Controllability of periodic linear systems, the Poincare sphere, and quasi-affine systems

For periodic linear control systems with bounded control range, an autonomized system is introduced by adding the phase to the state of the system. Here a unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior exists. It is determined by the spectral subspaces of the homogeneous part which is a periodic linear differential equation. Using the Poincar\'e sphere one obtains a compactification of the state space allowing us to describe the behavior near infinity of the original control system. Furthermore, an application to quasi-affine systems yields a unique control set with nonvoid interior.

1. Introduction.We study controllability properties for periodic linear control systems and give an application to quasi-affine control systems.Periodic linear control systems have the form where A ∈ L ∞ (R, R d×d ) and B ∈ L ∞ (R, R d×m ) are T -periodic for some T > 0.
We suppose that the controls u = (u 1 , . . ., u m ) have values in a bounded convex neighborhood U of the origin in R m .The set of admissible controls is We denote the solutions (in the Carathéodory sense) of (1.1) with initial condition x(t 0 ) = x 0 by ϕ(t; t 0 , x 0 , u), t ∈ R. The homogeneous part of (1.1) is the (uncontrolled) homogeneous periodic differential equation ẋ(t) = A(t)x(t). (1.2) Nonautonomous control systems can be autonomized by including time in the state of the system.This is useful, if recurrence properties can be exploited; cf.Johnson and Nerurkar [16].In the T -periodic case, it suffices to add the phases τ ∈ [0, T ) to the states in R d (cf.Gayer [15] for general periodic nonlinear systems) and we follow this approach.We give a spectral characterization of the reachable sets generalizing Sontag [24,Corollary 3.6.7]for autonomous linear control systems; cf. also [24, p. 139] for some historical remarks.The proof also uses arguments from Colonius, Cossich, and Santana [9, Theorem 15] for autonomous discrete-time systems.This yields a characterization of the unique control set (i.e., a maximal set of approximate controllability) with nonvoid interior.The Poincaré sphere from the global theory of nonlinear differential equations (introduced by Poincaré [23] for polynomial differential equations) provides a compactification of the state space; cf. the monograph Perko [22,Section 3.10] and, e.g., Valls [26] for a recent contribution.This leads us to a description of the behavior "near infinity" of the original control system.Since the induced system on the Poincaré sphere is obtained by projection of a homogeneous system it suffices to consider its restriction to the upper hemisphere.Alternatively one might consider the induced system on projective space.In [11] we have used the latter approach for autonomous affine control systems.We remark that Da Silva [12] has generalized [24,Corollary 3.6.7] in another direction, for linear control systems on solvable Lie groups.General background on control of periodic linear systems is contained in Bittanti and Colaneri [2].The present paper may also be considered as a contribution to a Floquet theory of periodic control systems.They involve two T -periodic matrix functions A(•) and B(•) and a periodic coordinate change can transform only one of them to a constant matrix, hence periodic linear systems cannot be conjugated to autonomous linear systems.But the formulation of Floquet theory in the framework of linear skew product flows can be generalized (cf., e.g., Colonius and Kliemann [8, Chapter 7], and Kloeden and Rasmussen [19] for the general theory of skew product flows).The spectral subspaces (the stable, center, and unstable subspaces) of (1.2) depending on the phase τ ∈ [0, T ) characterize controllability properties.
In the last part of this paper we introduce quasi-affine control systems which have the form with A(v) := A 0 + p i=1 v i A i for v ∈ V ⊂ R p , where A 0 , A 1 , . . ., A p ∈ R d×d , and B : V → R d×m is continuous.The controls (u, v) have values in a compact convex neighborhood U × V ⊂ R m × R p of (0, 0), and the set of admissible controls is Quasi-affine systems look similar to linear control systems but the coefficient matrices in front of x and u may depend on the additional controls v.If a periodic v ∈ V is fixed, one obtains a periodic linear control system with controls u.We use this relation to prove results for control sets of quasi-affine systems.A special case are affine control systems with separated additive and multiplicative control terms, ẋ(t) = A 0 x(t) + p i=1 v i (t)A i x(t) + Bu(t).
