Amit K. Sanyal, Christopher Moseley, and Anthony Bloch

18 Pages·2009·0.18 MB·English

COMMUNICATIONSININFORMATIONANDSYSTEMS (cid:13)c 2009InternationalPress
Vol.9,No.1,pp.59-76, 2009 003
ANALYTICAL AND NUMERICAL SOLUTION OF A
SUB-RIEMANNIAN OPTIMAL CONTROL PROBLEM WITH
APPLICATIONS TO QUANTUM SPIN SYSTEMS∗
AMIT K. SANYAL†, CHRISTOPHER MOSELEY‡, AND ANTHONY BLOCH§
Abstract. Experimentsinnuclearmagneticresonance(NMR)spectroscopyandNMRquantum
computing requirecontrol of ensembles of quantum mechanical systems. Thecontrolled transfer of
coherence along a one-dimensional chain of spin systems plays a key role in NMR spectroscopy of
proteins, and spin chains have also been proposed for NMR quantum information processing. The
problem of time-optimal or energy-optimal control of these systems corresponds to finding optimal
paths on Lie groups in which evolution in only certain directions on the group can be directly
controlled. In this paper, we consider energy-optimal control of a three-spin system; this turns
out to be a sub-Riemannian optimal control problem on SO(4). The goal of this optimal control
problemis: giventheinitialconfigurationandthedesiredfinalconfigurationastheidentityelement
in SO(4), design three control inputs that steer the system from the initial configuration to the
identityI∈SO(4)alonganextremaltrajectory. Wefirstobtainnecessaryconditionsforthenormal
extremal trajectories for both the continuous time system, and then for its discrete counterpart
obtained fromadiscretevariational scheme. Wealsoobtain expressionsforthecontrol inputs,and
provide a numerical algorithm for the system which can be used to carry out accurate numerical
simulations.
1. Introduction. Experiments in nuclear magnetic resonance (NMR), such as
NMR spectroscopy and NMR quantum computing, require control of ensembles of
quantum mechanical systems. In particular, the NMR spectroscopy of proteins in-
volves transfer of coherent states along one-dimensional chains of spin systems [1].
Spinchainshavebeenalsoproposedfortheimplementationofsolid-stateNMRquan-
tum computers [2]. In practice, the transfer time should be as short as possible to
minimize the effects of decoherence due to interaction with the laboratory environ-
ment, and to optimize the sensitivity of the experiments.
In [3], Yuan, Glaser and Khaneja present a method for efficient transfer of co-
herence along a linear chain of n-spin systems. The initial step in this method is
∗DedicatedtoRogerBrockett ontheoccasionofhis70thbirthday.
† Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822.
E-mail: [email protected]
‡ Department of Mathematics and Statistics, CalvinCollege, Grand Rapids, MI 49546. E-mail:
[email protected]
§ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109. E-mail:
[email protected]
59
60 AMITK.SANYAL,CHRISTOPHERMOSELEY,ANDANTHONYBLOCH
represented by the equation
x 0 −1 0 0 x
1 1
    
