geometric and numerical methods in space and quantum...

Geometric and numerical methods in space andquantum control

SADCO summer school and workshop onOptimal and Model Predictive Control

Bayreuth, September 2013

Jean-Baptiste CaillauInstitut Math.Univ. Bourgogne & CNRSOn leave at Lab. J.-L. LionsUniv. Paris VI & CNRS

SADCO European NetworkANR Geometric Control MethodsINRIA McTAO

Controllability

Poisson stability. On X of dim 4,

x(t) = 1·F0(x(t))+u1(t)F1(x(t))+u2(t)F2(x(t)), |u(t)|=√

u21 +u2

2 ≤ 1,

s.t.

(i) F1(x),F2(x),F01(x),F02(x) of rank 4, x ∈ X ,

(ii) F0 positively Poisson stable: ∃(tn)n → ∞ s.t.

e tnF0(x)→ x, n → ∞ (dense).

Thene−tF0(x) = lim

n→∞exp (tn− t)︸︷︷︸

>0

F0(x)

whence controllability.

Lemma. q(t) = f0(q(t), q(t))+u(t) =⇒ (i).

Controllability

CR3BP (1). With q ∈ Qµ = C\−µ,1−µ, (q, q) ∈ Xµ = T Qµ and

q(t) = ∇Ωµ(q(t))−2iq(t)+ εu(t), |u(t)| ≤ 1,

Ωµ(q) =12|q|2 +

1−µ

|q+ µ|+

µ

|q−1+ µ|=⇒ (i).

CR3BP (2). With p = q+ iq and (q, p) ∈ Xµ = T ∗Qµ ' Qµ ×C,

q(t) =∂Jµ

∂ p(q(t), p(t)), p(t) =−

∂Jµ

∂q(q(t), p(t))+ εu(t), |u(t)| ≤ 1,

Jµ(q, q) =12|q|2−Ωµ(q), (Jacobi)

=12|p|2 + p∧q− 1−µ

|q+ µ|− µ

|q−1+ µ|

=⇒ F0 =−→Jµ .

Controllability

Hill regions. Jµ(q, p) = j =⇒ Ωµ(q)+ j ≥ 0. Lagrange points:−→Jµ(Li) = 0, i =

1, . . . ,5.

L2 L1L3

L4

L5

µ 1 µ

x

z

/(2c)

(1 1/(2c))

0

0

1

ControllabilityTheorem. [C-Daoud’2012] For any µ ∈ (0,1) and ε > 0, the CR3BP is controllableon X1

µ , the connex component of (q, p) ∈ Xµ | Jµ < Jµ(L1) containing L2.

L2

L1

q = µ

q = 1 µ

x0

xf

2

Singularities of the extremal flow

Bang and singulars. On X of dim 4, sub-Riemannian (D ,g) and F0,

x(t) = F0(x(t))+u1(t)F1(x(t))+u2(t)F2(x(t)), |u(t)| ≤ 1,

t f → min .

PMP =⇒ u(t) = ψ(t)/|ψ(t)| si ψ(t) 6= 0 with

ψ(t) := (H1,H2)(x(t), p(t)), Hi(x, p) := pFi(x),

Σ := z = (x, p) ∈ T ∗X | H1(z) = H2(z) = 0.Assumption on D and F0:

(i) F1(x),F2(x),F01(x),F02(x) of rank 4, x ∈ X ,

Lemma. Singulars are of minimal order and the singular flow associated withHs(z) := H(z,us(z)) belongs to Σ, where

us(z) :=1

H12(z)(−H02(z),H01(z))·


Stratification. One has Σ = Σ−∪Σ0∪Σ+ with

Σ− := z ∈ Σ | H212(z) < H2

01(z)+H202(z),

Σ0 := z ∈ Σ | H212(z) = H2

01(z)+H202(z),

Σ+ := z ∈ Σ | H212(z) > H2

01(z)+H202(z).

Theorem. [C-Daoud’2012] In the nbd of z0 ∈ Σ−, every extremal has the formγbγsγb, and every admissible extremal is the concatenation of at most two bang arcs.In the nbd of z0 ∈ Σ+, every extremal is bang or singular, and every optimal extremalis bang. Optimal singulars are contained in Σ0 (saturating).

z0 ∈ Σ− z0 ∈ Σ+


Assumption on (D ,g) and F0:

(i’) D21(x)+D2

2(x) < D2(x), x ∈ X .

ouD(x) := det(F1(x),F2(x),F01(x),F02(x)),D1(x) := det(F1(x),F2(x),F12(x),F02(x)),D2(x) := det(F1(x),F2(x),F01(x),F12(x)).

