Null controllable sets and reachable sets for nonautonomous linear control systems

Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.

Abstract. Under the assumption of lack of uniform controllability for a family of time-dependent linear control systems, we study the dimension, topological structure and other dynamical properties of the sets of null controllable points and of the sets of reachable points. In particular, when the space of null controllable vectors has constant dimension for all the systems of the family, we find a closed invariant subbundle where the uniform null controllability holds. Finally, we associate a family of linear Hamiltonian systems to the control family and assume that it has an exponential dichotomy in order to relate the space of null controllable vectors to one of the Lagrange planes of the continuous hyperbolic splitting.
1. Introduction. In this paper we study a time-dependent linear control system of the form x = A 0 (t) x + B 0 (t) u(t) , (1.1) where x ∈ R n is the state vector and u(t) ∈ R m is a control vector. The functions A 0 , B 0 are assumed to be bounded and uniformly continuous on R, with values in the sets of real matrices of the appropriate dimensions.
We are concerned with the null controllability properties of the nonautonomous problem (1.1). Among the main questions that we pose, we point out two: to describe the dimension, topological structure and other dynamical characteristics of the null controllable set (composed of those states x 0 ∈ R n for which there exists a suitable control u steering x 0 to 0); and to do the same with the T -reachable sets (i.e., the sets of states x 0 ∈ R n for which there exists a suitable control u steering 0 to x 0 in time T ).
When dealing with a nonautonomous problem, it is frequent to embed it into a family which describes a flow, and which therefore allows one to make use of tools coming from the topological dynamics and the ergodic theory. By means of the standard Bebutov procedure of the hull construction, the system (1.1) becomes a particular one of the family where Ω is a compact metric space, σ : R × Ω → Ω, (t, ω) → ω·t is a continuous flow, and the matrix-valued functions A : Ω → M n×n (R) and B : Ω → M n×m (R) are continuous. Hence, each of the systems of this family is given by the evaluation of A and B along one of the orbits of the flow (Ω, σ). We will analyze the null controllability properties of the whole family, from where one can derive those of the initial system by means of an obvious "restriction" process. The dynamical and ergodic properties of the two flows that the linear family induces on the linear bundle and on the Grassmannian bundles will be one of the main tools in our analysis. We will also make use of another fundamental tool: often, the controllability properties of the family (1.2) are closely related to those of the family of time-reversed control systems for which the coefficient functions are evaluated along the orbits of the new timereversed flow σ − : R × Ω → Ω, (t, ω) → ω·(−t). It will be clear in what follows that the analysis of the systems (1.4) provides valid information on the family (1.2). As a matter of fact, part of the results will be formulated for the family (1.4), and later translated to the case of (1.2). Of course, the simplest situation corresponds to the case where the family of control systems (1.2) is uniformly null controllable (i.e., all its systems are null controllable), which is equivalent to say that the family of time-reversed control systems (1.4) is uniformly null controllable (see Fabbri et al. [5]). If this is the case, then the set of null controllable points of the system (1.2) coincides with R n for all ω ∈ Ω, and the same happens with the T -reachable sets if the time T is large enough. Therefore, in this paper we will always consider the case of existence of at least one system of the family (1.2) which is not null controllable.
We will now describe the structure of the paper and give a brief summary of our results. In Section 2 we recall the main notions of topological dynamics and ergodic theory which will be required in the rest of the paper. In particular, the linear and Grassmannian flows induced by (1.3) and the Lagrangian flow induced by a family of linear Hamiltonian systems are introduced here. The concepts of exponential dichotomy and rotation number are also given. Now we summarize the contents of Section 3, on which we give precise definitions of all the concepts involved. For a fixed ω ∈ Ω, we represent by E(ω) and E(ω) the sets of null controllable points for the systems (1.4) and (1.2), respectively. We will prove that E(ω) is a vector space which coincides with the T -reachable set of the system of (1.2) corresponding to ω·(−T ) for T large enough. When ω varies in Ω, the sets E(ω) present some properties of semi-invariance under the flow induced by (1.3) on the linear bundle. We will also prove that dim E(ω) is a lower semicontinuous function which turns out to be constant on the minimal subsets of Ω, and which attains its minimum value on one of these minimal subsets; moreover, dim E(ω) is locally constant on the residual subset of its continuity points, on which it attains its maximum value. In addition, if Ω 0 is a compact invariant subset of Ω on which dim E(ω) is constant (which is the case if Ω 0 is minimal), then E(ω) = E(ω) NULL CONTROLLABLE SETS AND REACHABLE SETS 3 for all ω ∈ Ω 0 , and the set {(ω, x) | ω ∈ Ω 0 , x ∈ E(ω)} ⊆ Ω 0 × R n defines a closed invariant subbundle for the linear flow with the following property: it is the greatest subset of Ω 0 × R n on which the families (1.2) and (1.4) are uniformly null controllable.