x 1 0 −u 0 x
d 2 2
(1) = ,
    
dtx  0 u 0 −1x 
 3   3
    
    
x  0 0 1 0 x 
 4   4
    
wherex ,...,x representexpectationvaluesofunitaryoperatorsthatplayakeyrole
1 4
inthetransfer,anduisthecontrol. Yuan,GlaserandKhanejatransformthissystem
into an equivalent one-control system in which the optimal control corresponds to
finding a geodesic on the 2-sphere endowed with a special Riemannian metric. This
is related to previous work by Khaneja et al [4].
Motivated by this prior work, we consider a generalization of the system repre-
sented by (1) involving three controls u ,u and u . The optimal control problem is
1 2 3
equivalenttofindingsub-Riemanniangeodesics[5]ontheLiegroupSO(4). Inthespe-
cialcasewhereu andu arefixedatunity,onerecoversthe systemconsideredin[3].
1 3
We obtainthe extremalsolutionstothisoptimalcontrolproblemforbothcontinuous
and discrete time. The discrete time results are then used to form an algorithmfor a
numerical solution to this sub-Riemannian optimal control problem. This algorithm
providesa wayto compute optimalcontrolsforthe state transferproblemconsidered
in the first part of [3] and other similar state transfer problems for the three-spin
system.”
This paper is organized as follows. Section 2 introduces the continuous sub-
Riemannian optimal control problem on SO(4). In Section 3, we obtain extremal
trajectoriesfor the continuous optimal controlproblemusing Pontryagin’smaximum
principle [6]. Then in Section 4, we introduce the discrete time sub-Riemannian op-
timal control problem corresponding to the continuous optimal controls introduced
in Section 2. We also obtain discrete extremal trajectories for this problem in this
section, using two alternate methods: that of direct discretization using variational
principles, and that of applying Pontryagin’s maximum principle. Because of the
Hamiltonian structure of the underlying system, there are certain scalar quantities
that are shown to be conserved in both the continuous time and discrete time ex-
tremals. Thesediscreteextremalsareusedtoobtainaniterativenumericalalgorithm
to numerically simulate the (continuous) extremal trajectories in Section 5. Section
6 presents a set of numericalsimulation results obtained using this numerical scheme
for the extremal trajectory and optimal control inputs. Finally, we summarize the
developments made in this paper in the concluding Section 7 and discuss possible
future developments.
SUB-RIEMANNIANOPTIMALCONTROLPROBLEM 61
2. TheContinuousOptimalControlProblem. Considerthecontrolsystem
defined by
x 0 u 0 0 x
1 1 1
    
x −u 0 u 0 x
d 2 1 2 2
(2) = ,
    
dtx   0 −u 0 u x 
 3  2 3 3
    
    
x   0 0 −u 0x 
 4  3  4
    
in which u : R → R is differentiable for i = 1,2,3. The system (2) can be regarded
i
equivalently as defining a family of differentiable curves in the matrix group SO(4).
Any such curve Q:R→SO(4) satisfies the kinematic equation
Q˙ = u (t)X1+u (t)X2+u (t)X3 Q(t)
1 2 3
(3) =U(cid:2) (t)Q(t) (cid:3)
whereX1,X2,X3areelementsofthebasis{X1,...,X6}forso(4)=T SO(4)defined
I
by
0 1 0 0 0 0 0 0
   
−1 0 0 0 0 0 1 0
X1 = , X2 = ,
   
 0 0 0 0 0 −1 0 0
   
   
   
 0 0 0 0 0 0 0 0
   
   
0 0 0 0 0 0 −1 0
   
0 0 0 0 0 0 0 0
X3 = , X4 = ,
   
0 0 0 1 1 0 0 0
   
   
   
0 0 −1 0 0 0 0 0
   
   
0 0 0 1 0 0 0 0
   
0 0 0 0 0 0 0 −1
(4) X5 = , X6 = ,
   
 0 0 0 0 0 0 0 0 
   
   
   