Corollary. Locally, every extremal has the form γbγsγb; The total switch angle forsuch a sequence is θb(z0)+θb(z′0)+π,

θb(z0) := arcsinH12√

H201 +H2

02

(z0).

Admissible extremals are concatenation of finitely many bang arcs.

Lemma. q(t) = f0(q(t), q(t))+u(t) =⇒ (i’) + involutivity (angle π switches).

Optimality of extremals

Quantum control. Magnetic field applied to molecules in liquid phase. Applica-tions in spectroscopy and medical imaging (Nuclear Magnetic Resonance).

Lindblad equations. Bilinear dynamics in the ”two-level” case:

x(t) = −Γx(t)+u2(t)z(t),y(t) = −Γy(t)−u1(t)z(t),z(t) = γ − γ z(t)+u1(t)y(t)−u2(t)x(t).

Parameters 2Γ ≥ γ ≥ |γ| model the interaction with the environment (dissipation).State q = (x,y,z) ∈ R3 is related to population density (relaxation from one level toanother). Magnetic field control u = (u1,u2).

Criterion. Minimization of the energy transferred to the system (fixed final time),∫ t f

0(u2

1(t)+u22(t))dt → min .


Spherical coordinates. Covering of S2−N,S

q = ρ(sinϕ cosθ ,sinϕ sinθ ,cosϕ).

Normal extremals. The maximized Hamiltonian is

H(x, p) = H0(x, p)+12(H2

1 (x, p)+H22 (x, p))

with

H0(x, p) =−(δ sin2ϕ + γ)ρ pρ −δ cosϕ sinϕ pϕ −

γ

ρ(ρ pρ cosϕ − pϕ sinϕ),

H1(x, p) =− pθ

tanϕ, H2(x, p) = pϕ ,

and δ := Γ− γ.

Optimality of extremalsLiouville integrability. Set r = lnρ,

H0(x, p) =−(δ sin2ϕ + γ)pr−δ cosϕ sinϕ pϕ − γe−r(pr cosϕ − pϕ sinϕ).

Proposition. For γ = 0, r and θ cyclic (symmetry of revolution) =⇒ pr and pθ

(Clairaut) linear first integrals.

Reduction to the sphere. γ = 0, pr parameter,

H(x, p) =−(δ sin2ϕ + γ)pr−δ cosϕ sinϕ pϕ +

12

(p2

θ

tan2 ϕ+ p2

ϕ

),

r(t) =−δ sin2ϕ(t)+ γ.

Algebraic curve. Set X = sin2ϕ and Y = X ; one can parameterize according to

Y 2 = 4(1−X)[(2h+2prδ +2prγ)+(δ 2− p2θ −2h−4prδ −2prγ)(1−X)

+(−2δ2 +2prδ )(1−X)2 +δ

2(1−X)3].

The genus is at most one, so ϕ is either rational or elliptic.


Cut point. First point along the extremal s.t. the extremal ceases to be (globally)minimizing.

Conjugate point. A point x(tc) on the extremal z = (x, p) is conjugate to x0 ifthere exists a Jacobi field δ z = (δx,δ p), solution of the linearized system along thisextremal,

δ z(t) = d−→H (z(t))δ z(t),

−→H = (∂pH,−∂xH),

which is non-trivial (δx not ≡ 0) and vertical at t = 0 and tc,

δx(0) = 0, δx(tc) = 0.


Cut and conjugate loci. γ = 0, pr = 0, δ = 0, ϕ0 = π/4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3

Optimality of extremalsTheorem. [Bonnard-C-Tanaka’2009] Sphere of revolution + convexity assumption=⇒ 4-cusp conjugate locus.

H(x, p) =−(δ sin2ϕ + γ)pr−δ cosϕ sinϕ pϕ +

12

(p2

θ

tan2 ϕ+ p2

ϕ

)

Optimality of extremals.

Averaging for two-body control problems The L2-minimization of the 2BPcontrolled problem

q(t) =− q(t)|q(t)|3

+u,∫ t f

0|u(t)|2 dt → min

can be analyzed by studying a suitable deformation on the previous Hamiltonian onthe sphere:

H(θ ,ϕ, pθ , pϕ) =12

(1−λ sin2

ϕ

sin2ϕ

p2θ + p2

ϕ

).

