Geometric and numerical methods in optimal control for the time minimal saturation in Magnetic Resonance Imaging DYNAMICS, CONTROL, and GEOMETRY In honor of Bronisław Jakubczyk’s 70th birthday 12.09.2018 - 15.09.2018 | Banach Center, Warsaw J. Rouot ∗ , B. Bonnard, O. Cots, T. Verron ∗ EPF :Troyes, France, jeremy.rouot@epf.fr J.Rouot, B.Bonnard, O.Cots, T.Verron 1 / 25
Magnetization vector • Bloch equation: M : magnetization vector of the spin-1/2 particle in a magnetic field B ( t ) . ˙ M ( t ) = − κ M ( t ) × B ( t ) z B ( t ) F. Bloch Nobel Prize (1952) M ( t ) y x J.Rouot, B.Bonnard, O.Cots, T.Verron 2 / 25
Experimental model in Nuclear Magnetic Resonance • Two magnetic fields : controlled field B 1 ( t ) and a strong static field B 0 J.Rouot, B.Bonnard, O.Cots, T.Verron 3 / 25
Experimental model in Nuclear Magnetic Resonance z • Two magnetic fields : controlled field B 1 ( t ) and B 0 a strong static field B 0 � 0 • � � � � M x � M x − Γ M x − ω 0 ω y = • + − Γ M y ω 0 0 − ω x M y M y y • − γ ( M 0 − M z ) − ω y ω x 0 M z M z x B 1 ( t ) Γ , γ are parameters related to the observed species ω 0 is fixed and associated to B 0 ω x , ω y are related to the controlled magnetic field B 1 ( t ) J.Rouot, B.Bonnard, O.Cots, T.Verron 3 / 25
Experimental model in Nuclear Magnetic Resonance z • Two magnetic fields : controlled field B 1 ( t ) and B 0 a strong static field B 0 � 0 • � � � � M x � M x − Γ M x − ω 0 ω y = • + − Γ M y ω 0 0 − ω x M y M y y • − γ ( M 0 − M z ) − ω y ω x 0 M z M z x B 1 ( t ) Γ , γ are parameters related to the observed species ω 0 is fixed and associated to B 0 ω x , ω y are related to the controlled magnetic field B 1 ( t ) • M ( t ) ∈ S ( O , | M ( 0 ) | ) , B 1 ≡ 0 ⇒ relaxation to the stable equilibrium M = ( 0 , 0 , | M ( 0 ) | ) . J.Rouot, B.Bonnard, O.Cots, T.Verron 3 / 25
• Normalized Bloch equation in the rotating frame ( ω 0 , ( Oz )) x ( t ) = − Γ x ( t ) + u y ( t ) z ( t ) , ˙ y ( t ) = − Γ y ( t ) − u x ( t ) z ( t ) , ˙ z ( t ) = γ ( 1 − z ( t )) − u y ( t ) x ( t ) + u x ( t ) y ( t ) . ˙ q = ( x , y , z ) = M / M ( 0 ) is the normalized magnetization vector, ( u x , u y ) is the control. J.Rouot, B.Bonnard, O.Cots, T.Verron 4 / 25
• Normalized Bloch equation in the rotating frame ( ω 0 , ( Oz )) x ( t ) = − Γ x ( t ) + u y ( t ) z ( t ) , ˙ y ( t ) = − Γ y ( t ) − u x ( t ) z ( t ) , ˙ z ( t ) = γ ( 1 − z ( t )) − u y ( t ) x ( t ) + u x ( t ) y ( t ) . ˙ q = ( x , y , z ) = M / M ( 0 ) is the normalized magnetization vector, ( u x , u y ) is the control. • Symmetry of revolution around ( Oz ) , we set: u y = 0 and we obtain the planar control system y ( t ) = − Γ y ( t ) − u ( t ) z ( t ) , ˙ z ( t ) = γ ( 1 − z ( t )) + u ( t ) y ( t ) , ˙ and u = u x is the control satisfying | u | ≤ 1. J.Rouot, B.Bonnard, O.Cots, T.Verron 4 / 25
