Optimal Control Problems with Symmetry Breaking Cost Functions

0.25 MB·

by  Anthony Bloch

OPTIMAL CONTROL PROBLEMS WITH SYMMETRY BREAKING COST
FUNCTIONS
ANTHONYM.BLOCH,LEONARDOJ.COLOMBO,ROHITGUPTA,ANDTOMOKIOHSAWA
Abstract. Weinvestigate symmetryreduction ofoptimalcontrol problemsforleft-invariant
7
control systems onLiegroups, withpartialsymmetrybreakingcost functions. Our approach
1
emphasizes the role of variational principles and considers a discrete-time setting as well as
0
thestandardcontinuous-timeformulation. Specifically,werecasttheoptimalcontrolproblem
2
asaconstrainedvariationalproblemwithapartialsymmetrybreakingLagrangianandobtain

n the Euler–Poincar´e equations from a variational principle. By applying a Legendre transfor-
a mationtoit,werecovertheLie–PoissonequationsobtainedbyA.D.Borum[Master’sThesis,
J UniversityofIllinoisatUrbana-Champaign,2015]inthesamecontext. Wealsodiscretizethe

4 variationalprincipleintimeandobtainthediscrete-timeLie–Poissonequations. Weillustrate
2 thetheorywithsomepracticalexamplesincludingamotionplanningprobleminthepresence
ofanobstacle.

]
C
O Key words: Euler–Poincar´e equations, Lie–Poisson equations, Optimal control, Symmetry re-
. duction
h
t AMS s.c.(2010): 70G45, 70H03, 70H05, 37J15,49J15
a
m
1. Introduction
[
Symmetry reduction of optimal control problems (OCPs) for left-invariant control systems

1 on Lie groups has been studied extensively over the past couple of decades. Such symmetries
v
are usually described as an invariance under an action of a Lie group and the system can be
3
reduced to a lower-dimensional one or decoupled into subsystems by exploiting the symmetry
7
9 (see, e.g., [1], [3], [11], [13], [22], [23], [27], , [29], [30]). Symmetry reduction of OCPs is desirable
6 from a computational point of view as well. Given that solving OCPs usually involves itera-
0 tive/numericalmethods suchas the shootingmethod(as opposedto solvinga singleinitialvalue
.
1 problem), reducing the system to a lower-dimensional one results in a considerable reduction of
0 the computational cost as well.
7 ThegoalofthisworkistostudysymmetryreductionofOCPsforleft-invariantcontrolsystems
1
onLiegroupswithpartiallybrokensymmetries,morespecifically,costfunctionsthatbreaksome
:
v of the symmetries but not all. Symmetry breaking is common in several physical contexts, from
Xi classical mechanics to particle physics. The simplest example is the heavy top dynamics (the
motion of a rigid body with a fixed point in a gravitational field), where due to the presence
r
a of gravity, we get a Lagrangian that is SO(2)-invariant but not SO(3)-invariant, contrary to
what happens for the free rigid body. Reduction theory for Lagrangian/Hamiltoniansystems on
Lie groups with symmetry breaking was developed in [16], [24], [26]. In the context of motion
planning, the symmetry breaking appears naturally in the form of a barrier function as we shall
see in Section 4.2.
TheresultsinthispaperaretheLagrangian/variationalcounterpartofthosein[5],[6];wealso
developa discrete-time versionof the results. From the Lagrangianpoint of view, we obtain the
Euler–Poincar´e equations from a variational principle and by using a Legendre transformation,
we obtainthe Lie–Poissonequations. By discretizing the variationalprinciple intime, we obtain
the discrete-time Lie–Poissonequations.
Wealsoworkoutsomepracticalexamplestoillustratethetheory,includinganoptimalcontrol
descriptionofthe heavyrigidtopequationsaswellasamotionplanningproblemwithobstacles;
the latter providesanexample ofthe symmetry breakingbya barrierfunction mentionedabove.
1.1. Outline. The paper is organized as follows. In Section 2, we introduce some preliminaries
about geometric mechanics on Lie groups. In Section 3, we study the Euler–Poincar´ereduction
1
2 A.M.BLOCH,L.J.COLOMBO,R.GUPTA,ANDT.OHSAWA
of OCPs for left-invariant control systems on Lie groups, with partial symmetry breaking cost
functionsandwealsoobtaintheLie–PoissonequationsbyusingaLegendretransformation. Fur-
thermore,weconsiderthe discrete-time caseandobtainthe discrete-time Lie–Poissonequations.
