About the continuity of reachable sets of restricted affine control systems

Chaos, Solitons and Fractals 94 (2017) 37–43

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

About the continuity of reachable sets of restricted affine control systems Víctor Ayala a,1,∗, Heriberto Román-Flores a,2, Adriano Da Silva b,3 a b

Universidad de Tarapacá, Instituto de Alta Investigación, Casilla 7D, Arica, Chile Instituto de Matemática, Universidade Estadual de Campinas, Cx. Postal 6065, 13.081-970 Campinas-SP, Brasil

a r t i c l e

i n f o

Article history: Received 30 August 2016 Revised 11 November 2016 Accepted 14 November 2016 Available online 26 November 2016 MSC: 93B03 93B99 93C15

a b s t r a c t In this paper we prove that for a restricted affine control system on a connected manifold M, the associated reachable sets up to the time t varies continuously in each independent variable: time, state and the range of the admissible control functions. However, as a global map it is just lower semi-continuous. We show a bilinear control system on the plane where the global map has a discontinuity point. According to the Pontryagin Maximum Principal, in order to synthesizes the optimal control the Hausdorff metric continuity is crucial. We mention some references with concrete applications. Finally, we apply the result to the class of Linear control systems on Lie groups. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Affine system Accessible sets Lower semi-continuity Hausdorff metric

1. Introduction A control system  = (M, D ) is determined by a manifold M and a family of differential equations D induced by a class of admissible control functions. For x ∈ M the accessible set of  from x, i.e., the set of points that can be reached from x through all possible D-trajectories in positive time, have been investigated in several works from different points of view. For instance, a control system has the accessibility property from x if the reachable set from x , has non-empty interior in the M topology, [25,42] . The description of this class of sets have been analyzed by, Darken [21] , Gronski [23], Lobry [34] and Sussmann and Jurdjevic [41]. Also, in [30,31] the author makes an effort to describe the structure of these sets for special systems on low dimension. Actually, the accessible sets are difficult to describe because they are boundary points that can only be reached by chattering controls, i.e., infinite number of switched of controls in finite time. From a particular state x ∈ M, the controllability property of means that starting from x it is possible to reach any point of the

∗

1 2 3

Corresponding author. E-mail address: [email protected] (V. Ayala). Supported by Proyecto Fondecyt no. 1150292, Conicyt, Chile Supported by Proyecto Fondecyt no. 1151159, Conicyt, Chile Supported by Fapesp grant no. 2016/11135-2.

http://dx.doi.org/10.1016/j.chaos.2016.11.006 0960-0779/© 2016 Elsevier Ltd. All rights reserved.

space state by using the available controls in positive time. In other words, the reachable set fromxmust be the wholeM. The study of controllability has been a subject of huge interest and has generated an enormous activity in research for different classes of control system. Specially, on Linear and Bilinear systems on Euclidean spaces, [20,24,43]. And Linear and Invariant systems on Lie groups. For linear systems we mention [1–9,12–14,16] and [27]. For invariant we refer to the father of this class of systems [19], and [38] and a complete list of references therein. Furthermore, for a restricted admissible class of control U, in [20] the authors introduce the notion of control set,a subset C of M where controllability holds at the interior int (C ) of C and approximately controllable at the boundary ∂ C of C. Then, they prove that the map

U (ρ ) →

ρ -control set

is lower semi-continuous. Here, ρ > 0 is a parameter which allows to increase (respect to ⊂ ) the admissible class of control function U by increment the range of the controls. See also, [17,36]. On the other hand, in his book [35], Pontryagin shows that for a restricted classical linear control system on Euclidean spaces, the accessible set up to the positive time t is compact, convex and having the form changed continuously on time with the Hausdorff metric. The Pontryagin Maximum Principal is a very powerful theorem for concrete applications in a broad spectrum of disciplines.

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V. Ayala et al. / Chaos, Solitons and Fractals 94 (2017) 37–43

For instance, for application in mechanics see [28], in control of rail vehicles [32], in aerospace systems [33,40], in economy, [39], etc. In our particular case, given an initial condition x and an arbitrary by fix compact and convex subset  of Rm , the continuity of the application

Rx, : t → R≤t, (x ) ⊂ M is crucial in the proof of the celebrated Lenin Price Pontryagin Theorem. Actually, in the classical optimal time for a Linear Control System on Rn , the continuity of Rx, allows to build the optimal control. In fact, in this particular case, R≤t, (x ) is also convex and if t∗ is the optimal time associated to the optimal control u∗ , then the ending point of the optimal curve ϕ (t, x, u∗ ), i.e., the point ϕ (t∗ , x, u∗ ), must belong to the boundary of R≤t ∗ , (x ), otherwise is interior! By applying the Banach Theorem, there exists a hyperplane Ht ∗ which leave the whole reachable set in one side of Ht ∗ . It turns out that there exists a covector ηt ∗ orthogonal to Ht ∗ such that

