Control sets of linear control systems on $\mathbb{R}^2$. The complex case

This paper explicitly computes the unique control set $D$ with non-empty interior of a linear control system on $\mathbb{R}^2$, when the associated matrix has complex eigenvalues. It turns out that the closure of $D$ coincides with the the region delimited by a computable periodic orbit $\mathcal{O}$ of the system.


Introduction
Let A be a real matrix of order two. A linear control system (LCS) on R 2 is given by the family of ODEṡ v = Av + uη, u ∈ Ω, where Ω = [u − , u + ] with u − < u + and η = 0.
This article explicitly describes a maximal region D of the system in which interior the controllability property holds. This region, called a control set, is relevant in applications. In fact, two arbitrary states in its interior can be connected by an integral curve of the system in positive time. In particular, by following an appropriate trajectory, it is possible to transform an initial condition into the desired state through the system in a finite time. Additionally, the existence of an optimal solution is also a warranty for a minimum time problem between these states.
Our approach is novel, and here we consider the drift A with a couple of complex eigenvalues. We describe the corresponding control set by the different possibilities of A's trace. And, we prove that D is limited by a specific periodic orbit O of the system.
For a linear control system, it is well known that the Kalman rank condition warrants the existence of a control set D with a non-empty interior. Furthermore, D is characterized by the positive and negative orbits, which allows for determining some topological properties of D. However, computing these orbits is a difficult task, and the same is true for D.
This article's main contribution through our approach, permits us to recover all the known results about the controllability and control sets properties for this class of systems without the extra assumptions 0 ∈ int Ω. Moreover, the most crucial issue is to compute the control set D explicitly as follow. The set, O := ϕ(s, P + , u − ), s ∈ 0, π µ ∪ ϕ(s, P − , u + ), s ∈ 0, π µ , is a periodic orbit of Σ R 2 . Here, the points P − , P + belong to R · A −1 η, a line determined by the drift and the control vector of the system. This orbit is obtained asymptotically by considering a solution starting on an equilibrium whose control function interchanges from u − and u + . With that there are three possibilities: tr A = 0 and the system is controllable; tr A < 0 and D is unique, closed and its boundary is O, or tr A > 0 and D is open and coincides with the region bounded by O. Moreover, Σ R 2 admits only D as control set when tr A < 0 and admits D and O as control sets when tr A > 0.
The article concludes with an asymptotic analysis through the parameters determining the dynamic of the system, i.e., the eigenvalues, and the size of the range determined by the controls u − and u + . In particular, controllability properties are recovered in some cases. We also mention that our method does not consider 0 in the range's interior as usual.
Notations: For any vector v ∈ R 2 we denote by R · v the line passing by the origin and parallel to v. We consider the natural order on R · v as For any τ ∈ R, we denote by R τ the rotation of τ -degrees which is clockwise if τ < 0 and counter-clockwise if τ > 0. In particular, we use define θ := R π/2 .
2 Geometric properties of spirals in R 2 .
This section analyzes the dynamics of spirals in the Euclidean space R 2 . In particular, we show that spirals with the center in the same line have a particular kind of invariance.
Let A ∈ gl(2, R) and denote by σ A the number The number σ A is related to the eigenvalues of A, and it is straightforward to see that A has a pair of complex eigenvalues if and only if σ A < 0.