Before describing the second group of results, which are the core of Section 4, we need some preliminary information. It is frequent to associate a family of quadratic functionals Q ω (t, x, u) = 1 2 ( x, G(ω·t) x + 2 x, g(ω·t) u + u, R(ω·t) u ) , ω ∈ Ω (1.5) to the control family (1.2), where G : Ω → M n×n (R), g : Ω → M n×m (R) and R : Ω → M m×m (R) are continuous, G and R are symmetric, and R > 0. Here, as usual, , represents the Euclidean inner product in R m and R n . If the pair (x(t), u(t)) solves the system (1.2) for a point ω ∈ Ω, then t2 t1 Q ω (t, x(t), u(t)) dt often represents the amount of "supply" (meaning in general energy) which has to be delivered to the system in order to transfer it from its state in time t 1 to its state in time t 2 . For this reason Q ω is called supply rate or power function. We will refer to the pair given by (1.2) and (1.5) as a linear-quadratic (or LQ, for short) control problem.
Many question involving LQ control problems have been extensively analyzed during the last decades. One of the more classical is that of fixing a point ω ∈ Ω and a initial state x 0 ∈ R n , and finding, among all the L 2 -pairs (x(t), u(t)) which solve (1.2) and satisfy x(0) = x 0 , that for which the quantity ∞ 0 Q ω (t, x(t), u(t)) dt is minimum. Another classical problem is to determine conditions ensuring the (normal or strict) dissipativity of the LQ problems as well as the existence of (normal or strong) storage functions. Roughly speaking, a dissipative system requires energy coming from the environment to move from its equilibrium position to another one, and the storage function, if it exists, bounds from below the energy that the system requires to pass from the state of minimum storage to a given state.
A fundamental tool for the analysis of these two problems is the description of the dynamics induced by the family of linear Hamiltonian systems associated to the LQ problem given by The family (4.3) was firstly associated to the family of infinite-horizon minimization problems by means of the Pontryagin Maximum Principle. This association requires an additional condition of uniform stabilization for the systems (1.2). When this condition holds, the minimization problem is solvable if and only if the family (1.6) has an exponential dichotomy over Ω (that is, it satisfies the so-called Frequency Condition) and none of the Lagrange planes l + (ω) associated to the solutions which are bounded at +∞ lies in the vertical Maslov cycle (that is, the family satisfies the so-called Nonoscillation Condition; or, in other words, for each ω ∈ Ω, l + (ω) has a basis whose elements are the columns vectors of a 2n × n matrix L + 1 (ω) L + 2 (ω) with det L + 1 (ω) = 0 for all ω ∈ Ω). This result was previously proved by Yakubovich [24,25] in the periodic case and extended by Fabbri et al. [4] to the general nonautonomous case. The interested reader can find in Chapter 7 of Johnson et al. [10] an exhaustive description of these results. Similar conditions are used in the description of the dissipativity of the LQ problems in the papers [26,5,9,8] and in Chapter 8 of [10].
It is obvious that we can simply define (1.6) from (1.2) and (1.5), without any extra assumption on stabilization. This is what we do in this paper. Recall that we consider the case of existence of systems of the family (1.4) which are not null controllable. With the focus put on obtaining results which involve as many situations as possible, we will impose the Frequency Condition, but not the Nonoscillation Condition. Roughly speaking, our results relate the set of null controllable points for the system (1.4) to the image space Im L − 1 (ω), for each ω ∈ Ω. Here is any matrix representing the Lagrange plane l − (ω) of the solutions which are bounded at −∞. Clearly, different quadratic forms Q ω will give rise to families (1.6) with or without exponential dichotomy and, even in the case of existence, with different properties of the Lagrange planes l − (ω). Therefore, if the only purpose is to know as much as possible of the null controllable sets, then we can play with the choice of the supply rate: changing it may allow us to go deeper in the analysis.
The main results of Section 4 are now summarized. Under the Frequency Condition and the additional assumption of the existence of an ergodic measure m 0 on Ω with full topological support and for which the rotation number of the family (1.6) is zero, we prove that the set E(ω) is a vector subspace of Im L − 1 (ω). (Note that the space Im L − 1 (ω) and dim Ker L − 1 (ω) are independent of the basis chosen for l − (ω).) We check that the vector space Im L − 1 (ω) has properties of semi-invariance under the linear flow induced by (1.6), and that dim L − 1 (ω) is a lower semicontinuous function with analogous properties to those previously obtained for dim E(ω). Finally, we assume that the dimension of Im L − 1 (ω) is constant on Ω and show that the Lagrange subbundle L − = {(ω, z) | ω ∈ Ω, z ∈ l − (ω)} determined by the exponential dichotomy contains a closed invariant subbundle whose sections intersect the vertical Lagrange plane l v ≡ 0n In ; and we also assume that dim E(ω) is constant on Ω and prove that L − contains another closed invariant subbundle such that the first components of the vectors of its sections belong to E(ω) for all ω ∈ Ω. As pointed out before, similar results can be obtained for the vector space E(ω) of the null controllable points for (1.2), which due to the time-reversion is related to the vector space Im L + 1 (ω). The setting that we consider throughout the paper, that is, the existence of systems of the family (1.2) which are not null controllable, is closely related to the existence of abnormal systems of the family. It is proved in Johnson et al. [8] that there are minimal subsets Ω * ⊆ Ω for which all the systems (1.6) are abnormal (i.e., they have solutions of the form 0 z2(t) for t ∈ R), and such that at least one of the associated Lagrange planes l ± (ω) lies on the vertical Maslov cycle C, defined in Subsection 2.2.1, for all ω ∈ Ω * . A more precise description of this connection will also be included at the beginning of Section 4.