< ηt ∗ , z > ≤ 0 for any z ∈ R≤t ∗ , (x ) and the maximum equals to zero is attainable exactly on the boundary point ϕ (t∗ , x, u∗ ). By the Bellman Maximum Principle, any point of the curve must be optimal. Hence, the existence of a 1-parameter curve of covectors follows, which is the main ingredient of the PMP to synthesize the optimal control and solve the problem. Our work is the first attempt to prove a similar result for the class of Linear Control Systems on Lie Groups introduced in [12]. In this article we just take care of the Hausdorff continuity part. But, for a more general class of systems. In the near future we expect to analyze convexity through some notion of geodesic of the system and to try to get the same Pontryagin result for linear system on Lie groups. Precisely, consider a restricted affine control system on a connected Riemannian C ∞ -manifold M, determined by the family of differential equations

 :

x˙ (t ) = f0 (x(t )) +

m 

ui (t ) fi (x(t )), with u ∈ U .

i=1

Where

U = {u ∈ L∞ (R, Rm ); u(t ) ∈ } is the class of restricted admissible control functions with  being a compact and convex subset of Rm with 0 ∈ int(). If x ∈ M and u ∈ U , we denote by ϕ (t, x, u) the   -solution satisfying ϕ (0, x, u ) = x. The reachable set R≤t, (x ) of   is built with the points of M which are possible to reach starting from the initial condition x, through all   -solutions in nonnegative time less or equal than t. It is well known that the map

(t, x, u ) ∈ R × M × U → ϕ (t, x, u ) ∈ M is continuous. Furthermore, the set U is a compact metrizable space in the weak∗ topology of L∞ (R, Rm ) = L1 (R, Rm )∗ (see for example [29]). As usual V∗ denotes the dual of the vector space V. In this paper we give a direct proof that for a restricted affine control system   on a connected manifold M, the associated reachable sets up to time t varies continuously on each variable separately by fixing the others. Precisely, the maps

and dH is the Hausdorff metric. Moreover, (C (M ), H ) is the metric space of all non-empty compact subsets of M with the Hausdorff metric. As a consequence, every continuous functional J defined on the accessible set R≤t, (x ) has a minimum and   maximum at any continuity point (t, x, ). In fact, J R≤t, (x ) ⊂ R is compact. The main theorem of the paper establish that the map

(t, x, ) ∈ R × M× Co(Rm ) → R≤t, (x ) is lower semi-continuous. Finally, we notice that no preliminary knowledge of control system is required to read the paper. 2. Control affine systems Let M be a connected Riemannian f0 , f1 , . . . , fm ∈ X ∞ (M ), m + 1 vector fields.

C ∞ -manifold

and

Definition 1. An affine control system is determined by the family of ordinary differential equations

 :

x˙ (t ) = f0 (x(t )) +

m 

ui (t ) fi (x(t )),

where u ∈ U .

i=1

The set of the control functions U is defined as

U = {u ∈ L∞ (R, Rm ); u(t ) ∈ } with  being a compact and convex subset of Rm . It is well known that the set of the control functions is a compact metrizable space in the weak∗ topology of L∞ (R, Rm ) = L1 (R, Rm )∗ , (see for instance Proposition 1.14 of [29]). As usual, V∗ means the dual of the vector space V. For a given initial state x ∈ M and u ∈ U we denote the solution of   by ϕ (t, x, u). The curve t→ϕ (t, x, u) is the only solution of   satisfying ϕ (0, x, u ) = x in the sense of Caratheodóry. That is, it is an absolutely continuous curve satisfying the corresponding integral equation. Throughout the paper we assume that all the solutions are defined in the whole real line. Even though this assumption is in general restrictive, there are several cases where the assumption of completeness goes without loss of generality, such as the class of linear systems on Lie groups, [15], and control affine systems on compact manifolds, [26]. Moreover, the map

(t, x, u ) ∈ R × M × U → ϕ (t, x, u ) ∈ M is a continuous map (see for instance Theorem 1.1 of [29]). For a given state x ∈ M and a positive time t let us introduce the sets

R≤t, (x ) = {y ∈ M; and

R ( x ) =



∃u ∈ U , s ∈ [0, t] with y = ϕ (s, x, u )},

R≤t, (x ).

