 0 ∈ intU , one verifies that for Then every point x ∈ D ′ can be steered to any other point z ∈ D ′ (first steer x to the origin in time k and then the origin to z in time k) and 0 ∈ int(C(0)). As in the proof of Proposition 6 one finds that D ′ is contained in a control set D.
Thus we have established the existence of a control set D with nonvoid interior, and 0 ∈ D 0 ∩ intD. It remains to show uniqueness. LetD ⊂ R d be an arbitrary control set with nonvoid interior. By Proposition 6 its transitivity setD 0 is nonvoid and for x 0 ∈D 0 By linearity, we have ϕ(k, x 1 , u) = x 2 for k ∈ N and x 1 , x 2 ∈ R d implies ϕ(k, αx 1 , αu) = αx 2 for any α ∈ (0, 1]. Here the control αu has values in U , since U is convex and 0 ∈ U . This implies that αD is contained in some control set D α and int(αD) is contained in the interior of D α . Now choose any x ∈ intD and suppose, by way of contradiction, that Then α 0 x ∈ ∂D and α 0 x ∈ intD α0 . ThereforeD ∩ intD α0 = ∅, and it follows thatD = D α0 and α 0 x ∈ intD. This is a contradiction and so α 0 = 0. Choosing α > 0 small enough such that αx ∈ D, we obtain αx ∈D ∩ D = ∅. Now it follows thatD = D.
The following theorem gives a spectral characterization of boundedness of the control set. Recall that A is called hyperbolic if all eigenvalues λ of A satisfy |λ| = 1.
Theorem 19 Assume that (A, B) is controllable. Then the control set D with nonvoid interior of system (9) is bounded if and only if A is hyperbolic.
Proof. By Theorem 15 there are compact sets K ⊂ E s , F ⊂ E u such that By Proposition 6, D = R(0) ∩ C(0), because 0 ∈ D 0 ⊂ intD, and hence every element x ∈ D can be represented in the following two ways: Remark 20 We know that in the hyperbolic case where K 0 and F are compact sets with 0 ∈ K 0 ∩F . In particular, it follows that K 0 , F ′ ⊂ D.
Next we present a simple example illustrating control sets.
Example 21 Consider for d = 2 and m = 1 We claim that for this hyperbolic matrix A the unique control set with nonvoid interior is D = (−1, 1) × [−2, 2]. The stable subspace associated with the eigenvalue 1 2 of A is the y-axis, the unstable subspace associated with the eigenvalue 2 is the x-axis. For a constant control u ∈ [−1, 1], one computes the equilibrium as (x(u), y(u)) ⊤ = (u, 2u) ⊤ . In particular. for u = 1 and u = −1 one obtains the equilibria resp. It is clear that for all u ∈ (−1, 1) the equilibrium (−u, 2u) ⊤ is in the interior of the control set D. Furthermore, observe that for x 0 > 1 one has in the next step 2x 0 + u > x 0 and for x 0 < −1 one has 2x 0 + u < x 0 . If y 0 > 2, then 1 2 y 0 + u < 1 2 y 0 + 1 ≤ y 0 and if y 0 < −2, then 1 2 y 0 + u ≥ 1 2 y 0 − 1 > y 0 . Hence solutions starting left of the vertical line x = −1 and right of x = 1 have to go to the left and to the right, respectively. Solutions which start above the horizontal line y = 2 and below y = −2, have to go down and up, respectively. This shows that the control set must be contained in (−1, 1) × [−2, 2]. The controllability property within D can be seen by the following analysis. If we start in an equilibrium (x(α), y(α)) ⊤ = (−α, 2α) ⊤ , α ∈ (−1, 1), we get e.g.