Let (Ω, σ) be a continuous flow. The σ-orbit of a point ω ∈ Ω is the set {σ t (ω) | t ∈ R}. Restricting the time to t ≥ 0 or t ≤ 0 leads to the definition of forward or backward σ-semiorbit. A subset Ω 1 ⊆ Ω is σ-invariant (resp. positively σ-invariant or negatively σ-invariant) if σ t (Ω 1 ) = Ω 1 for every t ∈ R (resp. t ≥ 0 or t ≤ 0). A σ-invariant subset Ω 1 ⊆ Ω is minimal if it is compact and does not contain properly any other compact σ-invariant set; or, equivalently, if the two semiorbits of any of its elements are dense in it. The continuous flow (Ω, σ) is recurrent or minimal if Ω itself is minimal.
If the forward semiorbit of a point ω 0 ∈ Ω is relatively compact, its omega-limit set O(ω 0 ) is given by those points ω ∈ Ω such that ω = lim m→∞ σ(t m , ω 0 ) for some sequence (t m ) ↑ ∞. This set is nonempty, compact, connected and positively σinvariant. The definition and properties of an alpha-limit set A(ω 0 ) are analogous, working now with sequences (t m ) ↓ −∞.
The summary of the most basic notions required in the present paper is completed with some definitions concerning the measures on Ω. Let m be a normalized Borel measure on Ω; i.e. a finite regular measure defined on the Borel subsets of Ω and with m(Ω) = 1. The measure m is σ-invariant if m(σ t (Ω 1 )) = m(Ω 1 ) for every Borel subset Ω 1 ⊆ Ω and every t ∈ R. If, in addition, m(Ω 1 ) = 0 or m(Ω 1 ) = 1 for every σ-invariant subset Ω 1 ⊆ Ω, then the measure m is σ-ergodic. From now on Ω will indicate a compact metric space and σ : R × Ω → Ω a continuous flow, and we will represent ω·t = σ(t, ω).

Linear Flow. Consider the family of linear systems
where A is a continuous real n × n matrix-valued function on Ω. We use the notation (2.1) ω to refer to the system of the family (2.1) corresponding to the point ω ∈ Ω; and we will do the same with the remaining families of systems appearing in the paper. Denote by U A (t, ω) the fundamental matrix solution of (2.1) ω with U A (0, ω) = I n , which is globally defined and nonsingular, and jointly continuous on R × Ω. By the uniqueness of solutions, and hence the map defines a continuous flow on Ω×R n , which is of linear skew-product type: it preserves the flow on Ω, which can be considered as the base of the bundle Ω × R n ; and it is linear on the fiber component. Frequently, a family of this type comes from a single nonautonomous linear system z = A 0 (t) z by means of the well-known Bebutov-type construction: if A 0 is bounded and uniformly continuous on R, then its hull Ω, defined by Ω = cls{A t | t ∈ R} is a compact metric space and the time-translation defines a continuous flow σ on it. Here A t (s) = A 0 (t + s) and the closure is taken in the compact-open topology of the set of bounded and uniformly continuous n × n matrix-valued functions. The base space Ω can be understood as the space in which the nonautonomous law varies with respect to time. Under additional recurrence properties on A 0 , the base flow is minimal. This is the case if A 0 is almost periodic or almost automorphic.
Weaker conditions on A 0 may provide a non minimal hull, which can contain different minimal subsets. In some of these cases, the solutions of the different linear systems of the family show a significatively different qualitative behavior.
The same Bebutov procedure can be carried out for the pair of matrix-valued functions (A 0 , B 0 ) giving rise to the initial control system (1.1) in order to include it in the family (1.2). 0 C ] for B ∈ GL(k, R) and C ∈ GL(d − k, R). Here * represents any k × (d − k) matrix and 0 represents the zero (d − k) × k matrix. With this identification, which provides G k (W ) with a differentiable structure, G k (W ) is the Grassmannian manifold of the k-dimensional linear subspaces of W . The set G k (W ) is a compact and connected manifold, which agrees with the real projective line if k = 1. We refer the reader to Matsushima [14] for the proofs of these properties.
Then the family (2.1) defines a continuous flow determines an isomorphism of R n for any t ∈ R and ω ∈ Ω.
2.2. Linear Hamiltonian systems. Consider now the family of linear Hamiltonian systems where H is a continuous real 2n×2n matrix-valued function on Ω and H 2 and H 3 are n×n symmetric matrices. Let U (t, ω) = U1(t,ω) U3(t,ω) U2(t,ω) U4(t,ω) represent the fundamental matrix solution of the system (2.5) ω with U (0, ω) = I 2n , which is globally defined. Then, as before, the map defines a continuous skew-product flow on Ω × R 2n . The symplectic nature of the matrix U provides this flow with some additional properties, which we now describe.