t>0

R≤t, (x ) is called the set of reachable point from x up to time t and R (x ) the set of reachable points from x. Our goals include first to prove the partial continuity of the map R

(t, x, ) → R≤t, (x ).

are continuous. The variable  belongs to the metric space (Co(Rm ), dH ) where

Means, continuity in each variable: time t, state x ∈ M and the range  of the admissible class of control U . Secondly, we prove that the global map R is lower semi-continuous. First, we notice that it is possible to reduce the proof by considering a special class of control. In fact, let us consider the set PC ⊂ U of the piecewise control functions U  and define the corresponding reachable sets as

Co(Rm ) = { ⊂ Rm ;  is a non-empty compact convex subset}

RPC ≤t, (x ) = y ∈ M;

t → R≤t, (x ),

x → R≤t, (x ) and  → R≤t, (x )



∃u ∈ UPC , s ∈ [0, t] with y = ϕ (s, x, u )



V. Ayala et al. / Chaos, Solitons and Fractals 94 (2017) 37–43

and

R ( x ) = PC



where

RPC ≤t,

= {xi, j : 1 ≤ j ≤ ki , 1 ≤ i ≤ m}.

( x ).

t>0

In fact, for v ∈ U let v be a positive number such that

It turns out that





cl RPC ≤t, (x ) = R≤t, (x ), for any t > 0, x ∈ M





(1)

where cl RPC (x ) stands for the closure of RPC (x ) (Proposition ≤t, ≤t, 1.16 of [29]). The closure of a set A will be denoted by cl(A). Remark 2. Let 1 and 2 two compact and convex subsets of Rm with the property that 1 ⊂ 2 . We consider the control affine systems 1 and 2 associated with the corresponding set of control functions U1 and U2 , respectively. By the very definition it follows that U1 ⊂ U2 . The uniqueness of the solutions means that any 1 -solution is also a 2 -solution. Furthermore, in the weak∗ -topology U1 is a compact subset of U2 . Finally, we recall that for arbitrary u ∈ U and γ > 0 the sets

Wu,γ (x1 , . . . , xk )

 = u ∈ U :

R

 u(s ) − u (s ), xi (s ) ds < γ for i = 1, . . . , k ,

where k ∈ N and xi ∈ L1 (R, Rm ) for 1 ≤ i ≤ k, form a sublessee for the weak∗ -topology (see [22]). 3. Partial continuity This section is devoted to show that the global map

(t, x, ) ∈ R × M× Co(Rm ) → R≤t, (x ) ⊂ Rm is continuous in each variable. Precisely, continuous with respect to the usual topology for the time t ∈ R, with respect to the topology of the manifold M for the state x and for the topology determined by the Hausdorff metric relative to the compact subsets of Rm . It is important to mention here that in the last case the fixed time t is finite. We begin by proving the continuity of R on the third variable Proposition 3. Respect to the Hausdorff metric the map  → Rt, (x ) is continuous. such that  ⊂ int  . We already Proof. Fix  and consider  know that

(t, x, u ) ∈ R × M × U → ϕ (t, x, u ) ∈ M is a continuous map. Fix x ∈ M. From the compactness of [0, t] and U it follows that the map

(s, u ) ∈ [0, t] × U → ϕx (t, u ) = ϕ (t, x, u ) ∈ M is uniformly continuous. By Proposition 1.6 of [29], for any u ∈ U there exists a piecePC with the property that wise constant function u¯ ∈ U

 (ϕx (s, u ), ϕx (s, u¯ )) < ε /2, for any s ∈ [0, t]. By the compactness of [0, t] × U and the continuity of the map PC and (t, u)→ϕ x (t, u), there are u1 , . . . , um ∈ U , u¯ 1 , . . . , u¯ m ∈ U γ1 , . . . , γm > 0 such that

U ⊂

m 

39

Wui ,γi (xi,1 , . . . , xi,ki ),

for xi, j ∈ L1 (R, Rm )

i=1

and for any s ∈ [0, t] and u ∈ Wui ,γi (xi,1 , . . . , xi,ki )

 (ϕx (s, u ), ϕx (s, u¯ i )) < ε /2, 1 ≤ i ≤ m. Claim 1: There exists  > 0 with the property that for each u ∈ U , ∃i∗ ∈ {1, . . . , m} with

Wu, ( ) ⊂ Wui∗ ,γi∗ (xi∗ ,1 , . . . , xi∗ ,ki∗ )

Wv,v ( ) ⊂ Wui ,γi (xi,1 , . . . , xi,ki ) for some 1 ≤ i ≤ m. Since U is compact, there exist v1 , . . . , vl ∈ U and 1 , . . . , l > 0 with