−1 0 0 0 0 1 0 0 
   
   
and
(5) U(t)=u (t)X1+u (t)X2+u (t)X3.
1 2 3
In Section 3, we obtain necessary conditions for a differentiable path Q(t) satis-
fying (3) and the boundary conditions
Q(0)=Q , Q(T)=I,
0
62 AMITK.SANYAL,CHRISTOPHERMOSELEY,ANDANTHONYBLOCH
to be a normal extremal trajectory (defined below) with respect to the functional
T
(6) E(Q)= L(Q(t),U(t))dt,
Z0
where
1
(7) L(Q,U)= (u )2+(u )2+(u )2 .
1 2 3
2
(cid:2) (cid:3)
We will refer to this functional as the action functional, following Roger Brockett’s
usage [7]. Extremals of this functional are called sub-Riemannian geodesics [5], since
theyareoptimalwithrespecttoasmoothinnerproductrestrictedtothe3-planefield
spanned by right translations of {X1,X2,X3}. We also obtain evolution equations
for a discrete version of this problem in Section 4, which are then used to obtain
a numerical integration algorithm in Section 5 to numerically simulate the normal
extremal trajectories in Section 6.
3. Normal extremals for the Continuous Optimal Control Problem.
We use the notationhA,Bi to denote the inner productoftwo matricesA,B ∈Rn×n
defined by
1
(8) hA,Bi= tr(ATB).
2
The basis {X1,...X6} is orthonormal with respect to this inner product.
Let U be defined as above, and let V denote a vector of the form V = v X4+
1
v X5+v X6 ∈ so(4), so that hU,Vi = 0. Note that for any such V we may rewrite
2 3
the action Lagrangianas
1
(9) L(Q,U,V)= hU, Ui+hU, Vi.
2
ItturnsoutthatthisformulationdeterminesachoiceofV foreachU alongextremals
of the action functional (Proposition 1).
According to the maximum principle of optimal control theory, there exists a
matrix function P, called the costate associated with the state function Q, so that
the extremals of (6) minimize the Hamiltonian
(10) H(P,Q,U,V)=hP, Q˙i−p L(Q,U,V)
0
1
=hP, UQi−p hU, Ui+hU, Vi
0
2
(cid:18) (cid:19)
where p ∈ {0,1}. Solutions to the optimal control problem with p = 1 are called
0 0
normal extremals, and solutions with p = 0 are abnormal extremals. In this paper,
0
we obtain necessary conditions for a curve in SO(4) to be a normal extremal.
Proposition 1. Any extremal Q of (6) with costate P satisfies Hamilton’s
equations:
(11) Q˙ =UQ, P˙ =−U⊤P.
SUB-RIEMANNIANOPTIMALCONTROLPROBLEM 63
Proof. The first equation is merely (3). The second equation follows from
P˙ =−grad H =−grad hP, UQi
Q Q
(12) =−grad hU⊤P, Qi=−U⊤P.
Q
Bythe generaltheoryofthe maximumprinciple,thetermhP, UQiintheHamil-
tonian H represents the pairing of a right-invariant 1-form with the right-invariant
vector field UQ. Indeed,
1 1
hP, UQi= tr(P⊤UQ)= tr(QP⊤U)
2 2
1
= tr (QP⊤−PQ⊤)U
4
= 1tr(cid:0)(PQ⊤−QP⊤)⊤U(cid:1))
4
(13) =h1(P(cid:0) Q⊤−QP⊤), Ui.
2
Therefore, hP, UQi is equivalent to the pairing of the linear functional h1(PQ⊤ −
2
QP⊤), ·i∈so(4)∗ with U ∈so(4).
Define λ =hP,XiQi=hM, Xii, for i=1,...6. Then by (13), we have
i
6
1
(14) PQ⊤−QP⊤ = λ Xi.
i
2
i=1
(cid:0) (cid:1) X
Proposition 2. Along any extremal curve Q:[0,T]→SO(4),
d 1
(15) M =[U,M], M , PQ⊤−QP⊤ ,
dt 2
(cid:0) (cid:1)
where the bracket denotes the matrix commutator.
Proof. If Q:[0,T]→SO(4) is an extremal, then
d
(PQ⊤)=P˙Q⊤+P(Q˙⊤)
dt
=−U⊤PQ⊤+PQ⊤U⊤ (by (11))
=U(PQ⊤)−(PQ⊤)U
= U,PQ⊤ ,
(cid:2) (cid:3)
as given by (15). Similarly, one can show that
d