2.2.1. The Lagrangian flow. Recall that two vectors z and w in R 2n are isotropic if z T Jw = 0, where J = 0n −In In 0n . Any linear subspace l ⊂ R 2n whose vectors are pairwise isotropic satisfies dim l ≤ n, since l is contained in the Euclidean subspace orthogonal to J·l = {J z | z ∈ l}. An n-dimensional linear subspace l ⊂ R 2n is a (real ) Lagrange plane if z T Jw = 0 for all z and w in l. The space L R of all real Lagrange planes of R 2n is a compact orientable manifold of dimension n(n + 1)/2: see [14] and Mishchenko et al. [15]. An element l of L R can be represented by a 2n × n real matrix L1 L2 of range n with L T 1 L 2 = L T 2 L 1 . The representation means that the column vectors form a basis of the Lagrange subspace. Therefore, L1 L2 and G1 G2 represent the same Lagrange plane if and only if there exists a nonsingular n × n real matrix Q such that L 1 = G 1 Q and L 2 = G 2 Q. We will write l ≡ L1 L2 in what follows.
The matrix-valued function H belongs to the symplectic Lie algebra sp(n, is the 2n×2n zero matrix 0 2n for all ω ∈ Ω and t ∈ R. As a consequence of this fact, the vector space U (t, ω)·l = {U (t, ω) z | z ∈ l} is a new Lagrange plane for all t ∈ Ω and ω ∈ Ω: it has dimension n since U (t, ω) defines an isomorphism on R 2n ; and if z, w ∈ l then z T U T (t, ω) JU (t, ω) w = z T J w = 0. This property implies that the map defines a continuous skew-product flow on Ω × L R . In addition, if l ≡ L1 L2 , then . Consider the open and dense subset D of L R defined by Obviously, each l ∈ D admits a unique representation of the form In M , and the n × n matrix M has to be symmetric. In addition, l ≡ L1 L2 belongs to D if and only if det where l v is the Lagrange plane generated by the n last coordinate vectors: l v ≡ 0n In .

2.2.2.
Exponential dichotomy. Now we recall the definition of exponential dichotomy (see, e.g., Chapter 1 of [10] for more details). The Euclidean norm in R 2n is fixed and represented by · .
Remark 2.3. In general it is not possible to ensure the existence of continuous functions R. FABBRI, S. NOVO, C. NÚÑEZ AND R. OBAYA 2.2.3. Rotation number. The section is completed by recalling the definition of the rotation number of the family (2.5) with respect to a given σ-ergodic measure m 0 on Ω. This concept will be used in Section 4, in which will just work with families of Hamiltonian systems z = H1 H3 H2 −H T 1 z for which H 3 takes positive semidefinite values (H 3 ≥ 0). This fact allows us to choose, among the many equivalent definition of the rotation number (see Chapter 2 of [10], which contains and extend the previous results of [16], [3] and [8]), that based on the characteristics of the so-called proper focal points, which is valid just for those Hamiltonian systems with H 3 ≥ 0. This definition involves several concepts and properties which will be useful later.
Take a conjoined basis for the system (2.5) ω ; i.e., a 2n × n matrix solution , which means that this solution intersects the vertical Maslov cycle C (defined by (2.8)) at t 0 . Among these points, the so-called proper focal points, defined below, are fundamental in the analysis of the oscillatory properties of the Hamiltonian systems when H 3 ≥ 0. Under this hypothesis, it is shown in Lemma 2.34 of [10] and in Theorem 3 of [13] that, given any interval [a, b] ⊂ R, there exists a finite number of points a = t 1 < . .
for all s 1 , s 2 ∈ (t j , t j+1 ). The last contention is a trivial consequence of the piecewise constant character. All this justifies the equivalence stated in the next definition (see, e.g., Definition 1.1 of Wahrheit [23]).
The next result is proved in Section 3.1 of [8].
consider the conjoined basis L1(t,ω) L2(t,ω) ≡ U (t, ω)·l, and denote by J ω,l (t) the set of its proper focal points contained in the interval (0, t] and by m(t * ) the multiplicity of t * ∈ J ω,l (t). Then there exists an α(m 0 ) ≥ 0 such that for m 0 -almost every ω ∈ Ω and all l ∈ L R . Definition 2.6. Assume that H 3 ≥ 0. The rotation number of the family (2.5) with respect to the σ-ergodic measure m 0 is the quantity α(m 0 ) given in Theorem 2.5.
Note that the rotation number is zero in the particular case in which for all the points ω in a subset of Ω with positive measure m 0 there exists a conjoined basis of (2.5) ω for which the set of proper focal points is upper-bounded. 3. Null controllability and reachable sets. Let Ω be a compact metric space with a continuous flow σ(ω, t) = ω·t. Consider the family of time-dependent linear control systems Denote by x(t, ω, x 0 , u) the solution of the system (3.1) ω for control u satisfying  of Ω contains at least one point ω 1 such that the corresponding system (3.1) ω1 is null controllable. The proof of this property appears in Johnson and Nerurkar [6]. In particular, the family is uniformly null controllable if and only if all its systems are null controllable.