Definition:
For any A ∈ gl(2, R) with σ A < 0 we define the spiral ϕ A to be the function Since σ A < 0, there exists an orthonormal basis of R 2 such that where 2λ = tr A and µ 2 = |σ A |.
Consequently, the spiral ϕ A can be written on such basis, as where R µτ is the rotation of µτ -degrees with relation to the previous basis, which is clockwise if µτ < 0 and counter-clockwise if µτ > 0.
The spiral ϕ A intersects the line passing by v 1 and v 2 for any τ ∈ k π µ Z. Moreover, showing that ϕ A (τ, v 1 , v 2 ) belongs to the circumference with center v 2 and radius e τ λ |v 1 − v 2 |. In particular, 2.2 Remark: Note that, by reverting the time, we can relate the spirals associated with λ and −λ. Also, if B : R 2 → R 2 is the linear map whose matrix on the previous basis is implying that the spirals associated with µ and −µ are related by conjugation.
By the previous Remark let us assume w.l.o.g. that λ < 0 and µ > 0 and , the region (see Figure 1) delimited by the line passing through v 1 and v 2 , and the curve By our choices, such a region can be algebraically described as where θ is the counter-clockwise rotation of π/2-degrees. The next result analyzes a kind of invariance for the region C A (v 1 , v 2 ).
Proof: Since, Therefore, it is enough to show the result assuming that v 2 = 0. For this case, we have that Moreover, in this case w 2 ∈ (0, v 1 ) and σ is the angle between v 1 and w 1 − w 2 . Thus, we already have that, showing that Define now the function In order to conclude the result, it is enough to show that g is nonnegative, that is, which we will do in the next steps.
Step 1.: g is nonnegative on critical points in int C σ ; By simple calculations, we get that if and only if, there exists γ ∈ R such that, The relation µs + σ = µτ gives us that On the other hand, In particular, if g admits a critical point (s, τ ) ∈ int D σ , by the previous arguments, we get showing the assertion.
By the previous calculations, implying that, As a consequence, it follows that Since C σ is a compact subset and g is smooth, the Weierstrass Theorem assures the existence of a global minimum for g on C σ . Since the possible candidates for such minimum were calculated in Steps 1. and 2. we conclude that g is nonnegative on C σ , ending the proof.
The set Ω is called the control range of the system Σ R 2 . The family of the control functions U is, by definition, the set of all piecewise constant functions with image in Ω. The solution of Σ R 2 starting at v ∈ R 2 and associated control u ∈ U is the unique piecewise differentiable curve s ∈ R → ϕ(s, v, u) satisfying It is not hard to see that the solutions of Σ R 2 are given by concatenations of the curves associated with constant control functions.
For any v ∈ R 2 , the positive and the negative orbits of Σ R 2 are given, respectively, by the sets If a control set D of Σ R 2 satisfies D = R 2 we say the Σ R 2 is controllable. From here we assume that the matrix A ∈ gl(R, 2) satisfies σ A < 0, and fix an orthonormal basis of R 2 such that

Remark
Since det A = 0 it holds that are the equilibria of the system. In particular, the solutions of Σ R 2 for constant control functions coincide with the spirals ϕ A (s, v, v(u)) if tr A = 0 and lie on circumferences if tr A = 0.
In what follows we analyze the dynamics of the solutions of Σ R 2 in order to obtain a full characterization of the control sets of the system. Moreover, all the results that follows do not need the assumption that 0 ∈ int Ω.

The control set with nonempty interior
In this section, we construct explicitly the control set of Σ R 2 with a non-empty interior by considering the possibilities for the trace of the matrix A.

The case tr A = 0
In this case, the solutions of Σ R 2 for constant controls have the form and they lie on the circumferences C u,v with center v(u) and radius |v − v(u)|.

Theorem:
If the associated matrix A of Σ R 2 is such that tr A = 0 and det A > 0, then Σ R 2 is controllable.
Proof: In order to show the result, it is enough to construct a periodic orbit between an arbitrary point v ∈ R 2 and some fixed v(u 0 ) ∈ v(Ω), which we do as follows: ] is a compact interval on the line R · θη; (b) The circumference C u + ,v intersects the line R · θη in two points. Denote by v 1 the point in this intersection close to v(u − ). In particular, v 1 = ϕ(s 1 , v, u + ) for some s 1 > 0; (c) If v 1 / ∈ v(Ω), we repeat the process in the previous item for the circumference C u − ,v1 , obtaining a point v 2 .
(d) Repeating the previous process, if v n / ∈ v(Ω), we obtain in the same way, a point v n+1 belonging to the intersection of the circumference Cū ,vn , and the line R · θη, whereū = u + if n is even andū = u − if n is odd. By induction, we quickly see that the radius R n of Cū ,vn satisfies Therefore, there exists N ∈ N such that v N ∈ v(Ω).
(e) Now, since v N ∈ v(Ω) there exists, by continuity, u N ∈ Ω satisfying |v(u N ) − v(u 0 )| = |v N − v(u N )|. The circumference C u N ,v N passes through v N and by the point v(u 0 ). Therefore, there exists s N > 0 such that ϕ(s N , v N , u N ) = v(u 0 ) and by concatenation we get a trajectory from v to v(u 0 ) (blue paht in Figure 3).
(f) By choosing the complementary path (red path in Figure 3) on the circumferences constructed on the previous items, we obtain a trajectory from v(u 0 ) to v, which gives us a periodic orbit as desired ( Figure  ??).

The case tr A = 0
Next, we construct a periodic orbit for Σ R 2 . The main result in this section will show that such orbit is the boundary of the unique control set of Σ R 2 .
A simple inductive process allows us to obtain On the other hand, as m → +∞.

Figure 4: Periodic Orbit
Note that both of the points P − , P + belong to the line RA −1 η and satisfy Moreover, it holds that and analogously, showing the following:

Proposition:
The subset of R 2 given by is a periodic orbit of Σ R 2 .
Let us denote by C the closure of the region delimited by the periodic orbit O. The next result shows that C, or its interior, is a control set of the system.