U ⊂

l 

Wvk ,k /2 ( )

with

Wvk ,k ( ) ⊂ Wui ,γi (xi,1 , . . . , xi,ki ),

k=1

for 1 ≤ i ≤ m. Let us select  such that 0 <  <  k /2 for all k ∈ {1, . . . , l }. For any u ∈ U let u¯ ∈ Wu, ( ). Consider k ∈ {1, . . . , l } such that u ∈ Wvk ,k /2 ( ). Then, we get



¯ u(s ) − vk (s ), xi, j (s ) ds ≤ R



¯ u(s ) − u(s ), xi, j (s ) ds R

+ u(s ) − vk (s ), xi, j (s ) ds <  + k /2 < k , R

showing that for some i∗ ∈ {1, . . . , m}

Wu, ( ) ⊂ Wvk ,k ( ) ⊂ Wui∗ ,γi∗ (xi∗ ,1 , . . . , xi∗ ,ki∗ ) as stated. Claim 2: For any ε > 0 there exists δ 1 > 0 with the property that

f or any  and

γ ∈ (0, δ1 ) with Nγ ( ) ⊂ int 

it follows that

 ⊂ Nγ ( ) ⇒ R≤t, (x ) ⊂ Nε (R≤t, (x )). In fact, by the continuity of (s, u)→ϕ x (s, u) in [0, t] × U there exists δ 1 > 0 such that for any s ∈ [0, t] and i = 1, . . . , m,

 (ϕx (s, u ), ϕx (s, u¯ i )) < ε /2 if

|u − u¯ i |∞ < δ1 .

. The Let  and γ ∈ (0, δ 1 ) satisfying  ⊂ Nγ ( ) ⊂ int control u¯ i is piecewise constant thus we can construct a piecewise constant function u i ∈ U such that |u i − u¯ i |∞ < γ , for any i = 1, . . . , m. Let us consider y ∈ R≤t, (x ), u ∈ U , s ∈ [0, t] and i ∈ {1, . . . , m} with the property that

y = ϕx (s, u )

with

u ∈ Wui ,γi (xi,1 , . . . , xi,ki ).

The triangular inequality shows that

 (ϕx (s, u i ), ϕx (s, u )) ≤  (ϕx (s, u i ), ϕx (s, u¯ i )) +  (ϕx (s, u¯ i ), ϕx (s, u )) < ε . Because ϕx (s, u i ) ∈ R≤t, (x ) we obtain y ∈ Nε (R≤t, (x )) and consequently

R≤t, (x ) ⊂ Nε (R≤t, (x )) as stated. Claim 3: For any ε > 0 there exists δ 2 > 0 with Nδ2 () ⊂ int and

 ⊂ Nδ2 () ⇒ R≤t, (x ) ⊂ Nε (R≤t, (x )).  PC 

We know that R≤t, (x ) = cl R≤t, (x ) . Hence, in order to prove Claim 3 it is enough to show that for any ε > 0 there is δ 2 > 0 and such Nδ2 () ⊂ int

 ⊂ Nδ2 () ⇒ RPC ≤t, (x ) ⊂ Nε R≤t, (x )). . For any  ⊂ Take δ 2 > 0 with δ 2 <  /2M and Nδ2 () ⊂ int  PC

PC Nδ2 () let z ∈ R≤t, (x ), s ∈ [0, t] and u ∈ U such that z =

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V. Ayala et al. / Chaos, Solitons and Fractals 94 (2017) 37–43

are the values assumed by u in [−T , T ], ϕx (s, u ). If c1 , . . . , cm there are elements c1 , . . . , cm ∈  such that |ci − ci | < δ2 . Define

the piecewise constant function u ∈ U by

u (s ) =

ci , p,

 (ϕx (s , u ), ϕx (t, u )) < ε ,

if u (s ) = ci and s ∈ [0, t] if s ∈ R-[−T , T ],



where p ∈  is an arbitrary point. Therefore, for 1 ≤ j ≤ ki , 1 ≤ i ≤ m, we get



u (s ) − u(s ), xi, j (s ) ds ≤

T

xi, j (s ) ds + diam

for any u ∈ Vi



implying that z ∈ Nε R≤t, (x ) . Since z ∈ R≤s, (x ) was arbitrary we conclude

R≤s, (x ) ⊂ Nε (R≤t, (x )) 2. If s < t we obtain

R

δ2

b) s > t ⇒ s ∈ [t, s] ⊂ (t − δ, t + δ ) Thus, by taking Vi with u ∈ Vi , the Eq. (2) gives us

R≤s, (x ) ⊂ R≤t, (x ) ⊂ Nε (R≤t, (x ) ).

xi, j (s ) ds < 

 (ϕx (s, u ), ϕx (s, ui )) < ε and z ∈ Nε (R≤t, ).