Next consider the time-reversed linear control family For each point ω ∈ Ω we consider the set of points which are null controllable for the time-reversed linear control system (3.3) ω , that is, According to Remark 3.3, the family of control systems (3.1) is uniformly null controllable (which according to Proposition 2.5 of [5] is equivalent to say that the family of time-reversed control systems (3.3) is uniformly null controllable) if and only if the set E(ω) is equal to R n for all ω ∈ Ω. The situation which is interesting for the purposes of this paper is the case in which the family of control systems (3.1) is not uniformly null controllable, or equivalently, E(ω) R n for some ω ∈ Ω.
The set E(ω) can be related to the set of those points of R n which can be reached from 0 at time T for the system (3.1) ω·(−T ) , that is, as the following result proves.
Proposition 3.4. Fix a point ω ∈ Ω. Then, , which according to (3.2) and the equality 2)) is equivalent to say that In other words, and according to (i), x 0 ∈ E(ω) if and only if there exists a T ≥ 0 such that x 0 ∈ E T (ω). This proves the first equality in (iii), which in turn makes it immediate to deduce from (ii) that E(ω) is a vector space too. In addition, since E T (ω) is nondecreasing in T and has a bounded dimension, there exists a minimum T (ω) such that E(ω) = E T (ω) for every T > T (ω), which proves the remaining assertions.
(iii) Since 0 ≤ d E (ω·t) ≤ n for all t ∈ R, the assertion is a consequence of (ii).
Next we consider the maps where α(ω) and β(ω) satisfy the conditions of Proposition 3.5(iii), and the quantities The following result collects several properties of these functions and quantities.
(i) It is an easy consequence of the definitions and Proposition 3.5(ii)&(iii).
(ii) From the definitions of d ± E (ω) and Proposition 3.5(ii), we have that d + E (ω) = max s∈R d E (ω·s) and d − E (ω) = min s∈R d E (ω·s), from where the statements follow. (iii) We will only prove that d + , because the other inequality is proved in an analogous way. First we prove that d E (ω 1 ) ≤ d + E (ω). If d E (ω 1 ) = 0 the inequality is obvious, so let us assume that d 1 = d E (ω 1 ) ≥ 1, take T > T (ω 1 ) and consider an n × d 1 matrix X(ω 1 ) of rank d 1 with columns x(T, ω 1 ·(−T ), 0, u j ) for some controls u j : [0, T ] → R m for j = 1, . . . , d 1 . Since ω 1 ∈ O(ω), we take a sequence (t k ) ↑ ∞ with ω 1 = lim k→∞ ω·t k and the n × d 1 matrix X(ω·t k ) with columns x(T, ω·(t k − T ), 0, u j ) for j = 1, . . . , d 1 which are d 0 vectors of E(ω·t k ). Since lim k→∞ X(ω·t k ) = X(ω 1 ), the lower semicontinuity of the rank function on matrices provides a k 0 such that rank X(ω·t k ) ≥ rank X(ω 1 ) = d 1 for each k ≥ k 0 ; that is, the columns are d 1 linearly independent vectors of E(ω·t k ). Consequently, if k is large enough, then d + e. In addition, the Poincaré Recurrence Theorem (see [1]) ensures that there exists a subset Ω 0 of full measure such that ω ∈ O(ω) ∩ A(ω) for ω ∈ Ω 0 , and hence the coincidence of d E , d + E and d − E on a set of full measure follows from (iii). (v) The constant character of d E , d + E and d − E on Ω * follows immediately from (iii), (i), and the minimal character of Ω * . In particular, d E (ω) = d E (ω·t) for ω ∈ Ω * and t ∈ R. Hence Proposition 3.5(i) ensures that U A (t, ω)·E(ω) = E(ω·t) because both spaces have the same dimension.
(vi) Take ω 0 with d m E = d E (ω 0 ) and a minimal subset Ω * ⊆ A(ω 0 ). From (iii) and (i) we deduce that d + for each ω ∈ Ω * . (vii) Proposition 3.5(iv) states that d E is lower semicontinuous. Consequently, the set Ω c ⊆ Ω of its continuity points is a residual set which is necessarily open because d E only takes integer values. Hence, for each ω ∈ Ω c there exists an open ball B(ω, δ ω ) ⊂ Ω c on which d E is constant, that is, d E is locally constant on Ω c . Finally, we check that each ω ∈ Ω with d E (ω) = d M E is a continuity point. Let (ω k ) be a sequence such that lim k→∞ ω k = ω. From the definition of d M E and the lower semicontinuity of d E , we deduce that and hence lim k→∞ d E (ω k ) = d E (ω), as claimed.
(viii) It follows from (iii) and (i) that d + , and therefore the three values coincide. In addition, d E (ω) ≤ d + E (ω) ≤ d − E (ω 0 ) for all ω ∈ Ω, and taking the maximum in ω ∈ Ω we deduce that d M E ≤ d − E (ω 0 ) ≤ d M E , which proves the first part of the statement. Note also that, since d + E (ω 0 ) = max s∈R d E (ω 0 ·s), d − E (ω 0 ) = min s∈R d E (ω 0 ·s), and they coincide with d E (ω 0 ), we deduce that d E is constant along the orbit of ω 0 . Next we check that Ω c ⊆ {ω ∈ Ω | d E (ω) = d M E } to prove the second part of the statement. We take ω ∈ Ω c .