Saturation of a single spin in minimum time • Aim . Steer the North pole N = ( 0 , 1 ) of the Bloch ball {| q | ≤ 1 } to the center O in minimum time. N = q ( 0 ) Bloch Ball Inversion sequence | q | ≤ 1 O = q ( t f ) σ v s u = 0 u = − 1 σ + − The inversion sequence σ N − σ v s is not optimal in many physical cases J.Rouot, B.Bonnard, O.Cots, T.Verron 5 / 25
• Pontryagin Maximum Principle . Pseudo-Hamiltonian: H ( q , p , u ) = p · ( F ( q ) + u G ( q )) = H F + u H G u ( · ) optimal ⇒ ∃ p ( · ) ∈ R 2 \ { 0 } : q = ∂ H p = − ∂ H ˙ ∂ p , ˙ ∂ q H ( q ( t ) , p ( t ) , u ( t )) = max | v |≤ 1 H ( q ( t ) , p ( t ) , v ) = cst ≥ 0 Regular and bang-bang controls: u ( t ) = sign ( H G ( q ( t ) , p ( t ))) , H G ( q ( t ) , p ( t )) � = 0 Singular trajectories are contained in { q , det( G , [ F , G ])( q ) = 0 } : z = γ/ ( 2 δ ) = z s ( γ, Γ) , δ = γ − Γ and y = 0 . J.Rouot, B.Bonnard, O.Cots, T.Verron 6 / 25
Computations: D ′ ( q ) + u D ( q ) = 0 with D = det( G , [ G , [ F , G ]]) and D ′ = det( G , [ F , [ F , G ]]) . We obtain: u s = γ ( 2 Γ − γ ) / ( 2 δ y ) on the horizontal singular line. u s = 0 on the vertical singular line Symmetry: u ← − u corresponds to y ← − y Collinearity set: C = { q | det( F , G )( q ) = 0 } C Switching function: Φ( t ) = p ( t ) · G ( q ( t )) and outside the set C , sign ( ˙ Φ( t )) = sign ( α ( q )) , α ( q ) � = 0 where α ( q ( t )) = det( G , [ F , G ])( q ) . det( G , F )( q ) J.Rouot, B.Bonnard, O.Cots, T.Verron 7 / 25
Definition of the points S 1 , S 3 C The singular trajectory q ( · ) is called d 2 ∂ d t 2 ∂ H Hyperbolic if p ( t ) · [ G , [ F , G ]]( q ( t )) = ∂ u ( q ( t ) , p ( t )) > 0. ∂ u d 2 ∂ d t 2 ∂ H Elliptic if p ( t ) · [ G , [ F , G ]]( q ( t )) = ∂ u ( q ( t ) , p ( t )) < 0. ∂ u J.Rouot, B.Bonnard, O.Cots, T.Verron 8 / 25
Optimal synthesis depends on the ratio γ Γ . Case 2: S 1 exists and Case 1: S 1 exists and Case 3: S 1 doesn’t exist S 2 / ∈ S 1 S 3 S 2 ∈ S 1 S 3 J.Rouot, B.Bonnard, O.Cots, T.Verron 9 / 25
Case 1: S 1 exists and S 2 ∈ S 1 S 3 Optimal trajectory from N to O : σ N + σ h s σ b + σ v s J.Rouot, B.Bonnard, O.Cots, T.Verron 10 / 25
Case 2 : S 1 exists and S 2 / ∈ S 1 S 3 Optimal trajectory from N to O : σ N + σ v s J.Rouot, B.Bonnard, O.Cots, T.Verron 11 / 25
Case 3: S 1 doesn’t exists Optimal trajectory from N to O : σ N + σ v s J.Rouot, B.Bonnard, O.Cots, T.Verron 12 / 25
Theorem The time optimal trajectory for the saturation problem of 1-spin is of the form: σ N σ h σ b σ v + s + s � �� � empty if S 2 ≤ S 1 J.Rouot, B.Bonnard, O.Cots, T.Verron 13 / 25
Numerical validations using Moments/LMI techniques Aim: Provide lower bounds on the global optimal time. • Numerical times obtain with the HamPath software to validate : Case Γ γ t f 9.855 × 10 − 2 3.65 × 10 − 3 42.685 C 1 2.464 × 10 − 2 3.65 × 10 − 3 C 2 110.44 1.642 × 10 − 2 2.464 × 10 − 3 164.46 C 3 9.855 × 10 − 2 9.855 × 10 − 2 C 4 8.7445 J.Rouot, B.Bonnard, O.Cots, T.Verron 14 / 25