Also, for any z ∈ R≤t, (x ) we get z = ϕ (t , x, u ) for some t ∈ [0, t] and u ∈ U . We get two possibilities  a) t ≤ s ⇒ z ∈ R≤s, (x ) ⊂ Nε R≤s, (x ) b) t > s ⇒ t ∈ [s, t] ⊂ (t − δ, t + δ ) So, by taking u ∈ Vi the Eq. (2) gives

Since z ∈ RPC (x ) was arbitrary we can conclude that ≤t,

 (ϕx (t , u ), ϕx (t, u )) < ε ,

−T

R\[−T,T ]

which shows that u ∈ Wu,  and u ∈ Wui ,γi (xi,1 , . . . , xi,ki ) for some 1 ≤ i ≤ m . Thus,

PC Rt, 



(x ) ⊂ Nε (Rt, (x ))

as claimed. Claim 4: The map  → R≤t, (x ) is continuous in the Hausdorff metric. For a given  and ε > 0 let δ = min{δ1 , δ2 /2} where δ 1 , δ 2 are defined as in the claims 2 and 3 respectively. It turns out that

dH (,  ) < δ

⇔

R≤t, (x ) ⊂ Nε (R≤s, (x )); Finally, by the preceding analysis if s ∈ (t − δ, t + δ ) we obtain

By Claim 3. we get

which is equivalent to

 ⊂ Nδ () ⇒ R≤t, (x ) ⊂ Nε (R≤t, (x )).

H (R≤t, (x ), R≤s, (x ) ) < ε

In addition, guaranteed by the previous analysis we obtain



implying that z ∈ Nε R≤s, (x ) . Since z ∈ R≤t, (x ) was arbitrary we conclude that

R≤t, (x ) ⊂ Nε (R≤s, (x ))

 ⊂ Nδ ( ) and  ⊂ Nδ ().

for any u ∈ Vi

and

R≤s, (x ) ⊂ Nε (R≤t, (x ))

showing the desired result.

.  ⊂ Nδ () ⇒ Nδ ( ) ⊂ N2δ () ⊂ Nδ2 () ⊂ int

To end this section we prove the continuity of the global map on the state space M.

On the other hand, Claim 2 implies that

 ⊂ Nδ ( ) ⇒ R≤t, (x ) ⊂ Nε (R≤t, (x )).

Proposition 5. The map x → R≤t, (x ) is continuous.

Hence,

dH (,  ) < δ ⇒ H (R≤t, (x ), R≤t, (x )) < ε finishing the proof. Next, for a given state x ∈ M we show the continuity of the global map on time.  Proposition 4. The map t → R≤t, (x ) is continuous.

Proof. Let x ∈ M, fix t > 0 and consider ε > 0. By continuity of the solutions and the compactness of the set [0, t] × U we can find δ > 0 with the property that

y ∈ B(x, δ ) ⇒  (ϕs,u (x ), ϕs,u (y )) < ε ,

for all (s, u ) ∈ [0, t] × U

By continuity, for any u ∈ U there exists δ u > 0 and a neighborhood Vu of u in U with the property

where ϕt,u (x ) = ϕ (t, x, u ). Then, for z ∈ R≤t, (x ) let s ∈ [0, t] and u ∈ U such that z = ϕs,u (x ). If y ∈ B(x, δ ) we know that ϱ(ϕ s, u (x), ϕ s, u (y)) < ε. Therefore,

|s − t | < γu and v ∈ Vu ⇒  (ϕx (t, u ), ϕx (s, v )) < ε /2.

z ∈ R≤t, (y ) and R≤t, (x ) ⊂ Nε (R≤t, (y ) ).

The set U is compact, so there exist V1 , . . . , Vn such that

In an analogous way we can show that

U =

n 

R≤t, (y ) ⊂ Nε (R≤t, (x ) ).

Vn .

i=1

By taking γ = min1≤i≤n {γui } we can conclude that for any s, s ∈ (t − γ /2, t + γ /2 ) and u, u ∈ Vi

 (ϕx (s, u ), ϕx (s , u )) < ε , for some 1 ≤ i ≤ n.