(x) Fix T > T (ω) and take an n × d matrix X(ω) of rank d with columns x(T, ω·(−T ), 0, u j ) for some controls u j : [0, T ] → R m and j = 1, . . . , d, that is, the columns of X(ω) are a basis of the d-dimensional subspace E(ω). Let X(ω k ) be the n × d matrix with columns x(T, ω k ·(−T ), 0, u j ) for j = 1, . . . , d, which, according to Proposition 3.4(iv), are d vectors of E(ω k ). Since lim k→∞ X(ω k ) = X(ω), the lower semicontinuity of the rank provides a k 0 such that rank X(ω k ) ≥ rank X(ω) = d for each k ≥ k 0 ; that is, the columns of X(ω k ) are d linearly independent vectors of E(ω k ), and hence, by hypothesis, they form a basis of E(ω k ). This implies that lim k→∞ E(ω k ) = E(ω) in G d (R n ) (see, e.g., Proposition 1.25 of [10]).
(xii) The proof is analogous to that of (xi).
Next we consider the null controllable set for the control system (3.1) ω , The following result shows that E(ω) can be related to the reachable set for the system (3.1) ω from 0 at time T defined by Note that F T (ω) = E T (ω·T ) (see (3.5)). (ii) There exists a minimum time T (ω) ≥ 0 such that, for every T > T (ω), In addition, if T > T (ω), then F T (ω) = U A (T, ω)· E(ω), or equivalently, Proof.
shown in (i), finishes the proof.
(ii) From (3.5), (3.7) and Proposition 3.4(iii), we deduce that The next result characterizes some cases in which the sets E(ω) and E(ω) coincide, or in other words, the null controllable points for (3.1) ω coincide with the null controllable points for the time-reversed system (3.3) ω . Theorem 3.9. Fix ω ∈ Ω. Let T (ω) and T (ω) be the nonnegative times for E(ω) and E(ω) provided by Proposition 3.4(iii) and Proposition 3.7(ii) respectively.
(ii) If there exists a constant T such that T (ω·t) ≤ T and T (ω·t) ≤ T for each t ∈ R, then E(ω) = E(ω).
(ii) Proposition 3.5(ii) and the corresponding result for E (with the time reversed flow (t, ω) → ω·(−t)) prove that dim E(ω·t) is nondecreasing in t and dim E(ω·t) is nonincreasing in t. From Proposition 3.8, since T > T (ω) and T > T (ω), we obtain that is, dim E(ω·t) = dim E(ω·(t − T )) for each t ∈ R, which together with the monotonicity of the dimensions implies that dim E(ω·t) and dim E(ω·t) are constant, and the result follows from (i).
The following result provides conditions under which the hypotheses of Theorem 3.9(ii) hold. Proposition 3.10. Let T (ω) and T (ω) be the nonnegative times for E(ω) and E(ω) provided by Proposition 3.4(iii) and Proposition 3.7(ii) respectively.
To prove the second one, we deduce from Proposition 3.8(ii) that dim E(ω) ≤ d for all ω ∈ Ω. Hence, Proposition 3.8(i) ensures that if ω ∈ Ω and T > T * ≥ T (ω), then which proves that dim E(ω) = d for all ω ∈ Ω. Now the same argument as above completes the proof of (i).
(ii) This assertion follows from (i), Theorem 3.6(v) and the corresponding result for E(ω) showing that dim E(ω) is constant on each minimal subset.
Finally, as a consequence of the previous results, we prove that in some cases, even if the family (3.1) is not uniformly null controllable, there exists a subbundle of Ω × R n , which could be the trivial one Ω × {0}, on which the uniform null controllability holds, in the sense explained in the following statements. Note that the set Ω can be replaced for any of its σ-invariant subsets on which the main hypothesis (that is, the constant character of dim E(ω)) holds.
Theorem 3.11. Consider the subsets of Ω × R n given by and suppose that dim E(ω) is constant on Ω. Then E = E is a closed τ A -invariant subbundle of Ω × R n , and there exists a positive time T 0 > 0 such that, if (ω, x 0 ) ∈ E, then x 0 is null controllable in time T 0 for the systems (3.1) ω and (3.3) ω . In particular, this happens if Ω is minimal.

Null controllability and exponential dichotomy.
Let Ω be a compact metric space with a continuous flow σ(ω, t) = ω·t. As it was explained in the Introduction, in this section we consider a family of time-dependent linear control systems where A : Ω → M n×n (R) and B : Ω → M n×m (R) are continuous, together with a family of time-dependent quadratic forms where G : Ω → M n×n (R), g : Ω → M n×m (R) and R : Ω → M m×m (R) are continuous, G and R are symmetric, and R > 0. We will also consider the family of linear Hamiltonian systems defined from the LQ problems by where z = [ x y ] for x, y ∈ R n and .