Context t f = inf u ( · ) T x ( t ) = f ( x ( t ) , u ( t )) , ˙ x ( t ) ∈ X , u ( t ) ∈ U , x ( 0 ) ∈ X 0 , x ( T ) ∈ X T X , U , X 0 , X T are subsets of R n which can be written as X = { ( t , x ) : p k ( t , x ) ≥ 0 , k = 0 , . . . , n X } , U = { u : q k ( u ) ≥ 0 , k = 0 , . . . , n U } X 0 = { x : r 0 k ( x ) ≥ 0 , k = 0 , . . . , n 0 } , X T = { ( t , x ) : r T k ( t , x ) 0 , k = 0 , . . . , n T } Objective: Compute min u ( · ) T when f , p k , q k , r 0 k , r T k are polynomials and the above sets are compacts. Result: [J. B. Lasserre, D. Henrion, C. Prieur, E. Trélat, 2008] Converging monotone nondecreasing sequence of lower bounds of t f . J.Rouot, B.Bonnard, O.Cots, T.Verron 15 / 25
J.Rouot, B.Bonnard, O.Cots, T.Verron 16 / 25
T J.Rouot, B.Bonnard, O.Cots, T.Verron 17 / 25
T � T � T � � v ∈ C 0 ([ 0 , T ] × X ) v ( t , x ( t )) d t = v ( t , x ) d µ ( t , x , u ) , 0 0 X U J.Rouot, B.Bonnard, O.Cots, T.Verron 18 / 25
Liouville’s equation Linear equation linking the measures µ 0 , µ and µ T . � � � ∂ v v ( T , x ) d µ T ( x ) − v ( 0 , x ) d µ 0 ( x ) = ∂ t + ∇ x · f ( x , u ) d µ ( t , x , u ) X T X 0 [ 0 , T ] × Q × U for all test functions v ∈ C 1 ([ 0 , T ] × X ) . Optimization over system trajectories ⇔ Optimization over measures satisfying Liouville equation. J.Rouot, B.Bonnard, O.Cots, T.Verron 19 / 25
Relaxed controls: u ( t ) is replaced for each t by a probability measure ω t ( u ) supported on U . Relaxed problem: T R = min T ω � s . t . x ( t ) = ˙ f ( x ( t ) , u ) d ω t ( u ) U x ( 0 ) ∈ X 0 , x ( t ) ∈ X , x ( T ) ∈ X T J.Rouot, B.Bonnard, O.Cots, T.Verron 20 / 25
Relaxed controls: u ( t ) is replaced for each t by a probability measure ω t ( u ) supported on U . Relaxed problem: T R = min T ω � s . t . x ( t ) = ˙ f ( x ( t ) , u ) d ω t ( u ) U x ( 0 ) ∈ X 0 , x ( t ) ∈ X , x ( T ) ∈ X T Linear Problem on measures: d µ ( t , x , u ) = d t d δ x ( t ) ( x ) d ω t ( u ) ∈ M + ([ 0 , T ] × X × U ) � T LP = min d µ T µ,µ T ,µ 0 � � ∂ v ∂ t + ∂ v � s . t . ∂ x f ( x , u ) d µ � � = v ( · , x T ) d µ T − v ( 0 , x 0 ) d µ 0 , ∀ v ∈ R [ t , x ] , µ ∈ M + ([ 0 , T ] × X × U ) , µ T ∈ M + ( X T ) , µ T ∈ M + ( X 0 ) J.Rouot, B.Bonnard, O.Cots, T.Verron 20 / 25
Notation: α = ( α 1 , . . . , α p ) ∈ N p , z = ( z 1 , . . . , z p ) ∈ R p . We denote by z α the monomial z α 1 1 . . . z α p and by N p d the set p { α ∈ N p , | α | 1 = � p i = 1 α i ≤ d } . � z α d ν ( z ) . Moment of order α for a measure ν ∈ M + ( Z ) : y ν α = Riesz linear functional: l y ν : R [ z ] → R s.t. l y ν ( z α ) = y ν α . i + j , ∀ i , j ∈ N p Moment Matrix: M d ( y ν )[ i , j ] = y ν d . J.Rouot, B.Bonnard, O.Cots, T.Verron 21 / 25
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