In Section 4 some practical examples are considered to illustrate the theory. Finally, in Section
5, we make some concluding remarks.
2. Preliminaries
Let Q be an n-dimensional differentiable manifold with local coordinates (q1,...,qn), the
configuration space of a mechanical system. Denote by TQ its tangent bundle with induced
local coordinates (q1,...,qn,q˙1,...,q˙n). Given a Lagrangian L: TQ → R, the corresponding
Euler–Lagrangeequations are
d ∂L ∂L
− =0, i=1,...,n,
dt(cid:18)∂q˙i(cid:19) ∂qi
which determine a system of implicit second-order differential equations.
Anindispensabletoolinthestudyofmechanicalsystemsissymmetryreduction. Particularly,
whentheconfigurationspaceisaLiegroup,onecanreducetheEuler–Lagrangeequations,which
generallygiveasystemofsecond-orderequations,toasystemoffirst-orderonesonitsLiealgebra.
Let G be a Lie group and g be its Lie algebra. Let L : G → G be the left translation by
g
an element g ∈ G, i.e., L (h) = gh, for any h ∈ G. Note that the map L is a diffeomorphism
g g
on G and is also a left action of G on itself. The tangent map of L at h ∈ G is denoted by
g
T L : T G→T G. Similarly, the cotangent map of L at h∈G is denoted by T∗L : T∗ G→
h g h gh g h g gh
T∗G. A vector field X ∈ X(G) is called left-invariant, if X(L (h)) = T L (X(h)), for any g,
h g h g
h ∈ G. Let Φ: G×Q → Q be a left action of G on Q. A function f: Q → R is said to be
G-invariant, if f ◦Φ =f, for any g ∈G.
g
If we assume that the Lagrangian L: TG → R is G-invariant under the left action of G on
TG, then we can obtain a reduced Lagrangian ℓ: g → R, where ℓ(ξ) = L(g−1g,TgLg−1(g˙))
= L(e,ξ). We can now obtain the reduced Euler–Lagrange equations, commonly known as the
Euler–Poincar´eequations (see, e.g., [2], [17], [25]), which are given by1
d ∂ℓ ∂ℓ
=ad∗ . (2.1)
dt∂ξ ξ ∂ξ
The Euler–Poincar´e equations together with the reconstruction equation ξ = TgLg−1(g˙) are
equivalent to the Euler–Lagrange equations on G. If we assume that the reduced Lagrangian ℓ
is hyper-regular, then we can obtain the reduced Hamiltonian h: g∗ → R (by using a Legendre
transformation) given by
h(µ)=hµ,ξi−ℓ(ξ),
where µ = ∂ℓ ∈ g∗. The Euler–Poincar´eequations (2.1) can now be written as the Lie–Poisson
∂ξ
equations (see, e.g., [2], [17], [25]), which are given by
µ˙ =ad∗ µ. (2.2)
∂h
∂µ
3. Optimal Control Problems on Lie Groups
Wewillfirstdefinealeft-invariantcontrolsystemonG,whichisassumedtoben-dimensional.
Definition 1. A left-invariant control system on G is given by
g˙ =T L (u),
e g
where g(·) ∈ C1([0,T],G) and u is a curve in the vector space g. More precisely, if g =
span{e ,...,e ,e ,...,e }, then u is given by
1 m m+1 n
m
u(t)=e + ui(t)e ,
0 i
Xi=1
1For an infinite-dimensional Lie algebra, one generally uses the δ notation instead of the ∂ notation (see,
e.g.,[4]).
OPTIMAL CONTROL PROBLEMS WITH SYMMETRY BREAKING COST FUNCTIONS 3
where e ∈g and the m-tuple of control inputs [u1...um]T take values in Rm.
0
Remark. If m < n, then the left-invariant control system is under-actuated otherwise it is
fully-actuated. ♦
In what follows, we will fix k = span{e ,...,e }, p = span{e ,...,e }, with e ∈ p and
1 m m+1 n 0
E :=G×k. Consider the following OCP
T
minJ :=min [C(g(t),u(t))+V(g(t))]dt
u(·) u(·) Z0
subject to (P1)
g˙(t)=T L (u(t)), g(0)=g , g(T)=g ,
e g(t) 0 T
where the following assumptions hold:
(i) C :E →R is a G-invariant function (under a suitable left action of G on E, which will be
defined shortly) and is also sufficiently regular.
(ii) V :G→R(potentialfunction)isnotaG-invariantfunctionandisalsosufficiently regular.