(QP⊤)= U,QP⊤ .
dt
(cid:2) (cid:3)
Therefore the result holds.
For any normal extremal of the action functional, M determines U ∈ span{X1,
X2,X3}⊂so(4) and an associated element V ∈so(4) in the orthogonalcomplement
of span{X1,X2,X3} in the following way.
64 AMITK.SANYAL,CHRISTOPHERMOSELEY,ANDANTHONYBLOCH
Theorem 1. Any normal extremal Q : [0,T] → SO(4) necessarily satisfies U +
V =M, so that
u =λ , u =λ , u =λ ,
1 1 2 2 3 3
(16)
v =λ , v =λ , v =λ .
1 4 2 5 3 6
Moreover, the component functions u and v satisfy
i i
u˙ =u v , v˙ =u v ,
1 2 1 1 3 2
(17) u˙ =u v −u v , v˙ =−u v +u v ,
2 3 3 1 1 2 3 1 1 3
u˙ =−u v , v˙ =−u v .
3 2 3 3 1 2
The system of ordinary differential equations (17) is the Eulersystem associated with
the extremal trajectory.
Proof. By the maximum principle, a necessary condition for Q : [0,T] → SO(4)
to be an extremal is that grad H = M − (U + V) = 0, so that U + V = M.
U
Since {X1,...,X6} is anorthonormalbasis for so(4), this establishes equations (16).
Equations (17) then follow from Proposition 2 and equations (16).
Asacorollary,werecoverthesymmetricequationobtainedbyBloch,Crouchand
Ratiu in [8] for normal extremals of a generalization of this sub-Riemannian optimal
control problem to a semisimple Lie group.
Corollary 1. For any normal extremal,
(18) U˙ −[U,V]+V˙ =0.
Proof. From Proposition 1 and Proposition 2, d(U +V) = d(M) = [U,M] =
dt dt
[U,V].
Note that equations (18) and (17) and equivalent. Systems of equations similar
to(18)wereobtainedinearlierworkbyBrockettin[7],andalsothemorerecentwork
in [9].
Although closed-form solutions of the Euler system (17) cannot be obtained, the
following conservation laws are easy to check.
Proposition 3. The quantities
(19) hU,Ui=(u )2+(u )2+(u )2 =C ,
1 2 3 1
(20) hV,Vi=(v )2+(v )2+(v )2 =C ,
1 2 3 2
(21) u u −v v +u v =C ,
1 3 1 3 2 2 3
are constant along any normal extremal.
Proof. By equations (17), the derivative of each expression on the left is 0.
Inthefollowingsections,wederiveanumericalalgorithmtonumericallysolvefor
the normal extremal trajectories of this system given boundary conditions.
SUB-RIEMANNIANOPTIMALCONTROLPROBLEM 65
4. The Discrete Optimal Control Problem. Fornumericalcomputationsof
extremal trajectories, we use direct discretization of the optimal control problem of
minimizingtheactionfunctional(6). Thisisanapplicationofthediscrete variational
principle [10, 11], which leads to a symplectic integration algorithm for numerical
computation of the extremal trajectories.
Consider a discrete counterpart to the kinematic equation (3):
Q =exp(hU )Q
k+1 k k
(22) with Q =Q , Q =I
0 I N
where Q ∈ SO(4), U ∈ span{X1,X2,X3} for each k = 0,1,2,...,N, for some
k k
positive integer N, Q ∈ SO(4) is the given initial state and h is a time stepsize
I
for the discretization given by h = T/N where Q(t), t ∈ [0,T] is the corresponding
continuous time trajectory. We introduce the action sum
N
1
(23) E =h hU , U i+hU , V i
d k k k k
2
k=0(cid:18) (cid:19)
X
whereV ∈span{X4,X5,X6}foreachk =0,1,2,...,N. Thediscreteoptimalcontrol
k
problem corresponds to minimizing this cost function subject to (22).
4.1. Discrete Variational Approach. We examine necessaryconditions from
the variational viewpoint. An admissible variation of Q has the form