Invariance pressure
In this section we recall the concept of invariance pressure considered in [1], [2], [18] where potentials are defined on the control range. Furthermore, we introduce the generalized version of total invariance pressure, where the potentials are defined on the product of the state space and the control range. Again we consider the general system (1). A pair (K, Q) of nonvoid subsets of M is called admissible if K ⊂ Q is compact and for each x ∈ K there exists u ∈ U such that ϕ(N, x, u) ⊂ Q. For an admissible pair (K, Q) and τ > 0, a (τ, K, Q)-spanning set S of controls is a subset of U such that for all x ∈ K there is u ∈ S with ϕ(k, x, u) ∈ Q for all k ∈ {1, . . . , τ }. Denote by C(U, R) the set of continuous function f : U → R which we call potentials.
For a potential f ∈ C(U,

Definition 22
The invariance pressure P inv (f, K, Q) of control system (1) is defined by For the potential f = 0, this reduces to the notion of invariance entropy, P inv (0, K, Q) = h inv (K, Q).
In order to define the total invariance pressure associate to every control u in a (τ, K, Q)-spanning set S of controls an initial value x u ∈ K with ϕ(k, x u , u) ∈ Q for all k ∈ {1, . . . , τ }. Then a set of state-control pairs of the form is called totally (τ, K, Q)-spanning. Denote by C(Q×U, R) the set of continuous function f : Q × U → R which we again call potentials. For a potential f ∈ Definition 23 The total invariance pressure P tot (f, K, Q; Σ) of control system (1) is defined by Note that by continuity and monotonicity of the logarithm, Furthermore −∞ < a τ (f, K, Q) ≤ ∞ for every τ ∈ N, every admissible pair (K, Q), and every potential f if every countable totally spanning set contains a finite totally spanning subset, cf. [2,Remark 7]. If f (x, u) is independent of x, i.e., it is a continuous function on U , the total invariance pressure coincides with the invariance pressure.
Remark 24 The definition of totally (τ, K, Q)-spanning sets is inspired by the definition of spanning sets for (K, Q) in Wang, Huang, and Sun [15, p. 313], where a similar notion is introduced in the context of invariant partitions which provide an alternative definition of invariance entropy..

The next elementary proposition presents some properties of the function
Proposition 25 The following assertions hold for an admissible pair (K, Q), functions f, g ∈ C(Q × U, R) and c ∈ R: (i) For f ≤ g one has P tot (f, K, Q) ≤ P tot (g, K, Q).
The following proposition shows that, in the definition of total invariance pressure, we can take the limit superior over times which are integer multiples of some fixed time step τ ∈ N. The proof is analogous to the proof given in [2, Theorem 20] for invariance pressure of continuous-time systems.
Proof. For every f ∈ C(Q × U, R), the inequality is obvious. For the converse note that the function g(x, u) Then for every k ≥ 1 there exists n k ∈ N 0 such that n k τ ≤ τ k < (n k + 1)τ and n k → ∞ for k → ∞. Since g ≥ 0 it follows that and consequently 1 τ k log a τ k (g, K, Q) ≤ 1 n k τ log a (n k +1)τ (g, K, Q).
Together with Proposition 25 (ii) and (13) applied to f − inf f , this shows that The following result is given in [2, Corollary 15] for continuous-time systems. The discrete-time case is proved analogously.
Proposition 27 Let K 1 , K 2 be two compact sets with nonvoid interior contained in a control set D ⊂ M and assume that every point in D is accessible. Then (K 1 , D) and (K 2 , D) are admissible pairs and for all f ∈ C(U, R) we have

Invariance pressure for linear systems
The main result of this section presents a formula for the invariance pressure of the unique control set with nonvoid interior for hyperbolic linear control systems of the form (9).
We start with a proposition providing an upper bound for the total invariance pressure of the unique control set with nonvoid interior, cf. Theorems 18 and 19. The proof uses arguments from [3] which in turn are based on a construction by Kawan [9,Theorem 4.3], [10, Theorem 5.1] (for the discrete-time case cf. also [10,Remark 5.4] and Nair, Evans, Mareels, Moran [12,Theorem 3]).
Let A + be the restriction of A to the unstable subspace E u . The unstable determinant of A is where n λ denotes the algebraic multiplicity of an eigenvalue λ of A.