Theorem:
For the LCS Σ R 2 with σ A < 0 and tr A = 0 it holds that 1. tr A < 0 and D = C is a control set; 2. tr A > 0 and D = int C is a control set.
Proof: Let us start by showing, in the next steps, that Since both cases are analogous, we will assume w.l.o.g. that λ < 0 and µ > 0.
As a consequence, the region C can be decomposed in two regions which are delimited, by the line passing through v(u + ) and v(u − ) and the curves ϕ(s, P + , u − ), s ∈ 0, π µ and ϕ(s, P − , u + ), s ∈ 0, π µ , respectively.
Consequently, the compactness of C shows the existence of s 0 > 0 such that ϕ(−s 0 , v, u) ∈ ∂C = O. Moreover, there exists n ∈ N and t 0 > 0 such that By construction the points P m , m ∈ N are attained from v(u − ) in positive time. Therefore, the previous arguments show that v is attained from v(u − ), or equivalently v ∈ O + (v(u − )). Furthermore, s → ϕ(s, v, u − ) is a curve that revolves around v(u − ) and s → ϕ(s, v(u − ), u) revolves around v(u). Then, for any u = u − , there exist s 0 , t 0 > 0 such that proving the claim.
As a consequence, On the other hand, by Step 2, controllability holds inside int C. Consequently, for any v, w ∈ int C we obtain showing the desired.
By the previous, it is straightforward to see that int C satisfies conditions (a) and (b) of Definition 3.1. Therefore, there exists a control set D such that int C ⊂ D and we have that: 1. If λ < 0, the positively invariance on item (a) implies that showing that D = C is in fact the control set of Σ R 2 .
2. If λ > 0 let v ∈ R 2 and assume that In particular, there exists s > 0, u ∈ U such that implying the maximality of int C and hence D = int C, concluding the proof.

Remark:
The previous result implies that, if tr A = 0, the LCS admits a bounded control set with nonempty interior which is closed if tr A < 0 and open when tr A > 0. Moreover, from Step 1 in the proof of Theorem 3.5, it holds that ∀v ∈ C, u ∈ U ϕ(s, v, u) ∈ C if s · tr A < 0 (1)

The possible control sets of a LCS
As is well stated in the literature, if 0 ∈ int Ω, the control set D previously obtained is the only control set of Σ R 2 with non-empty interior. This section shows that D is in fact the only control set with non-empty interior, even without the condition 0 ∈ int Ω. Moreover, if the trace of the associated matrix A is positive, the periodic orbit O = ∂D is also a control set of Σ R 2 .
In order to show the previous claim, the following statement will be crucial.
We can now prove the main result concerning the control sets of a LCS on R 2 .

Theorem:
Let Σ R 2 be a LCS satisfying σ A < 0 and tr A = 0. It holds: 1. If tr A < 0 the only control set of Σ R 2 is D; 2. If tr A > 0 then int D and ∂D are the only control sets of Σ R 2 .
Proof: Since ∂D = O is a periodic orbit, it satisfies conditions (a) and (b) of Definition 3.1 and is therefore contained in a control set of Σ R 2 . By Theorem 3.5, we know that D = C is a control set if tr A < 0 and D = int C is a control set if tr A > 0. Therefore, the result follows if we show that no control set of Σ R 2 intersects R 2 \ C.
Since the solutions of Σ R 2 are given by concatenations of the solutions for constant controls, it is not hard to show by induction that for all u ∈ U and v ∈ R 2 \ C, |ϕ(s, v, u) − C| ≤ e sλ |v − C| if λs < 0, and |ϕ(s, v, u) − C| ≥ e sλ |v − C| if λs > 0.
In particular, if |v − C| = > 0 we have that Let us assume that Σ R 2 admits a second control set D satisfying |v − C| = > 0 for some v ∈ D .
By condition (a) in Definition 3.1, there exists u ∈ U such that ϕ(s, v, u) ∈ D for all s > 0. If v / ∈ C, we have by invariance (see equation (1)), that ϕ(s, v, u) / ∈ C for all s > 0. Moreover, by condition (b) in Definition 3.1 and the previous calculations, it holds that D ⊂ O + (ϕ(s, v, u)) and by the previous Therefore, any control set D of Σ R 2 satisfies D ⊂ C concluding the proof.

Remarks on continuity and asymptotic behavior of control sets
The construction of the periodic orbit O allows us to analyze the asymptotic behavior of the control set D as the control range grows.

Remark:
To obtain controllability, the previous result only requires that Ω is unbounded and not necessarily the whole real line (see [11]).