(1.4) Controllability properties of affine systems are a classical topic in control.We only refer to the monographs Mohler [21], Elliott [14], and Jurdjevic [17].Our recent paper [11] proves results on control sets of general affine systems; cf. also [10] for control sets about equilibria.The contents of this paper are the following.After preliminaries in Section 2 on T -periodic linear control systems, Section 3 introduces the autonomized control system with state space S 1 ×R d , where the unit circle S 1 is parametrized by τ ∈ [0, T ).In Section 4, Theorem 4.3 characterizes the reachable and controllable subsets using the spectral subbundles of the periodic differential equation (1.2).Theorem 4.6 shows that a unique control set D a ⊂ S 1 × R d with nonvoid interior exists with unbounded part given by the center subbundle.Section 5 projects the control system to the open upper hemisphere S d,+ of the Poincaré sphere.Together with the equator S d,0 this constitutes a compactification where the behavior "near infinity" is mapped onto the behavior near the equator.The control set D a on S 1 ×R d projects onto the control set D a P on S 1 ×S d,+ and the intersection of intD a P with S 1 ×S d,0 is determined by the image of the center subbundle of (1.2).These results are also new for autonomous linear control systems.Section 6 presents some low dimensional examples, and, finally, Section 7 introduces quasi-affine systems.Theorem 7.2 characterizes their unique control set with nonvoid interior using the control sets of the periodic linear control systems for fixed periodic v ∈ V.
Notation: For a matrix A ∈ R d×d the set of eigenvalues is denoted by spec(A) and the real generalized eigenspace for µ ∈ spec(A) is GE(A, µ).The d × d identity matrix is I d and N = {0, 1, 2, . ..}.The interior of a set M in a metric space is intM .

Preliminaries.
In this section we introduce some notation and discuss consequences of the T -periodicity property.In particular, we recall a result on controllability for periodic systems without control restrictions.
The principal fundamental solution Here X(t, r)X(r, s) = X(t, s), t, r, s ∈ R, and by T -periodicity X(t + kT, s + kT ) = X(t, s) for all k ∈ Z.The variation-of-parameters formula for the solutions of (1.1) yields Denote for x ∈ R d the reachable set for t ≥ t 0 and the controllable set for t ≤ t 0 of (1.1) by ) resp., and let the reachable set and the controllable set be The reachable sets are convex and satisfy R t (t 0 , x) = R t+kT (t 0 + kT, x).(ii) The reachable sets R kT +t0 (t 0 , 0) are increasing with k ∈ N. Proof.Convexity of R t (t 0 , x) holds since the control range U is convex.The equality in (i) follows from For assertion (ii) let k ≥ ℓ and consider x ∈ R ℓT +t0 (t 0 , 0) with .
Then one obtains In order to clarify the relationship between the reachable and the controllable sets of the considered nonautonomous control systems it is convenient to introduce the following time reversed systems (cf.Sontag [24, Definition 2.6.7 and Lemma 2.6.8]).The reversal of (1.1) with trajectories denoted by ϕ − µ (t; t 0 , x 0 , u), t ∈ R. Lemma 2.2.For t 1 < t 0 the controllable set C t1 (t 0 , x) of (1.1) coincides with the reachable set R − t0 (t 1 , x) of the time-reversed system (2.3) at and y(t 0 ) = y and y(t 1 ) = x.Thus y(t) = ϕ − t0+t1 (t; t 1 , x, u − ), t ∈ [t 1 , t 0 ], and the assertion follows. Since 1 implies that also the controllable sets are convex and for k ≥ ℓ in N the inclusion

.4)
Proof.There are u, v ∈ U with Then one computes Thus (2.4) holds.Controllability criteria for periodic linear systems without control constraints are well known.The following theorem is due to Brunovsky [5], slightly reformulated.