(iii) The potential function also depends on a parameter α that may be considered to be an
0
element of the dual space X∗ of a vector space X. Hence, we define the extended potential
function as V :G×X∗ →R, with V (·,α )=V.
ext ext 0
(iv) ThereisaleftrepresentationρofGonX,i.e.,ρ :G→GL(X)isahomomorphism. Hence,
·
there is a left representation ρ∗ of G on X∗, i.e., ρ∗ : G→GL(X∗), defined as the adjoint
·
of ρ, i.e., for any g ∈G, we define ρ∗g−1 ∈GL(X∗) as the adjoint of ρg−1 ∈GL(X), i.e., for
any x∈X and α∈X∗
hρ∗g−1(α),xi=hα,ρg−1(x)i.
As a result, there is a left action of G on G×X∗ defined as
Φ:G×(G×X∗)−→G×X∗,
(g,(h,α))7−→(L (h),ρ∗ (α)). (3.1)
g g−1
(v) The extended potential function is G-invariant under (3.1), i.e., V ◦Φ = V , for any
ext g ext
g ∈G, or more concretely V (L (h),ρ∗ (α))=V (h,α), for any h∈G and α∈X∗.
ext g g−1 ext
(vi) The potential function is invariant under the left action of the isotropy group
G :={g ∈G|ρ∗(α )=α }.
α0 g 0 0
Remark. Note that E is a trivial vector bundle over G and define the left action of G on E as
follows
Ψ:G×E −→E,
(g,(h,u))7−→(L (h),u). (3.2)
g
Throughout the paper, we assume that C : E →R is G-invariant under (3.2), i.e., C◦Ψ =C,
g
for any g ∈G. ♦
3.1. Euler–Poincar´e Reduction. We can solve (P1) as a constrained variational problem
using the method of Lagrange multipliers (see, e.g, [2], [22]). Define the augmented Lagrangian
L :E×p∗×X∗ →R as follows
a
L (g,u,λ,α)=C(g,u−e )+V (g,α)+hλ,u−e i,
a 0 ext 0
where λ(·)∈C1([0,T],p∗). Define the left action of G on E×p∗×X∗ as follows
Υ:G×(E×p∗×X∗)−→E×p∗×X∗,
(g,(h,u,λ,α))7−→(L (h),u,λ,ρ∗ (α)). (3.3)
g g−1
Under the assumption (v), it follows that L : E ×p∗ ×X∗ → R is G-invariant under (3.3),
a
i.e., L ◦Υ = L , for any g ∈ G. In particular, under the assumption (vi), it follows that the
a g a
4 A.M.BLOCH,L.J.COLOMBO,R.GUPTA,ANDT.OHSAWA
augmented Lagrangian L (·,·,·,α ) =: L : E ×p∗ → R is G -invariant under (3.3), i.e.,
a 0 a,α0 α0
L ◦Υ = L , for any g ∈ G . We can now obtain the reduced augmented Lagrangian
a,α0 g a,α0 α0
ℓ :k×p∗×X∗ →R, which is given by
a
ℓ (u,λ,α):=L (e,u,λ,α)
a a
=C(u−e )+V (α)+hλ,u−e i,
0 ext 0
where α = ρ∗(α ) and with a slight abuse of notation, we write C(e,u−e ) = C(u−e ) and
g 0 0 0
V (α)=V (e,α). Anormalextremalfor(P1)satisfiesthefollowingEuler–Poincar´eequations
ext ext
(see Theorem 1 below for a proof)
d ∂C ∂C ∂V
+λ =ad∗ +λ +J ext,α , (3.4)
dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) X(cid:18) ∂α (cid:19)
α˙ =ρ′∗(α), α(0)=ρ∗ (α ), (3.5)
u g0 0
where J : T∗X ∼= X ×X∗ → g∗ is the momentum map corresponding to the left action of G
X
on X defined using the left representation ρ of G on X, i.e., for any x, ξ ∈X and α∈X∗
hJ (x,α),ξi=hα,ξ (x)i, (3.6)
X X
with ξ being the infinitesimal generator of the left action of G on X and ρ′∗ : g → gl(X∗) is
X ·
defined as the adjoint of ρ′, which is the representation induced by the left representation ρ of
G on X. Note that the solution to (3.5) is given by α(·) = ρ∗ (α ). Also, note that using the
g(·) 0
notationof[16], we haveJ (x,α)=x⋄α. For moredetails, see [9], [16],[17]. To summarize,we
X
have the following theorem.