Proposition 28 Consider a linear control system of the form (9) and assume that the pair (A, B) is controllable with a hyperbolic matrix A. Let D be the unique control set with nonvoid interior and let f ∈ C(D × U, R). Then there exists a compact set K ⊂ D with nonvoid interior such that the total invariance pressure satisfies where the infimum is taken over all τ ∈ N with τ ≥ d and all τ -periodic controls u with a τ -periodic trajectory ϕ(·, x, u) in intD such that u i ∈ intU for i ∈ {0, . . . , τ − 1}.
Proof. We will construct a compact subset K ⊂ D with nonvoid interior such that the inequality above holds. Observe that then by Proposition 27 the pair (K, D) is admissible.
We may suppose that A has real Jordan form R = T −1 AT . In fact, writing x = T x ′ one obtains with and D ′ := T −1 D the total invariance pressure P toz (f, K, D) coincides with the total invariance pressure P tot (f ′ , K ′ , D ′ ) of (14). Consider a τ 0 -periodic control u 0 (·) with τ 0 -periodic trajectory ϕ(·, x 0 , u 0 ) as in the statement of the theorem, hence Step 1: Choose a basis B of R d adapted to the real Jordan structure of R and let L 1 (R), . . . , L r (R) be the Lyapunov spaces of R, that is, the sums of the generalized eigenspaces corresponding to eigenvalues λ with the absolute value |λ| = ρ j . This yields the decomposition Let d j = dim L j (R) and denote the restriction of R to L j (R) by R j . Now take an inner product on R d such that the basis B is orthonormal with respect to this inner product and let · denote the induced norm.
Step 2: We fix some constants: Let S 0 be a real number which satisfies and such that ρ j < 1 implies ρ j + ξ < 1 for all j. Let δ ∈ (0, ξ). It follows that there exists a constant c = c(δ) ≥ 1 such that for all j and for all k ∈ N R k j ≤ c(ρ j + δ) k .
For every m ∈ N we define positive integers by , m ∈ N.
Since we assume controllability of (A, B) and τ 0 ≥ d there exists C 0 > 0 such that for every x ∈ R d there is a control u ∈ U with The inequality follows by the inverse mapping theorem. For the corresponding trajectory we find a constant C 1 > 0 such that for k ∈ {1, . . . , τ 0 } For b 0 > 0 let C be the d-dimensional compact cube C in R d centered at the origin with sides of length 2b 0 parallel to the vectors of the basis B. Choose b 0 small enough such that K := x 0 + C ⊂ D and B(u 0 (k), Cb 0 ) ⊂ U for all k ∈ {0, . . . , τ 0 }. This is possible, since x 0 ∈ intD and all values u 0 (k) are in the interior of U .
Step 3. Let ε > 0 and τ = mτ 0 with m ∈ N. By Theorem 19, the closure D is compact, hence for the continuous function f on the compact set D × U there is ε 1 > 0 such that for all (x, u), We may take m ∈ N large enough such that Furthermore, we may choose b 0 small enough such that Partition C by dividing each coordinate axis corresponding to a component of the jth Lyapunov space L j (R) into M j (τ ) intervals of equal length. The total number of subcuboids in this partition of C is r j=1 M j (τ ) dj . Next we will show that it suffices to take r j=1 M j (τ ) dj control functions to steer the system from all states in x 0 + C back to x 0 + C in time τ such that the controls are within distance ε 1 to u 0 and the corresponding trajectories remain within distance ε 1 from the trajectory ϕ(·, x 0 , u 0 ). Let y be the center of a subcuboid. By (17) there exists u = (u 0 , . . . , u τ 0 −1 ) such that ϕ(τ 0 , y, u) = 0 and u ∞ ≤ C 0 y ≤ C 0 b 0 < ε 1 .
Now consider an arbitrary point x ∈ C. Then it lies in one of the subcuboids and we denote the corresponding center of this subcuboid by y with associated control u = u(y). We will show in Step 4 that u 0 + u also steers x 0 + x back to x 0 + C and in Step 5 that the corresponding trajectory ϕ(k, x 0 + x, u 0 + u) remains within distance ε 1 of ϕ(k, x 0 , u 0 ), k ∈ {0, . . . , τ }.
