Geometric Control and Homotopic Methods for solving a Bang-Singular - - PowerPoint PPT Presentation

Geometric Control and Homotopic Methods for solving a Bang-Singular problem

Applied and Numerical Optimal Control Spring School & Workshop, Ensta ParisTech

• O. Cots

23-27 April 2012

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 1 / 40

Introduction - Collaborations

Figure: (LHS) Hard pulse of 90◦. (RHS) Optimal solution.

– Bernard Bonnard (Univ. Bourgogne) – Jean-Baptiste Caillau (Univ. Bourgogne) – Joseph Gergaud (N7 - IRIT) – Steffen J. Glaser (Univ. Munich) – Dominique Sugny (Univ. Bourgogne) – . . .

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 2 / 40

Contrast problem

(P)                            |q2(tf )|2 =

• y2

2(tf ) + z2 2(tf )

• → max
• ˙

y1 = − Γ1 y1 − u z1 ˙ z1 = γ1(1 − z1) + u y1

• ˙

y2 = − Γ2 y2 − u z2 ˙ z2 = γ2(1 − z2) + u y2 q1(0) = (0, 1) = q2(0) q1(tf ) = (0, 0) qi = (yi, zi), |qi| ≤ 1, i = 1, 2 γi = 1/(32.3Ti1.10−3) Γi = 1/(32.3Ti2.10−3) |u| ≤ 2π Spin 1 (y1, z1) - Deoxygenated blood : T11 = 1350 ms, T12 = 50 ms Spin 2 (y2, z2) - Oxygenated blood : T21 = 1350 ms, T22 = 200 ms Spin 1 (y1, z1) - Cerebrospinal fluid : T11 = 2000 ms, T12 = 200 ms Spin 2 (y2, z2) - Water : T21 = 2500 ms, T22 = 2500 ms Blood: Tmin = 6.7980978 Fluid: Tmin = 20.2301921

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 3 / 40

Pontryagin Maximum Principle

We have a (smooth) Mayer problem of the form: min

u∈U c(q(tf )),

˙ q = F(q) + uG(q), q = (q1, q2) ∈ R4, |u| ≤ 2π, u ∈ R, where

c(q(tf )) = −

• y2

2(tf ) + z2 2(tf )

• F(q) =

i=1,2(−Γiyi) ∂ ∂yi + (γi(1 − zi)) ∂ ∂zi

G(q) =

i=1,2 −zi ∂ ∂yi + yi ∂ ∂zi

The Hamiltonian is: H(q, p, u) = p, F(q) + uG(q) And the Maximum Principle gives: u = 2π sign(HG) is bang if HG = 0 u is singular if HG = 0 The boundary conditions are: q1(tf ) = 0 (zero magnetization of the first spin) p2(tf ) = −2p0q2(tf ), p0 ≤ 0.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 4 / 40

Simplest possible solution

The north pole N = ((0, 1), (0, 1)) = x(0), is an equilibrium point for the singular control system (us = 0): start with a bang. Due to symmetry of revolution: the first bang is either u = +2π or u = −2π. A bang solution amounts roughly to a rotation of each plane (yi, zi) around 0. At tf , q1(tf ) = 0, and there is no improvement by rotation (because T21 > T22) of the contrast: no bang at the end.

Lemma

The simplest BC-extremal in the contrast problem is of the form B+S.

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

Figure: Trajectories of spin 1, 2 and the control associated for the Blood case.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 5 / 40

Singular extremals

The Hamiltonian is H(q, p, u) = HF(q, p) + u HG(q, p), HF := p, F, HG := p, G. Let z(·) = (q(·), p(·)) be a singular extremal, then HG(z(.)) = 0 identically. Differentiating with respect to time, HG(z(t))= 0 dHG dt (z(t)) = {H, HF}(z(t)) = {HG, HF}(z(t))= 0    (2 constraints) d2HG dt2 (z(t)) = {{HG, HF}, HF}(z(t)) + us(t){{HG, HF}, HG}(z(t)) = 0 Besides, + we can restrict p to Hs = h because of the homogeneity : us(q, αp) = us(q, p). + generalized Legendre-Clebsch condition has to be satisfied that is {{HG, HF}, HG} ≥ 0. ⇒ for each q(0) ∈ R4, p(0) ∈ R4 lives in half a straight line and the singular flow starting from q(0) is a surface of dimension 2.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 6 / 40

Singular extremals in the Fluid case

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2

Figure: Singular surface with blowing up control in red ({{HG, HF}, HG} tends to 0).