Theorem 1. Anormalextremalfor (P1), undertheassumptions (i)–(vi), satisfiesthefollowing
Euler–Poincar´e equations
d ∂C ∂C ∂V
+λ =ad∗ +λ +J ext,α , (3.7)
dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) X(cid:18) ∂α (cid:19)
α˙ =ρ′∗(α), α(0)=ρ∗ (α ). (3.8)
u g0 0
Proof. The prooffollowsargumentssimilarto the onesgivenin[16],[22]. Considerthe following
variational principles:
(a) The variational principle
T
δ L (g(t),u(t),λ(t))dt=0,
Z a,α0
0
holds for all variations of g (vanishing at the endpoints) and u.
(b) The constrained variational principle
T
δ ℓ (u(t),λ(t),α(t))dt =0,
a
Z
0
holds using variations of u and α of the form
δu=η˙+ad η,
u
δα=ρ′∗(α),
η
where η(·)∈C1([0,T],g) vanishes at the endpoints.
We will first show that the variational principle (a) implies the constrained variational principle
(b). WebeginbynotingthatL :E×p∗×X∗ →RisG-invariantunder(3.3),i.e.,L ◦Υ =L ,
a a g a
for any g ∈ G and α = ρ∗(α ), so the integrand in the variational principle (a) is equal to the
g 0
integrand in the constrained variational principle (b). However, all variations of g vanishing
at the endpoints induce and are induced by variations of u of the form δu = η˙ +ad η, with
u
η(0) = η(T) = 0. The relation between δg and η is given by η = TgLg−1(δg) (see Proposition
5.1 in [4], which is Lemma 3.2 in [16]). So, if the variational principle (a) holds and if we define
η = TgLg−1(δg) and δu = TgLg−1(g˙), then by Proposition 5.1 in [4], we have δu = η˙ +aduη.
OPTIMAL CONTROL PROBLEMS WITH SYMMETRY BREAKING COST FUNCTIONS 5
In addition, we have δα = ρ′∗(α). Hence, the variational principle (a) implies the constrained
η
variational principle (b).
A normal extremal for (P1) satisfies the variational principle (a) and hence, the constrained
variational principle (b), by the above discussion. So, we have
T
0=δ ℓ (u(t),λ(t),α(t))dt
a
Z
0
T ∂C ∂V
= ,δu +hλ,δui+ δα, ext dt
Z (cid:20)(cid:28)∂u (cid:29) (cid:28) ∂α (cid:29)(cid:21)
0
T ∂C ∂V
= +λ,δu + δα, ext dt
Z (cid:20)(cid:28)∂u (cid:29) (cid:28) ∂α (cid:29)(cid:21)
0
T ∂C ∂V
= +λ,η˙+ad η + δα, ext dt
Z (cid:20)(cid:28)∂u u (cid:29) (cid:28) ∂α (cid:29)(cid:21)
0
T d ∂C ∂C ∂V
= − +λ +ad∗ +λ ,η + ρ′∗(α), ext dt
Z (cid:20)(cid:28) dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) (cid:29) (cid:28) η ∂α (cid:29)(cid:21)
0
T d ∂C ∂C ∂V
= − +λ +ad∗ +λ +J ext,α ,η dt,
Z (cid:20)(cid:28) dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) X(cid:18) ∂α (cid:19) (cid:29)(cid:21)
0
where we have used integration by parts along with the fact that η(0) = η(T) = 0 and so, we
have
d ∂C ∂C ∂V
+λ =ad∗ +λ +J ext,α .
dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) X(cid:18) ∂α (cid:19)
Finally, by taking the time derivative of α, we have
α˙ =ρ′∗(α), α(0)=ρ∗ (α ).
u g0 0
(cid:3)
There are two special cases of Theorem 1 that are of particular interest and we state them as
corollaries.
Corollary 1. Let X =g and ρ be the adjoint representation of G on X, i.e., ρ =Ad , for any
g g
g ∈G. Then, the Euler–Poincar´e equations (3.7)–(3.8) give the following equations
d ∂C ∂C
+λ =ad∗ +λ −ad∗ α,
dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) ∂Vext
∂α
α˙ =ad∗α, α(0)=Ad∗ α .
u g0 0
Proof. We first begin by noting that α ∈ X∗ = g∗ and ρ∗ is the coadjoint representation of G
on X∗, i.e., ρ∗ = Ad∗, for any g ∈ G. We also have ρ′ = ad , for any ξ ∈ g and it follows that
g g ξ ξ
ξ (x)=ad x, for any x∈X. From (3.6), we have
X ξ
hJ (x,α),ξi=hα,ad xi
X ξ
=hα,−ad ξi
x
=h−ad∗α,ξi,
x
which gives J (x,α)=−ad∗α. (cid:3)
X x
Corollary 2. Let X = g∗ and ρ be the coadjoint representation of G on X, i.e., ρg = Ad∗g−1,
for any g ∈G. Then, the Euler–Poincar´e equations (3.7)–(3.8) give the following equations
d ∂C ∂C ∂V
+λ =ad∗ +λ +ad∗ ext,
dt(cid:18)∂u (cid:19) u(cid:18)∂u (cid:19) α ∂α
α˙ =−aduα, α(0)=Adg0−1α0.