Theorem 2.4.For the periodic linear system in (1.1) without control restrictions, the following properties are equivalent: (i) For any two points x 1 , x 2 ∈ R d and any t 0 ∈ R there are t 1 > t 0 and u ∈ L ∞ ([t 0 , t 1 ], R m ) such that ϕ(t 1 ; t 0 , x 1 , u) = x 2 .
If any of the equivalent conditions above is satisfied, the system in (1.1) without control restrictions is called controllable.
Theorem 2.4 implies the following first result on controllability properties of the system with control restrictions.
Then linearity implies for α ∈ [0, 1] that The assertions for the controllable sets follow by time reversal from Lemma 2.2 and Lemma 2.1(i).
Consider the unit circle S 1 parametrized by τ ∈ [0, T ) and define the shift Here τ + t mod T denotes the unique element τ + t − kT ∈ [0, T ) for some k ∈ Z.Let ψ(t; τ 0 , x 0 ) be the solution of (1.2) with initial condition x(τ 0 ) = x 0 and define Then Ψ is a continuous dynamical system, a linear skew product flow, on The Floquet multipliers of equation (1.2) are the eigenvalues µ of The Floquet exponents are λ j := 1 T log |µ| (the Floquet exponents as defined here are the real parts of the Floquet exponents defined in [6] and [25]).Note that λ j < 0 if and only if |µ| < 1.The following result is [8, Theorem 7.2.9].
the linear skew product flow associated with the T -periodic linear differential equation (1.2).For each τ ∈ S 1 there exists a decomposition into linear subspaces L(λ j , τ ), called the Floquet (or Lyapunov) spaces, with the following properties: (i) The Floquet spaces have dimension d j := dim L(λ j , τ ) independent of τ ∈ S 1 .
(ii) They are invariant under multiplication by the principal fundamental matrix in the following sense: (iii) For every τ ∈ S 1 the Floquet (or Lyapunov) exponents satisfy The Floquet space L(λ j , τ ) is the direct sum of the real generalized eigenspaces for all Floquet multipliers µ with 1 Define for τ ∈ [0, T ) the stable, the center, and the unstable subspaces, resp., by L(λ j , τ ), E 0 τ := L(0, τ ), and the stable, the center, and the unstable subbundles resp.We also introduce analogous subbundles for the center-stable subspaces and the center-unstable subspaces given by Similarly as for periodic differential equations, it is convenient for linear periodic control systems of the form (1.1) to extend the state space by adding the phase τ ∈ [0, T ) to the state in order to get an autonomous system.We obtain the following autonomized control system on with solutions Observe that (3.3) is not a linear control system.Remark 3.2.If the matrix functions A(•) and B(•) are merely measurable, the general existence theory of ordinary differential equations does not apply to equation (3.3).Nevertheless, the solutions are well defined.
Denote the reachable and controllable sets for t ≥ 0 of (3.3) by resp.The time reversed autonomous system is The reachable sets R a,− t (τ, x) of the time-reversed autonomized system (3.4) coincide with the controllable sets C a t (τ, x) of system (3.3).Note the following relation to the reachable and controllable sets defined in (2.2) for the periodic system (1.1).
4. Spectral characterization of reachable and controllable sets.In this section we characterize the reachable and the controllable sets of the autonomized system (3.3) by the spectral bundles of the homogeneous part (1.2) introduced in Theorem 3.1.
We start with the following technical lemma.Lemma 4.1.Let δ > 0 and µ ∈ C with |µ| ≥ 1.Then there are n k → ∞ and 2 for all k.Proof.With µ n = x n + ıy n and a = α + ıβ we have If x n = 0 the product µ n a is real if and only if β = −α yn xn .According to Colonius, Cossich, and Santana [9, Lemma 13] there are n k → ∞ such that Im(µ n k ) Re(µ n k ) → 0 and hence, with , and k large enough, It follows for a This choice of a n k guarantees µ n k a n k ∈ R and using |µ| ≥ 1 The next lemma relates the reachable sets and the center-unstable subspaces of the homogeneous part.