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 7 / 40

Singular extremals in the Fluid case

−0.1 −0.05 0.05 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0.02 0.04 0.06 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2

Figure: Zoom of singular surface with blowing up control in red ({{HG, HF}, HG} tends to 0).

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 8 / 40

Local optimality of singular arcs

Définition

Let z(·) be a reference singular solution of # — Hs on [0, tf ]. The variational equation on the tangent space to Σs := {HG = {HF, HG} = 0} ˙ δz = d# — Hs(z(t))δz dHG = d{HF, HG} = 0 is called Jacobi equation. A Jacobi field J(t) = (δq, δp) is a non-zero solution of Jacobi equation. It is said semi-vertical at time t if δq(t) ∈ RG(q(t)). A time t ∈ (0, tf ] is said to be conjugate if there exists a Jacobi field J(t) semi-vertical at 0 and t.

Theorem

Under strict Legendre-Clebsch condition (i.e. {{HG, HF}, HG} > 0) and additional generic assumptions, the absence of conjugate times on (0, tf ) is necessary for local

• ptimality of a singular arc.
• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 9 / 40

Conjugate points fot singular extremals in the Fluid case

−0.1 −0.05 0.05 −0.1 −0.08 −0.06 −0.04 −0.02 0.02 y1 z1 −0.3 −0.2 −0.1 −0.85 −0.8 −0.75 −0.7 −0.65 −0.6 y2 z2

Figure: Zoom of singular surface with conjugate points in red.

⇒ We can expect more intricate structures than BS in the fluid case.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 10 / 40

The limit case tf = Tmin. Single spin problem (see [BCG+11])

(P1)                    tf → min ˙ y = − Γ y − u z ˙ z = γ(1 − z) + u y q(0) = (0, 1) q(tf ) = (0, 0) q = (y, z), |q| ≤ 1

3γ 2

< Γ |u| ≤ 2π The dynamics: ˙ q = F(q) + u G(q) The Hamiltonian: H(q, u, p) = HF(q, p) + u HG(q, p), Hi =< p, Fi >, i = F, G ⇒ The singular extremals are those contained in HG = 0.

[BCG+11]

• B. Bonnard, O. Cots, S. Glaser, M. Lapert, and D. Sugny.

Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance. math.u-bourgogne.fr, 2011.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 11 / 40

Single spin: singular extremals

⇒ The singular extremals are those contained in HG = 0. HG = < p, G > = ˙ HG = < p, [G, F] > =

• ⇒ det(G, [G, F]) = y(−2δz + γ) = 0

with δ = γ − Γ. The singular lines are y = 0 and z0 =

γ 2δ . The singular control us(q, p) is computed from

¨ HG = {{HG, HF}, HF}(z) + us{{HG, HF}, HG}(z) = 0. Horizontal line (z0 =

γ 2δ ): optimal, according to Generalized Legendre-Clebsch

condition (see [BC03]) and us(q) = γ(2Γ − γ)/(2δy) ⇒ us ∈ L1, us / ∈ L2 Vertical line (y = 0): optimal for z0 < z < 1 (GLC) and us(q) = 0.

[BC03]

• B. Bonnard and M. Chyba.

Singular trajectories and their role in control theory, mathématiques & applications, vol. 40. lavoisier.fr, Jan 2003.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 12 / 40

Single spin: global synthesis

Red: bang (u = 2π) Blue: bang (u = −2π) Green: singular (u = us) B: saturation

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 13 / 40

Summary: single spin => contrast problem

Solutions of the contrast problem are made of Bang-Singular sequences. We get Tmin. The solution for the limit case tf = Tmin is B+SB+S. To solve the contrast problem, we use an indirect method (multiple shooting) ⇒ we need to know the structure a priori. ⇒ we use an homotopic approach to capture the structure and initialize the multiple shooting method.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 14 / 40

Homotopy

(Pλ)                            −

• y2

2(tf ) + z2 2(tf )