Step 5. By linearity and formulas (17), (18), and (21) we can estimate for Together with (22) and (19) this shows that for k ∈ {0, 1, . . . , τ } Step 6. We have constructed r j=1 M j (τ ) dj control functions that allow us to steer the system from all states in K = x 0 + C back to x 0 + C in time τ and satisfy (24). By iterated concatenation of these control functions we obtain a totally (nτ, K, D)-spanning set S tot for each n ∈ N with cardinality This implies, using also (20), Since ε can be chosen arbitrarily small and S 0 arbitrarily close to log |det A + |, the assertion of the proposition follows. For the invariance pressure, we obtain the following consequence.
Corollary 29 Consider a linear control system of the form (9) and assume that the pair (A, B) is controllable with a hyperbolic matrix A. Let D be the unique control set with nonvoid interior and let f ∈ C(U, R). Then for every compact set K ⊂ D with nonvoid interior the invariance pressure satisfies where the infimum is taken over all τ ∈ N with τ ≥ d and all τ -periodic controls u with a τ -periodic trajectory ϕ(·, x, u) in intD such that u i ∈ intU for i ∈ {0, . . . , τ − 1}.
Proof. The assertion follows from Proposition 28, since every compact subset of D is contained in a compact subset K of D with nonvoid interior and the invariance pressure is independent of the choice of such a set K by Proposition 27.
Remark 30 Kawan [10, Theorem 3.1] derives for the outer invariance entropy h inv,out (K, Q), which is a lower bound for the invariance entropy, the formula h inv,out (K, Q) = log det A + .
Here (K, Q) is an admissible pair, K has positive Lebesgue measure, and Q is compact. For the potential f = 0, Corollary 29 shows that the invariance entropy satisfies implying that h inv (K, Q) = log det A + .
We proceed to prove a lower bound for the invariance pressure. Recall that with respect to A the state space R d can be decomposed into the direct sum of the center-stable subspace E sc and the unstable subspace E u which are the direct sums of all generalized real eigenspaces for the eigenvalues λ with |λ| ≤ 1 and |λ| > 1, resp. Let π : R d → E u be the projection along E sc .
Proposition 31 Let K ⊂ D be compact and assume that both K and D have positive and finite Lebesgue measure. Then for every f ∈ C(U, R) P inv (f, K, D) ≥ log det A + + inf (τ,x,u) where the infimum is taken over all (τ, x, u) ∈ N × D × U with τ ≥ d and πϕ(i, x, u) ∈ πD for i ∈ {0, 1, . . . , τ − 1}. Since for u ∈ S there is x ∈ K with πϕ(i, x, u) = 0 ∈ πD for i ∈ {0, 1, . . . , τ −1}, the assertion for trivial unstable subspace E − follows. Now suppose that E u is nontrivial. We may assume that P inv (f, K, Q) < ∞ and hence and all considered spanning sets are countable. Note that by invariance of E sc and E u the induced system on E u is well defined with trajectories πϕ(k, x, u), k ∈ N. For each u in a (τ, K, D)-spanning set S define Thus πK = u∈S πK u . Since D is measurable, each set πK u is measurable as the countable intersection of measurable sets. We denote the Lebesgue measure in R d by µ d and the induced measure on E u by µ. The linear part of the affine-linear map πϕ τ,u (x) is given by (A + ) τ , hence it follows that µ(πD) ≥ µ(πϕ τ,u (πK u )) = πϕτ,u(πKu) dµ = πKu det(A + ) τ dµ = µ(πK u ) det A + τ .
Theorem 32 Consider a linear control system of the form (9) and assume that the system without control restriction is controllable in R d , the matrix A is hyperbolic, and the control range U is a compact convex neighborhood of the origin with U = intU . Let D be the unique control set with nonvoid interior. Then D is bounded and for every compact set K ⊂ D with nonvoid interior and every potential f ∈ C(U, R), the invariance pressure is given by