k
(24) δQ =Y Q
k k k
where Y ∈span{X1,X2,X3} and Y =Y =0.
k 0 N
By equation (22), exp(U )=Q QT, from which we find that
k k+1 k
hδU exp(hU )=δ(exp(hU ))
k k k
=δQ QT +Q (δQ )T
k+1 k k+1 k
=Y Q QT +Q QTYT
k+1 k+1 k k+1 k k
(25) =Y exp(hU )−exp(hU )Y ,
k+1 k k k
so that admissible variations of the U have the form
k
(26) hδU =Y −Ad Y .
k k+1 exp(hUk) k
Asinthe continuouscase,there isaV ∈span{X4,X5,X6}associatedwitheachU ,
k k
k =0,...,N. AdmissiblevariationsofV taketheformδV =Z wherehY , Z i=0.
k k k k k
Theorem 2. Normal extremals corresponding to the discrete variational problem
of minimizing (23) satisfy
Q =exp(hU )Q
k+1 k k
(27) U −U +Ad V −V =0,
k+1 k exp(−hUk+1) k+1 k
66 AMITK.SANYAL,CHRISTOPHERMOSELEY,ANDANTHONYBLOCH
for k=0,1,2,...,N −1.
Proof. Thefirstofequations(27)is simplythediscretekinematicsequation. The
first variation of E is given by
d
N
δE =h (hU , δU i+hV , δU i+hU , δV i)
d k k k k k k
k=0
X
N
(28) = hU +V , Y −Ad Y i+hU , Z i ,
k k k+1 exp(hUk) k k k
k=0
X(cid:0) (cid:1)
using equation (26)and δV =Z . The necessaryconditions for extremality are then
k k
given by
N
(29) δE = hU +V −Ad (U +V ), Y i=0,
d k−1 k−1 exp(−hUk) k k k
k=1
X
andhU , Z i=0. Equation(29)isequivalenttothenecessaryconditioninthesecond
k k
of equations (27) for extremal trajectories.
These discreteextremaltrajectoriescanbe usedtonumericallysimulate the con-
tinuousextremaltrajectorieswithgiveninitialandfinalpointsQ andQ onSO(4).
0 N
In fact, we canshow thatthe firstorder (in h)approximationsofthe continuous(18)
and discrete (27) extremals are identical. For the discrete extremals (27), we obtain
the first order approximationas follows:
U −U + V −h[U ,V ] −V ≈0
k k−1 k k k k−1
U −U V −V
k k−1(cid:0) k(cid:1) k−1
(30) ⇒ −[U ,V ]+ ≈0.
k k
h h
Afirstorderapproximationofthe continuousextremals(18)willbe identicalto(30),
since the “velocities” to first order are approximated as
U −U V −V
U˙(t )= k k−1, V˙(t )= k k−1.
k k
h h
Therefore the discrete extremals are equivalent to the continuous extremals at least
upto the firstorder,andhence canbe usedto approximatethe continuousextremals
inanumericalsimulation. Thefirst-orderapproximationin(30)canbeusedtoobtain
discrete relations that are counterparts of equations (17) in the continuous case.
4.2. Applying Pontryagin’s Maximum Principle. We can also obtain the
extremals for the discrete optimalcontrolproblemof minimizing (23)using the max-
imum principle. Appending the kinematic and control constraints in (22) to the cost
function (23), we can form the discrete Hamiltonian for this problem as below:
1
(31) H(P ,Q ,U ,V )=H =hP ,exp(hU )Q i− hU ,U i−hV ,U i.
k+1 k k k k k+1 k k k k k k
2
SUB-RIEMANNIANOPTIMALCONTROLPROBLEM 67
This discrete Hamiltonian can also be expressed as:
1
(32) H(P ,Q ,U ,V )=hexp(−hU )P ,Q i− hU ,U i−hV ,U i.
k+1 k k k k k+1 k k k k k
2
The discrete extremals can now be obtained by applying the maximum principle
to the discrete Hamiltonian, as given in the following result.
Theorem 3. Extremals (Q ,P ) of the discrete optimal control problem of min-
k k
imizing (23) subject to (22) satisfy
Q =exp(hU )Q ,
k+1 k k
P =exp(hU )P , and
k+1 k k
1
(33) U +Ad V =hM , M = P Q⊤−Q P⊤ ,
k exp(−hUk) k k k 2 k k k k
(cid:0) (cid:1)