Lemma 4.2.Assume that the periodic linear system in (1.1) without control restrictions is controllable.Then for every τ ∈ [0, T ) the center-unstable subspace E +,0 τ of the homogeneous part (1.2) and the reachable sets of system (1.1) with controls u ∈ U satisfy Proof.The second inclusion follows from Proposition 2.6.It remains to prove the first inclusion.Since by Proposition 2.6 R NT +τ (τ, 0) is convex it suffices to prove that the real generalized eigenspaces for the eigenvalues (the Floquet multipliers) with absolute value greater than or equal to 1 are contained in R NT +τ (τ, 0).For each eigenvalue µ of X(T + τ, τ ) and q ∈ N let J q (µ) := ker(µI − X(T + τ, τ ) q ) and denote the set of real parts by Note that J R q (µ) ⊂ J R q+1 (µ).Since C d splits into the direct sum of the generalized eigenspaces q∈{0,1,...,d} ker(µI − X(T + τ, τ ) q ) and X(T + τ, τ ) is real it follows that R d splits into the direct sum of the subspaces q∈{0,1,...,d} J R q (µ) for µ ∈ spec(X(T + τ, τ )).
Now choose ℓ ∈ N with ℓ ≥ 2/δ.Taking real parts in (4.2) and choosing a = a n k one obtains where Re z(n k ) ∈ J R q−1 (µ) and Re(a n k w) ∈ R dT +τ (τ, 0).For k = 1 the variation-ofparameters formula (2.1) with u = 0 implies X(T + τ, τ ) n1 Re(a n1 w) = X(n 1 T + τ, τ ) Re(a n1 w) ∈ R (n1+d)T +τ (τ, 0).We may assume that n 2 ≥ n 1 + d and obtain With x = Re(a n1 w) ∈ R dT +τ (τ, 0) and Hence, using again formula (2.1) with u = 0 and Lemma 2.1(ii), In the next step we obtain for Proceeding in this way, we arrive at By the induction hypothesis the linear subspace J R q−1 (µ) is contained in the convex set R NT +τ (τ, 0), which is open by Proposition 2.6.This implies (cf.Sontag [24,Lemma 3.6.4] For the k with µ n k a n k < 0, replace a n k by −a n k to get the same conclusion.It follows that w 1 is a convex combination of the points 0 and ρw 1 in the convex set R NT +τ (τ, 0): It follows that w 1 ∈ R NT +τ (τ, 0) completing the induction step.We have shown that E +,0 τ ⊂ R NT +τ (τ, 0) proving the lemma.The following result characterizes the reachable and controllable sets of the autonomized system (3.3) by spectral properties of the homogeneous part (1.2).Recall that we denote the spectral subbundles of the T -periodic linear differential equation (1.2) by E − , E + , E +,0 , and E −,0 .For subsets K τ ⊂ R d and matrices Y (τ ), τ ∈ S 1 = [0, T ) we use the following notation: Theorem 4.3.Suppose that the periodic system in (1.1) with unconstrained controls is controllable and consider the autonomized system (3.3) with controls u ∈ U.
Define for τ ∈ S 1 a bounded linear map by Using the variation-of-parameters formula (2.1) and periodicity one computes
Next we define subsets of complete approximate controllability.Definition 4.4.A nonvoid set D a ⊂ S 1 × R d is a control set of the autonomized system (3.3) on S 1 × R d if it has the following properties: (i) for all (τ, x) ∈ D a there is a control u ∈ U such that ϕ a (t, (τ, x), u) ∈ D a for all t ≥ 0, (ii) for all (τ, x) ∈ D a one has D a ⊂ R a (τ, x), and (iii) D a is maximal with these properties, that is, if D ′ ⊃ D a satisfies conditions (i) and (ii), then D ′ = D a .