More on averaging (CNES 2013-).

– geostationary missions (navigation / telecommunication programs, NEOSAT)

– deorbitation issues, active debris removal

– time or consumption minimization

– refined dynamical models for averaging (J2 effect, atmosphere, other bodies...)



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3


Cut and conjugate loci. γ = 0, pr = 0, δ > 0, ϕ0 = π/4

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

2.5

3



3 2 1 0 1 2 30

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.6

0.7

0.8

0.9

1

1.1

1.2



3 2 1 0 1 2 30

0.5

1

1.5

2

2.5

3

1.03 1.035 1.04 1.045 1.05 1.055 1.06 1.065 1.07

0.7853

0.7853

0.7854

0.7854

0.7855



3 2 1 0 1 2 30

0.5

1

1.5

2

2.5

3

0.7 0.8 0.9 1 1.1 1.20.74

0.75

0.76

0.77

0.78

0.79

0.8

0.81

0.82

0.83

0.84


Local time minima in three-body problem. (with Farres, A.)

for the trajectories that we find in Figures 33, 35 and 37 respectively. As we can see, for type T1

trajectories Jc(t) decreases and gains energy while they spiral around the Earth and when Jc(t)is slightly larger than Jc(L1) this one starts to decreases to meet Jc of the Moon Orbit. On theother hand, for the type T2 trajectories, Jc(t) will reach much larger values than Jc(L1) beforedecreasing to get to the MO.

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

Tmax

= 10, !0 = 0, k=4

X

Y

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

Tmax

= 10, !0 = 0, k=8

X

Y

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

Tmax

= 10, !0 = 0, k=10

X

Y

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

Tmax

= 10, !0 = 0, k=11

X

Y

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

Tmax

= 10, !0 = 0, k=12

X

Y

0 0.2 0.4 0.6 0.8 1

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

0.5

Tmax

= 10, !0 = 0, k=13

X

Y

!0.5 0 0.5 1

!0.6

!0.4

!0.2

0

0.2

0.4

Tmax

= 10, !0 = 0, k=14

X

Y

!0.5 0 0.5 1

!0.4

!0.2

0

0.2

0.4

0.6

Tmax

= 10, !0 = 0, k=15

X

Y

Figure 33: For class C01: Transfer trajectories for Tmax = 10N and 0 = 0 fixed. The initial conditionon the GEO orbit is x0 = (r0 µ, 0, 0, v0).

0 0.5 1 1.5 2 2.5!10

!9

!8

!7

!6

!5

!4

!3

!2

!1

0

Tmax

= 10, !0 = 0, k=4

t

jc(t

)

0 0.5 1 1.5 2 2.5!8

!7

!6

!5

!4

!3

!2

!1

0

Tmax

= 10, !0 = 0, k=8

t

jc(t

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8!7

!6

!5

!4

!3

!2

!1

0

Tmax

= 10, !0 = 0, k=10

t

jc(t

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8!5.5

!5

!4.5

!4

!3.5

!3

!2.5

!2

!1.5

!1

!0.5

Tmax

= 10, !0 = 0, k=11

t

jc(t

)

0 0.5 1 1.5!5.5

!5

!4.5

!4

!3.5

!3

!2.5

!2

!1.5

!1

!0.5

Tmax

= 10, !0 = 0, k=12

t

jc(t

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8!5

!4.5

!4

!3.5

!3

!2.5

!2

!1.5

!1

!0.5

Tmax

= 10, !0 = 0, k=13

t

jc(t

)

0 0.5 1 1.5 2 2.5 3!5

!4

!3

!2

!1

0

1

Tmax

= 10, !0 = 0, k=14

t

jc(t

)

0 0.5 1 1.5 2 2.5 3!5

!4

!3

!2

!1

0

1

Tmax

= 10, !0 = 0, k=15

t

jc(t

)

Figure 34: For class C01: Variation of Jc for Tmax = 10N and 0 = 0 fuzzed. The initial condition onthe GEO orbit is x0 = (r0 µ, 0, 0, v0).

We recall that the main di↵erence between the behaviour of the three classes of transfertrajectories C01 (blue), C02 (red) and C03 (green) is the transfer time. As we can see in Figure 40for a fixed 0, the transfer time, tf , for class C02 orbits is always less than for class C01 and classC03. But there are three cases where these curves intersect each other. Hence, we have trajectoriesof a di↵erent class with the same transfer time (i.e. cost function). It might be interesting tostudy in more detail these intersections to study, as we have two di↵erent class of strategies withthe same cost. Here we have computed the transfer trajectories and the energy variations ofthe trajectories corresponding to the 3 intersections that we see in Figure 40, from left to rightC01 \ C02, C01 \ C03 and C02 \ C03. As usual the colour of the orbit is related to the class oforbit C01 are in blue, C02 are in red and C03 are in green.