The goal of this section is to relate the dynamical properties of the linear and Lagrangian flows induced by the family (4.3) to the null controllable sets E(ω) studied in the previous section. Recall that we can play with the choice of the supply rate in order to obtain Hamiltonian families (4.3) with different properties and hence to obtain a sharper analysis of the null controllability properties of (4.1). We refer the reader to Kratz [13] for a similar idea to study the reachable sets in the case in which B is symmetric, g ≡ 0 and R ≡ B.
Before stating the results, it is convenient to explain the relation between the lack of uniform null controllability and the presence of abnormal systems in the family (4.3). The interested reader can find in Reid [17] and Kratz [12] a previous analysis of abnormal systems, and in Reid [18] andŠepitka-Šimon Hilscher [19,20] a generalization of the concept and theory of principal solutions suitable for systems of this type.
Recall that U (t, ω) is the fundamental matrix solution of the system (4.3) for ω ∈ Ω with U (0, ω) = I 2n . We consider the linear subspaces for t in a positive half-line , for t in a negative half-line , for t ∈ R , and define d(ω) = dim Λ(ω) and d ± (ω) = dim Λ ± (ω). A complete analysis of the functions d, d + and d − is carried out in [8].
(iii) If x 0 ∈ Ker L 1 (t, ω), then the solution L1(t,ω) L2(t,ω) x 0 of (4.3) ω takes the form 0 z2(t) on (a, ∞), which proves (iii). Next, for a fixed (ω, l) ∈ Ω × L R , we define That is, these quantities measure the number of independent solutions of (4.3) ω with initial data in the subspace l which take the form 0 z2(t,ω) on a positive half-line, on a negative half-line and on the full line, respectively. Now assume that the family (4.1) is not uniformly null controllable (which is the setting we are interested in) and that the family (4.3) admits an exponential dichotomy over Ω (which will be one of the main hypotheses for the results of this section). Let Ω × R 2n = L + ⊕ L − be the corresponding decomposition with associated Lagrange planes l ± (ω) = {z | (ω, z) ∈ L ± } (see subsection 2.2.2). We define the functions d ± : Ω → {0, . . . , n}, ω → d ± (ω) = dim(Λ(ω) ∩ l ± (ω)) .
Let Ω * ⊆ Ω be one of the minimal sets for which all the systems are abnormal (see Proposition 4.2(ii)). Proposition 3.3 and Theorem 3.4 of [8], which relate d ± (ω) to d(ω), prove in particular that the functions d, d+ and d − are constant on Ω * , where d = d + + d − . Consequently, the number of linearly independent solutions of (4.3) ω of the form 0 z2(t,ω) can be calculated in terms of the number of linearly independent solutions of this form which have initial data in the subspaces l + (ω) and l − (ω); or, in other words, which are bounded as t goes to ±∞. And this number is the same for all the elements of Ω * . In addition, since d > 0 on Ω * , then either d + or d − (or both of them) is strictly positive on Ω * . That is, at least one of the associated Lagrange planes l ± (ω) lies on the vertical Maslov cycle C defined by (2.8) simultaneously for all ω ∈ Ω * . Now we are almost in a good position to begin with the analysis of the null controllable sets E(ω) for the time-reversed control family, defined by (3.4). In addition to the exponential dichotomy hypothesis, the analysis relies on the existence a σ-ergodic measure with full support for which the rotation number is zero. That is, we will work under these conditions: To formulate the first result we need to define the sets (4.5), which in turn requires the following perturbation result. Its proof is given in [8] (Theorem 4.18) and in [10] (Theorem 5.73). has an exponential dichotomy over Ω. Moreover, the Lagrange planes l ± ε (ω) lie in D = L R − C and if we represent l ± ε (ω) ≡ In M ± ε (ω) ∈ L R for each ω ∈ Ω, then we have Under Hypotheses 4.3 and with the notation established in Theorem 4.4 define, for each ω ∈ Ω, the set The following result shows that, if l − (ω) ≡ L − 1 (ω) L − 2 (ω) , then P (ω) = Im L − 1 (ω), and that E(ω) is a vector subspace of P (ω). It also proves that P (ω) has properties of semi-invariance under the linear flow induced by (4.3), and that dim P (ω) is a lower semicontinuous function with analogous properties to those previously obtained for dim E(ω).
Note that, in particular, P (ω) = R n (which happens in the particular case that the uniform null controllability property holds, since in this case E(ω) = R n ) if and only if M − (ω) exists (which is equivalent to say that L − 1 (ω) is nonsingular). So, the interesting case for the next result is that of non global existence of M − , since otherwise its points (ii)-(v) do not provide any valuable information. Nevertheless, even in the case of global existence of M − , the null controllable set E(ω) can be a proper subspace of P (ω) = R n .