for k=0,1,...,N −1.
Proof. The necessary conditions for extremality are obtained from the discrete
Hamiltonian (32) as:
Q =grad H =exp(hU )Q ,
k+1 Pk+1 k k k
P =grad H =exp(−hU )P
k Qk k k k+1
⇒P =exp(hU )P ,
k+1 k k
which give the first two of equations (33). Now applying the maximum principle to
the discrete Hamiltonian, we get
h
(34) grad H = P Q⊤exp(−hU )−exp(hU )Q P⊤ −U −V =0.
Uk k 2 k+1 k k k k k+1 k k
(cid:16) (cid:17)
Substituting P =exp(hU )P into (34), we obtain
k+1 k k
h
(35) Ad P Q⊤−Q P⊤ =U +V ,
2 exp(hUk) k k k k k k
(cid:0) (cid:1)
which can be rewritten as the third of equations (33).
Fromthefirsttwoofequations(33),wenotethatthediscretetrajectoryexpressed
as(Q ,P )canberestrictednaturallytotheproductspaceSO(4)×SO(4),aswasthe
k k
casefortheircontinuouscounterpartsin(11). However,sucharestrictionmayinclude
only some of the discrete extremals while other extremals are excluded. Further, Q
k
andP evolvealongthesamediscretevectorfieldU ,k =0,1,...,N. Torelatethese
k k
equations to equation (27) in the variables U and V , we rewrite equation (34) as:
k k
(36) hM =U +V ,
k+1 k k
by substituting the first of equations (33) into equation (34). Therefore, we have
(37) hM =U +V .
k k−1 k−1
68 AMITK.SANYAL,CHRISTOPHERMOSELEY,ANDANTHONYBLOCH
Substituting the above back into the third of equations (33), we get
U +Ad V =U +V ,
k exp(−hUk) k k−1 k−1
which is identical to equation (27). The expression (36) also implies that
M ∈so(4)∗,
k
orM Q ∈T∗ SO(4). Using equations(33),wecanshowthe followingresult,asthe
k k Qk
discrete counterpart to Proposition 3.
Proposition 4. The following quantities
D =hU ,U i, D =hV ,V i, and
1 k k 2 k k
(38)
D =u u −v v +u v
3 k1 k3 k1 k3 k2 k2
are constant along the extremals (Q ,U ,V ) of the discrete optimal control problem
k k k
of minimizing (23) subject to (22).
Proof. Using equations (35) and (37), we obtain:
hU +V ,U +V i=h2hAd M ,Ad M i
k k k k exp(hUk) k exp(hUk) k
=h2hM ,M i
k k
=hU +V ,U +V i.
k−1 k−1 k−1 k−1
Since the U and V are orthogonal in the trace inner product, the above equation
k k
gives
hU ,U i+hV ,V i=hU ,U i+hV ,V i,
k k k k k−1 k−1 k−1 k−1
whichleadstothe conservedquantitiesD andD in(38),asU andV areintrace-
1 2 k k
orthogonal complements in so(4). Similarly, it can be verified using equation (27)
that the quantity D is conserved along the discrete extremals.
3
Note that the quantities D , D and D are the discrete counterparts of the
1 2 3
continuous functions C , C and C given by Proposition 3.
1 2 3
These discreteextremaltrajectoriescanbe usedaspartofanumericalalgorithm
for numerically obtaining the continuous extremal trajectories. This involves solving
a discrete two-point boundary value problem. Generally, a numerical algorithm to
solve this problem would consist of applying a shooting method so that the initial
and final configurations of the computed extremal trajectory are “close to” (up to
an error tolerance bound of) the given initial and final configurations (Q = Q and
0 I
Q =I).
N
5. Derivation of Numerical Algorithm. Inthis section,weoutline anumer-
ical algorithm for simulating the extremals of the original continuous-time optimal
control problem with the cost function (6). This is done by using the corresponding