(2)

Let us take δ = γ /2 and s ∈ (t − δ, t + δ ). We have two cases depending on the relative position of s and t 1. If s ≥ t it follows that

R≤t, (x ) ⊂ R≤s, (x ) ⊂ Nε (R≤s, (x ) ). Also, if z ∈ R≤s, (x ) we get z = ϕ (s , x, u ) for some s ∈ [0, s] and u ∈ U . These are the possibilities   a) s ≤ t ⇒ z ∈ R≤t, (x ) ⊂ Nε R≤t, (x )

Hence,

 (x, y ) < δ ⇒ H (R≤t, (x ), R≤t, (y ) ) concluding the proof.



Remark 6. We know that the following map is continuous ϕ

(t, x, u ) ∈ R × M × U → ϕ (t, x, u ) ∈ M. So, by a fixed state x ∈ M the reachable set R≤t, (x ) under the continuous image [0, t ] × U by ϕ is compact. Hence, any continuous functional J : R≤t, (x ) → R has a minimum and maximum point.

V. Ayala et al. / Chaos, Solitons and Fractals 94 (2017) 37–43

4. Lower semi-continuity In this section we will be concerned with the global map R

It is shown that R is lower semi-continuous on the product. In order to do that, we use an equivalent concept called inner semicontinuity appears in (see [10]), as follows. We denote by P (Z ) the family of subsets of the metric space Z. Definition 7. Let X, Y be two metric spaces, and consider a setvalued map F : X → P (Y ). We say that F is inner semi-continuous at x0 ∈ X if x→x0

Here, X is a linear vector field, means that its flow {Xt : t ∈ R} is a subgroup of the group Aut(G) of G-automorphism. The vector fields Xj are right-invariant on G, for j = 1, . . . , m and

u = ( u1 , . . . , um ) ∈ U ⊂ L∞ ( R,  ⊂ Rm )

(t, x, ) ∈ R × M× Co(Rm ) → R≤t, (x ) ⊂ Rm .

F (x0 ) ⊂ lim inf F (x ) = {y ∈ Y, ∀xk → x0 ,

41

∃yn ∈ F (xn ) with yn → y}.

Now, we are in a position to prove the main result of the section. Theorem 8. The map (t, x, ) → R≤t, (x ) is lower semi-continuous. Proof. Let y0 ∈ R≤t0 ,0 (x0 ) and consider the convergent sequence

with  compact, convex and 0 ∈ int (). Linear control systems on Lie groups are important for at least two reasons. First, they are a natural generalization of the classical linear control system on the Euclidean space G = Rd , which is defined by

x˙ (t ) = Ax(t ) + Bu, A ∈ Rn×n , B ∈ Rn×m and u = (u1 , . . . , um ) ∈ U . In fact, just observe that etA ∈ Aut (Rd ) for any t ∈ R. And, any column vector bj of B is an invariant vector field on Rd . Besides that, in [26] Jouan shows that  G is relevant from theoretical and practical point of view. Actually, he shows that any affine control system on a connected Riemannian C ∞ -manifold M, as in Definition 1, whose dynamic generates a finite dimensional Lie algebra, i.e.





(tn , xn , n ) → (t0 , x0 , 0 ).

dim SpanLA X, Y 1 , . . . , Y m < ∞

Since y0 ∈ R≤t0 ,0 (x0 ) it holds that y0 = ϕ (s0 , x0 , u0 ) for s0 ∈ [0, t0 ] and u0 ∈ U0 . Actually, by the equality

is equivalent to a linear control system on a Lie group G or on a homogeneous space of G. Since the  G -solutions are defined for any time, [15], all the results of the previous sections apply to  G .





cl RPC ≤t, (x ) = R≤t, (x ) PC there is no loss of generality in assuming u0 ∈ U≤t,  . 0

Let us denote by εn = t0 − tn and consider sn = s0 − εn . If s0 = 0, the result follows trivially, since xn → x0 . If s0 > 0, the fact tn → t implies sn → s0 . And, if n ∈ N is large enough, sn ∈ [0, tn ]. Let c1,0 , . . . , cn,0 ∈ 0 the values assumed by u0 in [0, t0 ]. By hypothesis n → 0 then, as we did before, it is possible to build piecewise constant control functions un ∈ Un with un → u0 . By considering

yn = ϕ (sn , xn , un ) ∈ R≤tn ,n (xn ) we get by continuity that the sequence (yn ) converges to y0 . It turns out that

R≤t0 ,0 (x0 ) ⊂

lim inf

(t,x,)→(t0 ,x0 ,0 )

R≤t, (x )



concluding the proof.