• + (1 − λ)

tf

0 |u|2−λ(t)dt → min

• ˙

y1 = − Γ1 y1 − u z1 ˙ z1 = γ1(1 − z1) + u y1

• ˙

y2 = − Γ2 y2 − u z2 ˙ z2 = γ2(1 − z2) + u y2 q1(0) = (0, 1) = q2(0) q1(tf ) = (0, 0) qi = (yi, zi), |qi| ≤ 1, i = 1, 2 γi = 1/(32.3 Ti1) Γi = 1/(32.3 Ti2) |u| ≤ 2π Homotopy (Pλ): −

• y2

2(tf ) + z2 2(tf )

• + (1 − λ)

tf

0 |u|2−λ(t)dt → min

The Hamiltonian: H(q, u, p, λ) = HF(q, p) + u HG(q, p) + (1 − λ) |u|2−λ (Pλ)-extremals are admissible for (P).

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 15 / 40

Maximum principle

The maximization of the Hamiltonian gives: u(q, p, λ) = argmax H(q, w, p, λ) w ∈ U and we have λ < 1 ⇒ u = sign(HG) min

• 2π,
• 2|HG|

((2 − λ)(1 − λ))

• 1

(1−λ)

• We call true Hamiltonian the function (which does not depend on u)

Hr(q, p, λ) = H(q, u(q, p, λ), p, λ) In order to solve the problem (P) in λ = 1, we: 1 solve by simple shooting method the problem in λ = 0 easily, 2 use homotopic method to solve (Pλ), from λ = 0 to λ = λf < 1, to . . . 3 . . . capture the structure of (P) and initialize the multiple shooting method.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 16 / 40

Simple shooting method

Smooth optimal control problem (Pλ) (PMP) (BVPλ)    (˙ x, ˙ p) = H(x, p, λ) = ( ∂H(x,p,λ)

∂p

, − ∂H(x,p,λ)

∂x

) b0(x(0), p(0), λ) = 0 ∈ Rn bf (x(tf ), p(tf ), λ) = 0 ∈ Rn (Shooting method) Sλ(x0, p0) = b0(x0, p0, λ) bf (xf , pf , λ)

• = 0 (S : R2n → R2n)

with (xf , pf ) ≡ (x(tf , x0, p0, λ), p(tf , x0, p0, λ)) = exptf # — H(x0, p0, λ)

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 17 / 40

1: Solving in λ = 0 with simple shooting method: Blood case

The final time tf is equal to 1.1Tmin where Tmin is the minimal time to transfer the spin 1 from (0, 1) to (0, 0). Blood case: Spin 1 = (De)oxygenate blood : T11 = 1350 and T21 = 200 Spin 2 = Oxygenate blood : T12 = 1350 and T22 = 50

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u

Figure: Solution for λ = 0 with L2−λ regularization. Trajectories of spin 1 and 2 and the control associated.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 18 / 40

2: homotopic method

Let us define the homotopic function h : R2n × [0, 1] − → R2n (z0, λ) − → Sλ(z0) with z0 = (q0, p0). Assuming that 0 is a regular value for h, the level set {h = 0} is a one dimensional submanifold of R2n+1 called the path of zeros. We know a zero of h(., λ) for λ0 = 0 noted z0

0, and we want to follow this path to reach a

zero for a target value of the parameter λ close to 1.

λ z0 (zf

0, λf )

(z0

0, 0)

˙ c(s)

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 19 / 40

Differential algorithms

HAMPATH uses DOPRI5 from E. Hairer and G. Wanner [HrW93] [HW], for the numerical integration (without any correction) of: (IVP) ˙ c(s) = T(c(s)) c(0) = (z0

0, 0)

Until sf such that λ(sf ) close to 1 (dense output). ˙ c(s) = T(c(s)) is determined by

1

h′(c(s))˙ c(s) = 0

2

|˙ c(s)|=1

3

det h′(c(s))

t˙

c(s)

• is of constant sign

λ z0 (zf

0, λf )

(z0

0, 0)

˙ c(s)

[HrW93]

• E. Hairer, S.P

. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I, Nonstiff Problems, volume 8 of Springer Serie in Computational Mathematics. Springer-Verlag, second edition, 1993. [HW]

• E. Hairer and G. Wanner.