The following lemma shows that there is a control set around (0, 0).Lemma 4.5.Suppose that the periodic system in (1.1) with unconstrained controls is controllable.Then D a := R a (0, 0)∩C a (0, 0) is a control set and S 1 × {0} ⊂ intD a .
The following theorem characterizes the unique control set with nonvoid interior of the autonomized system.Recall that the center subbundle E 0 of the periodic linear differential equation (1.2) is non-trivial if and only if 0 is a Floquet exponent if and only if there is a Floquet multiplier of modulus 1, i.e., if spec(X(T, 0)) ∩ S 1 = ∅.
Theorem 4.6.Suppose that the periodic system in (1.1) with unconstrained controls is controllable.Then there exists a unique control set D a with nonvoid interior of the autonomized system (3.3) with controls u ∈ U.
Remark 4.7.A control system is called locally accessible, if the reachable and controllable sets up to time t > 0 have nonvoid interior for every t > 0. If this holds for the autonomized system (3.3), then Colonius and Kliemann [7, Lemma 3.2.13(i)]implies that D a = intD a .Even if the system in (1.1) without control restrictions is controllable, the autonomized system (3.3)need not satisfy intR a t (τ, x) = ∅ for small t > 0, hence, in general, it is not locally accessible.The example in Bittanti, Guarbadassi, Mafezzoni, and Silverman [3, p. 38] is a counterexample.
Remark 4.8.Gayer [15,Theorem 3] relates the control sets of autonomized (general nonlinear) control systems to control sets of discrete-time systems depending on τ ∈ S 1 defined by Poincaré maps.For system (3.3)these systems are defined by 5. The Poincaré sphere.This section describes the global controllability behavior of periodic linear control systems of the form (1.1) with homogeneous part (1.2) by projection to the Poincaré sphere.This allows us to determine the behavior "near infinity" by the induced system near the equator.
The system on the Poincaré sphere is obtained by attaching the state space R d to the north pole (0, 1) ∈ R d × R of the unit sphere S d in R d+1 and then taking the stereographic projection to S d .More formally, the extended system with scalar part ż = 0 is defined as where b i (t) denote the columns of B(t).For z ≡ 1 we get a copy of the original system (1.1).Abbreviate The projection of the homogeneous control system (5.1) on S d ⊂ R d+1 has the form (omitting the argument t) This is obtained by subtracting the radial components of the linear vector fields Â(t) and Bi (t).We compute This is the system equation for the induced control system on the Poincaré sphere.By adding the phase τ ∈ S 1 this induces an autonomous control system on S 1 × S d .Remark 5.1.The homogeneous control system (5.2) also induces a control system on projective space P d and a corresponding autonomized control system on S 1 ×P d .Parallel to the following developments on the unit sphere S d one may also work with P d .Here we prefer to work on the sphere since this allows us to write down everything explicitly.
On the "equator" of the sphere S d given by S d,0 := {s = (s 1 , . . ., s d , s d+1 ) ∈ S d |s d+1 = 0 }, the first d components of (5.2) reduce to the (uncontrolled) differential equation which leaves S d−1 ⊂ R d invariant.This coincides with the periodic differential equation obtained by projecting the homogeneous part (1.2) to S d−1 .Furthermore, the equator is invariant, hence also the upper hemisphere S d,+ := {s = (s 1 , . . ., s d , s d+1 ) ∈ S d |s d+1 > 0 } is invariant.When the phases τ ∈ [0, T ) are added to the states, the periodic differential equations (1.2) and (5.3) induce autonomous differential equations on S 1 × R d and S 1 × S d−1 , resp.A conjugacy of (autonomous) control systems on manifolds M and N can be defined as a map h : M → N which together with its inverse h −1 is C ∞ such that the trajectories ϕ(t; x 0 , u), t ∈ R, on M and ψ(t; y 0 , u), t ∈ R, on N with initial conditions ϕ(0; x 0 , u) = x 0 and ψ(0; y 0 , u) = y 0 (assumed to exist) satisfy Analogously, one can define conjugacies of differential equations.It is clear that reachable sets, controllable sets, and control sets are preserved under conjugacies.