Finally, we have done a small exploration on di↵erent characteristics for the minimum time -transfer trajectories that appear in Figures 30, 31 and 32, where for each orbit we have computedthe maximal value for Jc along the orbit, the norm of the ad-joint vector at t0, |p(t0)|, and

28



orbit takes much more time than the blue one. Moreover, the green orbit starts by approachingthe Moon with an anticlock-wise orbit and at some point it experience a drastic change.

Finally, if we look at the control law that produces these three transfer orbits (Figure 29) wesee how these one is very similar at the first part of the transfer while a big di↵erence appearswhen we approach the L1 neighbourhood. There we see how, di↵erent ways to decelerate thegrowth in energy, produce di↵erent outputs (i.e. transfer orbits).

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

x

y

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

x

y

0 0.2 0.4 0.6 0.8 1

!0.4

!0.3

!0.2

!0.1

0

0.1

0.2

0.3

0.4

x

y

Figure 29: Control Law for the transfer orbits form GEO to MO. Three di↵erent solutions found fromthe initial orbit from Figure 27

To fix notation, from now on we will call: C01 or C1+ to the the transfer trajectories that arrive

to the MO in a clockwise sense, that we will identify throughout this section by the colour blue;C02 or C1

to the transfer trajectories that arrive to the MO in a anti-clockwise sense, that wewill identify by the colour red; finally C03 or C2

+ to the transfer orbits similar to orbit 3 (green).

4.1.1 Homotopy w.r.t. 0

Here we have taken the three local minimum time transfer trajectories that appear in Figure 28(type C01, C02 and C03), and as in Section 3.1.1, we have done an homotopy with respect to theinitial position on the GEO orbit, 0. All three initial orbits are for Tmax = 10N and 0 = , werecall that the initial position on the GEO orbit is given by:

x0 = (r0 cos 0 µ, r0 sin 0,v0 cos 0, v0 sin 0).

To do this homotopy we proceed as we did in Section 3.1.1, we compute (if possible) the homotopicpath for = 7! 21 and = 7! 21, taking small intervals of size 2 to increase theprecision. In Figures 30, 31 and 32 we see the projection of these curves in the 0 vs tf space.The points in black on the 3 curves are the local minima of these curves, and will be candidatesfor local minimum time transfer trajectories when the initial condition on the GEO orbit is notfree (Section 4.2). The initial conditions for these local minima are summarised in Tables 29, 30and 31.

As we can see, the behaviour of these three curves has similarities to the results for the GEOto L1 transfer for Tmax = 10. As we can see as 0 increases so does the transfer time, and as0 decreases tf decreases up to a certain value 0, there the transfer time will start to increasesdrastically for small variations of 0 up to some point where the homotopic curve has a turningpoint. The only di↵erence is that for a given 0 the minimum transfer time is di↵erent for thethree kind of trajectories. In Figure 39 we have the three homotopic paths on the same figure,and we can appreciate how in terms of transfer time C01 is always below C02, which is alwaysbelow C03.

For each of the three homotopic curves, in Tables 17 to 17 we have the initial conditionstf , x0, p0 for the local minima for 0 = , 0, /2 and 3/2.

In Figures 33, 35 and 37 we have the transfer trajectories from the three homotopic curvesfor di↵erent initial conditions for 0( mod 2) = 0. As we can see, for each class, the trajectories

26



the second cut point (x1, p1). The three plots on the top show, from left to right: the x, yprojection of the transfer trajectory, the x, y projection of the transfer trajectory and Jc(t)along the transfer trajectory. The three plots on the bottom show, from left to right: the x, yprojection of the transfer trajectory and the control law u(t), the projection (H1, H2) and thevariation of |(H1, H2)| along the transfer trajectory. As we can see, the main di↵erence betweenboth trajectories appears on the control law at the beginning of the transfer, where the secondcut point (red curve) experiences a drastic change.