DOPRI5 http://www.unige.ch/~hairer/prog/nonstiff/dopri5.f.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 20 / 40

The path of zeros: λf = 0.915: Blood case

−2 −1.5 −1 −0.5 0.5 0.2 0.4 0.6 0.8 1 py1(0) λ −2.5 −2 −1.5 −1 −0.5 0.2 0.4 0.6 0.8 1 pz1(0) λ −0.08 −0.07 −0.06 −0.05 −0.04 −0.03 0.2 0.4 0.6 0.8 1 py2(0) λ 0.215 0.22 0.225 0.23 0.2 0.4 0.6 0.8 1 pz2(0) λ

Figure: Homotopic path with L2−λ regularization. Initial adjoint vector w.r.t. λ.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 21 / 40

Control: λ ∈ {0.5, 0.8, 0.9, 0.915}: Blood case

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u λ = 0.5 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u λ = 0.8 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u λ = 0.9 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u λ = 0.915

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 22 / 40

States-Control: λ = 0.915: Blood case

0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u λ = 0.915

− → Bang-Singular structure.

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2

Figure: Solution for λ = 0.915 with L2−λ regularization. Trajectories of spin 1 and 2 and the control associated.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 23 / 40

3: multiple shooting

We have a BS structure for tf = 1.1Tmin. We denote by t0, t1, tf the different instants and by z0, z1, zf the the associated state and costate variables (z = (q1, q2, p1, p2) ∈ R4 × R4). Hamiltonian : H(q, u, p) = HF(q, p) + u HG(q, p), u = 2π if t ∈ [t0; t1] u = using if t ∈ [t1; tf ] Equations (cf. [Mau76]) : Bang Sing (t0, z0) − → (t1, z1) − → (tf , zf ) q1 = (0, 1) HG = 0 q1 = (0, 0) q2 = (0, 1) ˙ HG = 0 q2 = p2 with the matching conditions z(t1; t0, z0) = z1.

[Mau76]

• H. Maurer.

Numerical solution of singular control problems using multiple shooting techniques. Journal of optimization theory and applications, Jan 1976.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 24 / 40

Trajectories of spin 1 and 2 and the control: blood case (λ = 1.0)

We used the solution at λ = 0.915 to initialise the multiple shooting. The time t1 is taken in [0.01, 0.2] and leads to the following solutions.

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

Figure: (TOP) First solution (contrast of 0.41). (BOTTOM) Second solution (contrast of 0.42). From the L2−λ regularization.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 25 / 40

Trajectories of spin 1 and 2 and the control: blood case (λ = 1.0)

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6

Figure: Best solution (contrast of 0.45) from the L2−λ regularization.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 26 / 40

Summary

The regularizations L2−λ permit to find solutions satisfying the first order necessary conditions of optimality and BS structure has been detected. To solve in λ = 0 is very easy. Homotopy is automatic: we do not need to provide any steps grid. But there are many local solutions.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 27 / 40

Contrast problem: sensitivity w.r.t. the final time tf

We use HAMPATH to perform a differential continuation on tf between Tmin + ε and 2Tmin. Tmin : the minimal time to transfer the spin 1 from (0, 1) to (0, 0). ֒ → there is an horizontal asymptote on the contrast plot. 99% of the optimal solution is

• btained for tf = 1.3Tmin.

1 1.2 1.4 1.6 1.8 2 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 tf/min tf

• y2

2(tf) + z 2 2(tf)

1 1.2 1.4 1.6 1.8 2 0.05 0.1 0.15 0.2 t1/tf tf/min tf

Figure: Best solution in black. (LHS) Contrast w.r.t the transfer duration. (RHS) Normalized switching time.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 28 / 40

Solution for the limit case: tf close to Tmin

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u

Figure: Best solution for tf = 1.000004 × min tf .