It is worth pointing out that about the continuity of reachable sets of affine control systems, Theorem 8 is the best that you can expect. Actually, in the next section we show that the global map R could has discontinuity points. In fact, we give an explicitly example of a two dimensional bilinear control system on R2 where the function R

+

R (x, ) ∈ M× Co(Rm ) → R ( x )

is not continuous (see [17]). This also show that we should not expect the continuity of the global map

R(t, x, ) = R≤t, (x ) as well. 5. An example and a counterexample In this section we show the potential of Theorem 8 through the class of linear control systems on Lie groups, [12]. On the other hand, based on the class of bilinear control systems we show an example where the global map has a discontinuity point. In [12] Ayala and Tirao introduced the notion of linear control system  on a connected Lie group G as the family of ordinary differential equations

G :

g˙ (t ) = X (g(t )) +

m  j =1

u j (t )X j (g(t )), g(t ) ∈ G.

Example 9. By definition a bilinear control system  in Rd is determined by a family of differential equations



:

x˙ (t ) =

A+

m 



ui (t )Bi x(t ), t ∈ R, x(t ) ∈ Rd

i=1

where A, B1 , . . . , Bm ∈ gl (d, R )

(3)

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V. Ayala et al. / Chaos, Solitons and Fractals 94 (2017) 37–43

with purely imaginary eigenvalues, and its exterior ext (C ) which is formed by the diagonalizable matrices of g with real eigenvalues. Furthermore, any matrix in C makes a 45°angle with the axis generated by A. They consider a positive real number r, the restricted bilinear system

r :

x˙ (t ) = (A + u(t )B )x(t ),

A, B ∈ sl (2, R ) and u ∈ U with r = [−r, r ]. and prove the following theorem. Theorem 11. Assume det [A, B] = 0. Then, the controllability of  r depend on the relative position of the segment

sr : A + uB : u ∈ r as follows 1. If det(A ) ≥ 0 the system is controllable 2. If det(A ) < 0 there are two possibilities a) If det [A, B] < 0 the line A + uB : u ∈ r crosses int (C ) and the system is controllable. The only bifurcation point is given by

r ∗ = inf {r : sr ∩ int (C ) = ∅} b) If det [A, B] > 0 the system in not controllable for any r > 0. The main geometric ingredient here is: the controllability property is equivalent to the fact that the segment sr must cross the bicone. In fact, the fundamental argument in the prove of the theorem above is the existence of r > 0 and a control u ∈ r with the property that

Spec (A + uB ) ∈ iR ⇔ sr ∩ int (C ) = ∅. Actually, for any r > r∗ controllability of  r follows from the fact that the only semigroup with non empty interior in the semisimple Lie group G is the whole group SL(2, R ), [37]. Fix x ∈ R2 . Respect to the continuity on the third variable of R

Rx,R+ :  → R (x ) we can conclude the following: under the condition (2.a) the set valuated map Rx,R+ is not continuous. Actually, ∗

Rx,R+ has a discontinuity at the point r . In fact, for r < r∗ , Proposition 3 implies that for any x = 0, the reachable set Rx,R+ (r ) is compact. However, Rx,R+ (r ) = R2 − {0} for any r > r∗ . 6. Conclusion In this paper we analyze the continuity of the global map R

(t, x, ) ∈ R × M× Co(Rm ) → R≤t, (x ) ⊂ Rm associated to any affine control system as in Definition 1. Precisely, we prove 1. R is continuous at any independent variable 2. R is Lower semi-continuous on its domain 3. Theorem 8 is the best results you can expect. In fact, we show a restricted bilinear control system on the plane where R has a discontinuity point A very powerful results called the Pontryagin Maximum Principle shows the compactness, convexity and Hausdorff metric continuity deformation of the reachable sets for the class of Restricted Linear Control Systems on Euclidean spaces. In order to compute the time optimal control the Hausdorff t-continuity property of R≤t, (x ) is crucial. Our paper is the first attempt to prove a similar result for the class of Restricted Linear Control Systems on Lie Groups (LCS) introduced in [12]. We apply Theorem 8 to the LCS class because its relevance due to the Equivalence Theorem of Jouan, [26].