!0.2 0 0.2 0.4 0.6 0.8 1

!0.6

!0.4

!0.2

0

0.2

0.4

GEO to LMO ! C01 ! ( CUT 03 )

X

Y

!3 !2 !1 0 1 2 3 4!3

!2

!1

0

1

2

3GEO to LMO ! C01 ! ( CUT 03 )

vx

v y

0 1 2 3 4 5 6 7!5

!4.5

!4

!3.5

!3

!2.5

!2

!1.5

!1GEO to LMO ! C01 ! ( CUT 03 )

tf

Jc

!0.2 0 0.2 0.4 0.6 0.8 1

!0.6

!0.4

!0.2

0

0.2

0.4


X

Y

!4 !3 !2 !1 0 1 2 3 4 5!3

!2

!1

0

1

2

3

4


H1

H2

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

4

4.5GEO to LMO ! C01 ! ( CUT 03 )

tf

|H12 +

H22|

Figure 57: C1 cut point no 3, blue orbits correspond to the first cut value and red orbits to the secondcut value. (top-left) XY projection of the transfer trajectory, (top-centre) VxVy projection of thetransfer trajectory, (top-right) t vs Jc (energy variation along the transfer trajectory), (bottom-left)control along the trajectory, (bottom-centre) H1 vs H2, (bottom-right) t vs |(H1, H2)|.

Finally, in Figure 58 we plot the integration backward in time (t 2 [tf : 0]) and forward intime (t 2 [0 : 2tf ]) for both transfer trajectories and the variation of Jc for each of the trajectories.As we can see, when we integrate backwards in time, we have a similar behaviours as the oneexperienced by the di↵erent cut points in the GEO to L1 transfer problem. We have that the firstcut points spirals away form the Earth (red curve) while the second cut points spirals towards theEarth (blue curve). This is also reflected on the behaviour of Jc(t) where in the first case will startto grow, while in the second case this one will decrease. Moreover, this behaviour is repeated forall the cut points of class C01. If we look at the behaviour of the two trajectories for t 2 [0 : 2tf ],it is true that there is a di↵erence between the two trajectories. But as we can see in Appendix C,this behaviour does not show a distinctive pattern between the four cut points of class C01.

If we look at the cut points of class C02 (Table 5) not all the cut points experience a similarbehaviour. As the plots in Appendix C show, we have cut point number 1, 2, 3, 4, 5, 8 and 9that present a similar behaviour and cut points 6 and 7 that show another. Here we show theresults for cut point 5 and cut point number 6 and we will briefly their main di↵erences. A moreextensive study on these two kind of cut points should be done in more detail.

In Figures 59 and 60 we plot the behaviour of the trajectories close to the cut point for cutpoint number 5 and 6 respectively. Where we have the on the right hand side the variation of NEand NZ for the di↵erent solutions.

In Figures 61 and 62 we see the behaviour of the two transfer trajectories for cut point number5 and 6 respectively. As usual, red is assigned to the first cut point (x0, p0) and blue to the secondone (x1, p1). On the top we have, form left to right, the x, y projection, the x, y projection

41

References[1] Metrics with equatorial singularities on the sphere. Ann. Mat. Pura Appl., to

appear (with Bonnard, B.)

[2] Conjugate and cut loci of a two-sphere of revolution with application to optimalcontrol. Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009), no. 4, 1081-1098(with Bonnard, B.; Sinclair, R.; Tanaka, M.)

[3] Energy minimization in two-level dissipative quantum control: The integrablecase. Discrete Contin. Dyn. Syst. suppl. (2011), 198–208 (with Bonnard, B.;Cots, O.) Proceedings of 8th AIMS Conference on Dynamical Systems, Differen-tial Equations and Applications, Dresden, May 2010.

[4] Differential pathfollowing for regular optimal control problems. Optim. MethodsSoftw. 27 (2012), no. 2, 177–196. apo.enseeiht.fr/hampath (with Cots,O.; Gergaud, J.)

[5] Minimum time control of the restricted three-body problem. SIAM J. ControlOptim. 50 (2012), no. 6, 3178–3202 (with Daoud, B.)

[6] Minimum time control in the Earth-Moon system. CELMEC VI, Viterbo, Septem-ber 2013 (with Farres, A.)

ControllabilityProof. X1

µ = X1µ ∩Jµ > j connex, invariant by the flow of F0 =

−→Jµ , and of finite

measure for dq∧dp: Almost every point of X1µ is Poisson stable (Poincare). (Remark:

subset of zero measure of initial conditions generating collisions).

L2

L1

q = µ

q = 1 µ

x0

xf

2

geometric and numerical methods in space and quantum...

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