(v) It follows from (iv) that the map t → dim Ker L − 1 (t, ω) = n − d P (ω·t) is nonincreasing, which together with relation (2.9) shows that Ker L − 1 (t, ω) is nonincreasing in t for the order given by the contention of vector spaces. This proves the first assertion in (v). (Incidentally, note that this proves that any conjoined basis for (4.3) ω representing the Lagrange planes l − (ω·t) has no proper focal points: see Definition 2.4). It also follows from (iv) the existence of a negative half-line (−∞, t − (ω)] on which the map t → d P (ω·t) attains its minimum value and of a positive half-line [t + (ω), ∞) on which it attains its maximum value. This fact together again with (2.9) ensures that the map t → Ker L − 1 (t, ω) is constant on (−∞, t − (ω)] and on [t + (ω), ∞). So (a) is proved. Properties (b) and (c) follow easily from (a), the definitions of d ± (ω, l − (ω)) and d(ω, l − (ω)), Proposition 4.2(iii), and, in the case of (c), the fact that Ker L − 1 (t + (ω), ω) ⊆ Ker L − 1 (t, ω) for all t ∈ R. (vi) Since d P (ω) = rank L − 1 (ω), the statement is a consequence of Remark 2.3 and the lower semicontinuity of the rank function on matrices.
Remark 4.6. Note that, as pointed out in the proof of the previous point (v), under Hypotheses 4.3 and with the notation established in the preceding result, the conjoined basis of (4.3) has no proper focal points. Therefore, an application of the Sturmian separation theory for linear Hamiltonian systems without controllability (see Theorem 1.5 ofŠimon Hilscher [21]) shows that any other conjoined basis of (4.3) has at most n proper focal points. In particular, Hypotheses 4.3 ensures that all the systems of the family (4.3) are nonoscillatory at +∞: see, e.g., [8] and [7].
Always under Hypotheses 4.3, we consider the maps where d P (ω) = dim P (ω) and t − (ω) and t + (ω) satisfy the conditions of Theorem 4.5(v). Next consider the quantities The following result collects several properties of these functions and quantities.
In particular, this is the case if Ω is minimal.
Proof. (i) This proof is completely analogous to that of Theorem 3.6.
(ii) Theorem 4.5(i) ensures that P (ω) is a linear space for all ω ∈ Ω, and by hypothesis its dimension is always d P . It follows from here and Theorem 4.5(iii) that U A (t, ω)·P (ω) = P (ω·t) for all t ∈ R and ω ∈ Ω. And finally, the analogous property of Theorem 3.6(x) for P (ω) proves the closed character of P .
The next result shows that, under the assumption of constant dimension for the subspaces Im L − 1 (ω) and E(ω) (which in particular holds if Ω is minimal), there exist two closed invariant subbundles L − 0 and L − E of L − such that the sections l − 0 (ω) of the first one intersect the vertical Lagrange plane l v ≡ 0n In , and such that the first components of the vectors of the sections l − E (ω) of the second one belong to E(ω) for all ω ∈ Ω. Recall that the flows τ A and τ H are defined by (2.3) and (2.6).
Proof. (i) It is clear that l − 0 (ω) is a vector space for each ω ∈ Ω. Note also that the vector 0 z2 belongs to l − (ω) if and only if there exists a vector x 0 ∈ R n such that ., x 0 ∈ Ker L − 1 (ω), which has constant dimension given by n − d P (ω) = n − d P . This together with the fact that the rank of (ω) is n shows that dim l − 0 (ω) = n − d P for all ω ∈ Ω, as stated.
Fix ω ∈ Ω and represent U (t, ω) ) is constant, and hence the equalities U (t, ω) Finally, we must check that lim k→∞ l − 0 (ω k ) = l − 0 (ω) in the Grassmannian manifold G n−d P (R 2n ) in the case that lim k→∞ ω k = ω. The compactness of G n−d P (R 2n ) provides a subsequence (ω kj ) such that lim j→∞ l − 0 (ω kj ) = g ∈ G n−d P (R 2n ). Moreover, for each j we can choose a representant of l − 0 (ω kj ) of the form 0 G k j for some n × (n − d p ) matrix G kj of rank n − d P ; here 0 denotes the n × (n − d P ) null matrix. Thus, we deduce from Proposition 1.24 of [10] that g has a representant of the same form, and hence g = l − 0 (ω), which shows that the subbundle is closed. (ii) It is also clear that l − E (ω) is a vector subspace for each ω ∈ Ω. Moreover, z1 z2 ∈ l − E (ω) if and only if there exists a vector c ∈ R n such that . . d E be a basis of E(ω), and consider the vectors . . , d E , which are linearly independent. We also take a basis of l − 0 (ω) given by Then, it is easy to deduce that the vectors for u(t) = R −1 (ω·t) −g T (ω·t) + B T (ω·t) c . Thus, L − 1 (t, ω) c = x(t, ω, L − 1 (ω) c, u) and since L − 1 (ω) c ∈ E(ω), we conclude from Proposition 3.4(iv) that L − 1 (t, ω) c ∈ E(ω·t), which proves the invariance.
The last assertion of the theorem follows immediately from (i), (ii), and Theorems 3.6(v) and 4.7(v).

4.1.
Null controllable sets for the initial family. The results seen so far in this section relate the properties of the null controllable sets E(ω) for the timereversed control systems (3.3) to the properties of the subbundle L − provided by the exponential dichotomy. But they can be easily translated to the null controllable sets E(ω) for the initial control systems (3.1) and the subbundle L + , as we will explain.
In particular, the assertions in (ii), (iii) and (iv) hold if Ω is minimal.