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 29 / 40

1: Solving in λ = 0 for the second example: the fluid case

The final time tf is equal to 1.5Tmin where Tmin is the minimal time to transfer the spin 1 from (0, 1) to (0, 0). Fluid case: Spin 1 = Cerebrospinal fluid : T11 = 2000 and T21 = 200 Spin 2 = Water : T12 = 2500 and T22 = 2500

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 t u

Figure: Solution for λ = 0 with L2−λ regularization. Trajectories of spin 1 and 2 and the control associated.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 30 / 40

2: Homotopy results (on λ): fluid case (λf = 0.954)

0.2 0.4 0.6 0.8 1 0.55 0.6 0.65 0.7 0.75 0.8 λ

• y2

2(T ) + z 2 2(T )

1.47 1.48 1.49 1.5 1.51 1.52 0.2 0.4 0.6 0.8 1 |p(0)| λ

Figure: λ = 0.0 → λ = 0.954, contrast ∈ [0.586, 0.765] and norm of the path.

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 −2 −1 1 2 3 4 5 6 t u

Figure: Solution for λ = 0.954 (contrast of 0.765) from the L2−λ regularization.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 31 / 40

3: Final solution for λ = 1 and homotopy results (on tf )

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 −6 −4 −2 2 4 6 t u

Figure: Solution for λ = 1, tf = 1.5Tmin (contrast of 0.783) from the L2−λ regularization.

1 1.5 2 2.5 3 0.65 0.7 0.75 0.8 tf/min tf

• y2

2(tf) + z 2 2(tf)

1.5 1.55 1.6 1.65 1 1.5 2 2.5 3 |p(0)| tf/min tf

Figure: (BLUE) 2-BS structure. (RED) 3-BS structure. Homotopy: tf = 3Tmin → tf = Tmin, contrast ∈ [0.69, 0.8] and norm of the path.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 32 / 40

Summary

1 1.5 2 2.5 3 0.65 0.7 0.75 0.8 tf/min tf

• y2

2(tf) + z 2 2(tf)

Figure: (BLUE) 2-BS structure. (RED) 3-BS structure. Contrast w.r.t. tf .

During the continuation we detect that for tf = 1.225Tmin, the last singular control does not respect the constraint (us <= 2π). Hence, we add a Bang arc, which consists in adding two switching times in the multiple shooting formulation and we have a 3BS-structure.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 33 / 40

New results in the blood case: homotopy on parameters

Spin 1 (y1, z1) - Deoxygenated blood : T11 = 1350 ms, T12 = 50 ms Spin 2 (y2, z2) - Oxygenated blood : T21 = 1350 ms, T22 = 200 ms Spin 1 (y1, z1) - Cerebrospinal fluid : T11 = 2000 ms, T12 = 200 ms Spin 2 (y2, z2) - Water : T21 = 2500 ms, T22 = 2500 ms We perform here a continuation on the physical parameters from the Fluid case to the Blood one.

−1 −0.5 0.5 1 −1 −0.5 0.5 1 y1 z1 −1 −0.5 0.5 1 −1 −0.5 0.5 1 y2 z2 0.2 0.4 0.6 0.8 1 −6 −4 −2 2 4 6 t u

Figure: 3BS solution for the blood case, obtained from the 2BS fluid solution. Here λ = 1.0, tf = 1.5Tmin, contrast is 0.484.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 34 / 40

New results in the blood case: homotopy on the final time

1 1.5 2 2.5 0.4 0.42 0.44 0.46 0.48 0.5 tf/min tf

• y2

2(tf) + z 2 2(tf)

1.424 1.426 1.428 1.43 1.432 1.434 1.436 1 1.5 2 2.5 |p(0)| tf/min tf

Figure: (BLUE) 2BS-structure. (RED) 3BS-structure. Contrast and norm of the path w.r.t. tf

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 35 / 40

Old and New results in the blood case

1 1.5 2 2.5 0.4 0.42 0.44 0.46 0.48 0.5 0.52 tf/min tf

• y2

2(tf) + z 2 2(tf)

1.42 1.425 1.43 1.435 1.44 1.445 1.45 1 1.5 2 2.5 |p(0)| tf/min tf

Figure: (BLACK) BS-structure. (BLUE) 2BS-structure. (RED) 3BS-structure. Contrast and norm of the path w.r.t. tf

⇒ The optimal structure varies according to tf which is not detected in this case, during the path following.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 36 / 40

Algorithmic aspects: HAMPATH package

From the true hamiltonian and the boundary, intermediate and transversality conditions, HAMPATH [CCG11]: produces automatically the state-costate equations (thanks to TAPENADE) computes the shooting function by numerical integration (thanks to DOPRI5) provides the variational equations used in the jacobian of the shooting function (thanks to TAPENADE) integrates the variational equations so that the diagram commutes (the step size control is only made on the state and co-state equations) (IVP)