In the following, we slightly abuse notation by identifying vectors and their transposes when it is clear from the context what is meant.
Proposition 5.2.(i) The map  is a conjugacy of the autonomized control system (3.3) on S 1 × R d and the restriction to S 1 × S d,+ of the autonomized system induced by (5.2).
(ii) The map e S : is a conjugacy of the autonomized differential equation induced by (5.3) on S 1 × S d−1 and the restriction to S 1 × S d,0 of the autonomized control system corresponding to (5.2).
(ii) This trivially holds since the solutions on S d,0 are obtained by adding the last component 0 to the solutions on S d−1 .
Since the image of the map e P is contained in the (open) upper hemisphere S d,+ a converging sequence of points e P (τ k , x k ) in the image converges to an element of S 1 × S d,0 if and only if x k → ∞.Hence Proposition 5.2 shows that the behavior near the equator reflects the behavior near infinity.
Next we discuss the projection of the reachable and controllable sets to the Poincaré sphere.Note that under the map x → (x,1) (x,1) the origin x = 0 ∈ R d is mapped to the north pole (0, 1) ∈ S d,+ ⊂ R d × R. The following theorem shows that for the autonomized system the closure of the reachable set from the north pole intersects the equator in the image of the center-unstable subbundle, and the closure of the controllable set to the north pole intersects the equator in the image of the centerstable subbundle.Furthermore the closure of the unique control set with nonvoid interior on S 1 × S d,+ intersects the equator in the image of the center subbundle.
Theorem 5.3.Suppose that the periodic system in (1.1) with unconstrained controls is controllable.
(ii) The induced system on S 1 ×S d,+ has a unique control set with nonvoid interior given by D a P = e P (D a ) satisfying intD a P ∩ (S 1 × S d,0 ) = e P (E 0 ) ∩ S 1 × S d,0 .
Using that the b k remain bounded, one finds → 0, and Then it follows that This shows that (τ, z, 0) ∈ e P (E +,0 ) ∩ S 1 × S d,0 .The assertions for the controllable set follow similarly.
(ii) By Theorem 4.6 it follows that the unbounded part of intD a is E 0 .Remark 5.4.Theorem 5.3(i) shows that for the autonomized system on the Poincaré sphere bundle S 1 × S d the closure of the reachable set from the north pole e P (0, 0) = (0, (0, 1)) ∈ S 1 × S d intersects the "equator" S 1 × S d,0 in the image under e P of the center-stable subbundle E +,0 .A closer look at the dynamics on the equator reveals a finer picture: Consider the Floquet bundles {(τ, x) ∈ S 1 × R d |x ∈ L(λ j , τ ) }.The projected flow on the projective bundle S 1 × P d−1 goes from the projected Floquet bundle for λ j to the projected Floquet bundles with λ i > λ j .This can be made precise by some notions from topological dynamics: the projected Floquet bundles form the finest Morse decomposition, in particular, they coincide with the chain recurrent components (cf.Colonius and Kliemann [8, Section 7.2 and Theorem 8.3.3]).For the relation to the flow on S 1 × S d−1 one can prove that for every chain recurrent component on S 1 × P d−1 there are at most two chain recurrent components on S 1 × S d−1 projecting to it.By Proposition 5.2(ii) this also describes the flow on the "equator" S 1 × S d,0 .Examples 6.2 and 6.4 illustrate some of these claims.

6.
Examples.First we note the following consequence of Theorem 4.6.In the scalar case with d = 1 one obtains from the inclusions in (4.8) that one of the following cases holds: The set intD a is contained either in K − or in K + (if the Floquet exponent is negative or positive, resp.) or D a = E 0 = S 1 × R (if 0 is the Floquet exponent).In the first two cases D a is bounded, in the third case it is unbounded.