The next step is to define an appropriate notion of convexity on Lie groups for LCS. We believe that this is possible through some notion of   -geodesics. Actually, we know how to compute explicitly the hamiltonian vector fields and the Hamiltonian equations for this class of control systems in our recently paper published by a SIAM Journal, [11]. In the near future we hope to get the proposed aims. References [1] Ayala V, Tirao J Linear control systems on Lie groups and controllability. In: Ferreyra G, et al., editors. Differential Geometry and Control. American Mathematical Society, Providence, RI; 1999. [2] Ayala V, San Martin L. Controllability properties of a class of control systems on lie groups. Lectures notes in control and information science; 2001. Nonlinear control in the Year 20 0 0. [3] Ayala V, Da Silva A. Controllability of linear systems on lie groups with finite semisimple center. Aceptted to, SIAM J 2016. [4] Ayala V, Silva A. Control sets of linear systems on lie groups. Submitted to, Nonlinear Diff Eq Appl 2016. [5] Ayala V, Kizil E, Tribuzy I. On an algoritm for finding derivations of lie algebras. Proyecciones Math J 2012;31(1):81–90. [6] Da Silva A. Controllability of linear systems on solvable lie groups. SIAM J Control Optim 2016;54(1):372–90. [7] Jouan P. Controllability of linear systems on lie group. J Dyn Control Syst 2011;17:591–616. [8] Jouan P, Dath M. Controllability of linear systems on low dimensional nilpotent and solvable lie groups. J Dyn Control Syst 2014. [9] Jouan P. Equivalence of control systems with linear systems on lie groups and homogeneous spaces. ESAIM 2010;16:956–73. [10] Aubin JP, Frankowska H. Set-valued analysis. Modern birkhäuser classics. Boston, MA: Birkhäuser Boston, Inc.; 2009. [11] Ayala V, Jouan P. Almost riemannian structures and linear control systems on lie groups. SIAM J Control Optim 2016;54(5):2919–47. [12] Ayala V, Tirao J. Linear control systems on lie groups and controllability. Am Math Soc Ser 1999;64:47–64. [13] Ayala V, Martin LS. Controllability properties of a class of control systems on lie groups. Lect Notes Control Inf Sci 2001;1(258):83–92. [14] Ayala V, Silva AD. Controllability of linear control systems on lie groups with semisimple finite center. Accepted at, SIAM J Control Optim 2016. [15] Ayala V, Silva AD, Kizil E. About the solutions of linear control systems on lie groups. Accepted in, Proyecciones J Math 2016;35(4):491–503. [16] Ayala V, Jouan P. Almost riemannian structures and linear control systems on lie groups. SIAM J Control Optim 2016;54(5):2919–47. [17] Ayala V, Mart’ın LABS. Controllability of two-dimensional bilinear systems: restricted controls, discrete time. Proyecciones J Math 1999;18:207–23. [18] Barros CJB, Goncalves JR, Rocio OD, Martin LS. Controllability of two dimensional bilinear systems. Proyecciones J Math 1996;15:111–39. [19] Brockett R. System theory on groups and coset spaces. SIAM J Control 1972;10:265–84. [20] Colonius F, Kliemann W. The dynamics of control. Birkhäuser; 20 0 0. [21] Darken J. Accessible sets for analytical control systems in R2 . J Math Anal Appl 1990;145:197–215. [22] Dunford N, Schwartz JT. Linear operators, part i: general theory. Wiley-Interscience; 1977. Communications on Pure and Applied Analysis, 2001; 10: 847–857 [23] Gronski J. Classification of closed sets of attainability in the plane. Pacific J Math 1978;77(1):117–29. [24] Elliot D. Bilinear Control Systems: Matrices in Action. Springer; 2009. [25] Hermes H. On local and global controllability. SIAM J Control 1974;12(2):252–61. [26] Jouan P. Equivalence of control systems with linear systems on lie groups and homogeneous spaces. ESAIM 2010b;16:956–73. [27] Jouan PH. Controllability of linear systems on lie group. J Dyn Control Syst 2011;17:591–616. [28] Jurdjevic V. Geometric control theory. New York: Cambridge University Press; 1997. [29] Kawan C. Invariance entropy for deterministic control systems - an introduction. Lecture notes in mathematics 2089. Berlin: Springer; 2013. [30] Krener AJ. The accessible sets of quadratic free nilpotent control systems. Commun Inf Syst 2011;11(1):35–56. [31] Krener AJ, Schaettler H. The structure of small time reachable sets in low dimensions. SIAM J Control Optim 1988;27:120–47. [32] Lee DH, Milroy IP, Tyler K. Application of pontryagin’s maximum principle to the semi-automatic control of rail vehicles. In: Second conference on control engineering 1982: merging of technology and theory to solve industrial automation problems; preprints of papers. Barton, ACT: Institution of Engineers, Australia; 1982. p. 233–6. [33] Leitmann G. Optimization techniques with application to aerospace systems. London: Academic Press Inc.; 1962. [34] Lobry C. Controllabilite des systemes non lineaires. SIAM J Control Optim 1970;8(4):573–605. [35] Pontryagin LS, Boltyanski VG, Gamkrelidze RS, Mishchenko EF. Mathematical theory of optimal processes. Interscience; 1962.

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