Numerical integration

− − − − − − − − − − → h(z0, λ)

Derivative

 

• 

 Derivative (VAR)

Numerical integration

− − − − − − − − − − →

∂h ∂z0 (z0, λ)

[CCG11] J.B. Caillau, O. Cots, and J. Gergaud. Differential pathfollowing for regular optimal control problems. HAMPATH apo.enseeiht.fr/hampath. Optimization Methods and Software, to appear, 2011.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 37 / 40

Global diagram of HAMPATH

efun b(ti, zi, λ) hfun Hr(t, z, λ) db

• H

( ∂Hr ∂p , − ∂Hr ∂x ) S(ti, zi, λ) d H S′(ti, zi, λ) T(S′(ti, zi, λ)) ssolve shooting method hampath continuation method expdhvfun variational equations TAPENADE TAPENADE DOPRI5 VAR DOPRI5 DOPRI5 VAR DOPRI5 HYBRJ MATLAB functions Available for use FORTRAN 90

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 38 / 40

Conclusion

The HAMPATH package [CCG11] gives tools to solve OCP by indirect methods:

Shooting methods: simple and multiple; Homotopic methods: differential without correction; Function which integrate the variationnal equations and permit to check sufficient second order conditions in the smooth case; ...

The derivatives and the functions are computed accurately and automatically. It is very easy to use since only 2 FORTRAN subroutines need to be implemented. Then it is interfaced with MATLAB. Homotopic method detects BS and 2BS-structures and gives good initial points for multiple shooting. It gives also some sensitivity w.r.t to the final time and parameters. Finding the global optimum is stil an open problem. Some work have been done about second order conditions for the contrast problem with BS structure (see [BC12] for details);

[BC12]

• B. Bonnard and O. Cots.

Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance. submitted to Mathematical Models and Methods in Applied Sciences (M3AS), 2012.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 39 / 40

[AG03] E.L. Allgower and K. Georg. Introduction to numerical continuation methods, volume 45 of Classics In Applied Mathematics. SIAM, 2003. [BC03]

• B. Bonnard and M. Chyba.

Singular trajectories and their role in control theory, mathématiques & applications, vol. 40. lavoisier.fr, Jan 2003. [BC12]

• B. Bonnard and O. Cots.

Geometric numerical methods and results in the control imaging problem in nuclear magnetic resonance. submitted to Mathematical Models and Methods in Applied Sciences (M3AS), 2012. [BCG+11]

• B. Bonnard, O. Cots, S. Glaser, M. Lapert, and D. Sugny.

Geometric optimal control of the contrast imaging problem in nuclear magnetic resonance. math.u-bourgogne.fr, 2011. [BCS09]

• B. Bonnard, M. Chyba, and D. Sugny.

Time-minimal control of dissipative two-level quantum systems: the generic case. Automatic Control, Jan 2009. [BS09]

• B. Bonnard and D. Sugny.

Time-minimal control of dissipative two-level quantum systems: the integrable case. SIAM J. Control Optim, Jan 2009. [CCG11] J.B. Caillau, O. Cots, and J. Gergaud. Differential pathfollowing for regular optimal control problems. HAMPATH apo.enseeiht.fr/hampath. Optimization Methods and Software, to appear, 2011. [HrW93]

• E. Hairer, S.P

. Nørsett, and G. Wanner. Solving Ordinary Differential Equations I, Nonstiff Problems, volume 8 of Springer Serie in Computational Mathematics. Springer-Verlag, second edition, 1993. [HW]

• E. Hairer and G. Wanner.

DOPRI5 http://www.unige.ch/~hairer/prog/nonstiff/dopri5.f. [LZB+10]

• M. Lapert, Y. Zhang, M. Braun, S. Glaser, and D. Sugny.

Singular extremals for the time-optimal control of dissipative spin 1/2 particles. Physical review letters, Jan 2010. [Mau76]

• H. Maurer.

Numerical solution of singular control problems using multiple shooting techniques. Journal of optimization theory and applications, Jan 1976.

• O. Cots (IMB Bourgogne)

Results on contrast problem April 2012 40 / 40