The following two examples are autonomous two dimensional linear control systems.Hence it is not necessary to autonomize the system and the results on the control sets in R 2 follow from Sontag [24,Corollary 3.6.7].These examples serve as illustrations for the projection to the Poincaré sphere.
Example 6.2.Consider ) Here the origin is a saddle for the uncontrolled system.For the control system induced on the Poincaré sphere S 2 a computation based on (5.2) yields For u = 0 the north pole (0, 0, 1) is the only equilibrium, and for u = 0 the equilibria move away from the north pole.Theorem 4.6 implies that there is a unique control set D a ⊂ S 1 × R 2 with nonvoid interior and that it is bounded.By Theorem 5.3(ii) D a P = e P (D a ) is the unique control set with nonvoid interior on the upper hemisphere S 2,+ .On the equator one has s 3 = 0 and the equation reduces to This coincides with the projection of the homogeneous part of the original equation in R 2 onto the unit circle S 1 .The equilibria are (±1, 0, 0) and (0, ±1, 0).Linearization on the equator S 2,0 yields in e 1 = (1, 0, 0) and e 2 = (0, 1, 0) ẋ2 = −2x 2 and ẋ1 = 2x 1 , resp.
If we linearize on the sphere S 2 we have to linearize (6.7) in e 1 and e 2 with respect to the second and third arguments only.We obtain ∓2 u 0 −1 with eigenvalues ∓2 and ∓1 with eigenvectors given by (x, 0) ⊤ and (±u • x, x) ⊤ , x = 0, resp.The orthogonal projection of the system on the upper hemisphere S 2,+ to the unit disk yields the global phase portrait with control set e P (D a ) sketched in Figure 2. Observe that near the equator s 3 is close to 0, hence the control vector field in (6.7) goes to 0 for s 3 → 0. Remark 6.3.Perko [22] considers the differential equation (6.6) with u = 0.In this case the formulas derived above coincide with his results.The global phase portrait in Figure 2 is similar to [22, Figure 5 on p. 275] with the additional feature that around the north pole of S 2 the image of the control set occurs.Perko [22], as well as Lefschetz [20, pp. 202], actually, does these computations for differential forms, i.e., in the cotangent bundle of the sphere.
The following example is a slight modification of Example 6.2.It illustrates Remark 5.4 since the flow on the intersection of e P (R(0)) with the equator is nontrivial.
On the other hand, Proposition 5.2(ii) shows that the flow on the equator S 2,0 is determined by the flow on the unit circle S 1 induced by the homogeneous part (with u ≡ 0).The Floquet subspaces L(1) = R × {0} and L(2) = {0} × R are given by the eigenspaces and intersect S 1 in the equilibria (±1, 0) and (0, ±1), resp.All other points s 0 ∈ S 1 satisfy lim t→−∞ s(t, s 0 ) = (±1, 0) and lim t→∞ s(t, s 0 ) = (0, ±1).The orthogonal projection of the system on the upper hemisphere S 2,+ to the unit disk yields the global phase portrait with control set e P (D a ) sketched in Figure 3.
7. Controllability properties of quasi-affine systems.In this section we apply the results above to the study of controllability properties for quasi-affine control systems of the form (1.3).Explicitly, system (1.3) may be written as ẋ(t) = A 0 x(t) + p i=1 v i (t)A i x(t) + B(v(t))u(t), (u, v) ∈ U × V. (7.1) We denote the solutions of (7.1) with initial condition x(0) = x 0 ∈ R d by ψ(t; x 0 , u, v), t ∈ R. The homogeneous part of (7.1) is the bilinear control system ẋ(t) = A(v(t))x(t), v ∈ V, ( and we denote the solutions of (7. 2) is control-affine.On the other hand, the affine flow Ψ on the vector bundle (U ×V)×R d is not continuous, in general, even if we suppose that B(v) := B 0 + p i=1 v i B i with B 0 , B 1 , . . ., B p ∈ R d×m .In fact, if products v i u j occur on the right hand side of (7.1), the system is not control-affine, and hence continuity does not hold.