Geometric and numerical methods in optimal control for the time - PowerPoint PPT Presentation

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