6 A.M.BLOCH,L.J.COLOMBO,R.GUPTA,ANDT.OHSAWA
Proof. We first begin by noting that α∈X∗ =g∗∗ ∼=g and ρ∗ is the adjoint representationof G
on X∗, i.e., ρ∗g = Adg−1, for any g ∈ G. We also have ρ′ξ = −ad∗ξ, for any ξ ∈ g and it follows
that ξ (x)=−ad∗x, for any x∈X. From (3.6), we have
X ξ
hJ (x,α),ξi=hα,−ad∗xi
X ξ
=h−ad α,xi
ξ
=hx,ad ξi
α
=had∗ x,ξi,
α
which gives J (x,α)=ad∗ x. (cid:3)
X α
TheEuler–Poincar´eequation(3.7)isnotparticularlyfeasiblefordescribingthetimeevolution
ofuandλbecausetheequationsforthemarecombinedintoasingleequation. However,assuming
an additional structure on the Lie algebra g, we may decouple the equations for u and λ (see [3]
for a similar approachapplied to the standard Euler–Poincar´eequation):
Proposition 1. Assume that g=k⊕p such that
[k,k]⊆p, [p,k]⊆k, [p,p]⊆p,
then the time evolution of u and λ in (3.7) are given by the following equations
d ∂C ∂C ∂V
=ad∗ +ad∗ λ+J ext,α ,
dt ∂u e0 ∂u u−e0 X(cid:18) ∂α (cid:19)(cid:12)
(cid:12)k
dλ ∂C ∂V (cid:12)
=ad∗ λ+ad∗ +J ext,α (cid:12) ,
dt e0 u−e0 ∂u X(cid:18) ∂α (cid:19)(cid:12)
(cid:12)p
(cid:12)
where u−e ∈k. (cid:12)
0
Proof. It is easy to verify that g∗ =k∗⊕p∗ such that
ad∗k∗ ⊆p∗, ad∗k∗ ⊆k∗, ad∗p∗ ⊆p∗, ad∗p∗ ⊆k∗. (3.9)
k p p k
It is also easy to verify that ∂C ∈ k∗ and by definition λ ∈ p∗. We now have a splitting of the
∂u
left hand side of (3.7) in k∗ and p∗. We have the following
∂C ∂C ∂C
ad∗ =ad∗ +ad∗ ,
u ∂u e0 ∂u u−e0 ∂u
ad∗λ=ad∗ λ+ad∗ λ,
u e0 u−e0
where u− e ∈ k. By using (3.9), we have ad∗ ∂C ∈ k∗, ad∗ ∂C ∈ p∗, ad∗ λ ∈ p∗ and
0 e0 ∂u u−e0 ∂u e0
ad∗ λ ∈k∗. We can also split J (∂Vext,α) ∈g∗ in k∗ and p∗. We now have a splitting of the
u−e0 X ∂α
right hand side of (3.7) in k∗ and p∗. So, (3.7) splits into the following equations
d ∂C ∂C ∂V
=ad∗ +ad∗ λ+J ext,α ,
dt ∂u e0 ∂u u−e0 X(cid:18) ∂α (cid:19)(cid:12)
(cid:12)k
dλ ∂C ∂V (cid:12)
=ad∗ λ+ad∗ +J ext,α (cid:12) .
dt e0 u−e0 ∂u X(cid:18) ∂α (cid:19)(cid:12)
(cid:12)p
(cid:12)
(cid:12) (cid:3)
Remark. Note thatsemisimpleLie algebrasadmitaCartandecomposition,i.e.,ifg issemisim-
ple, then g=k⊕p such that
[k,k]⊆p, [p,k]⊆k, [p,p]⊆p,
where k = {x ∈ g | θ(x) = −x} is the −1 eigenspace of the Cartan involution θ and p = {x ∈
g | θ(x) = x} is the +1 eigenspace of the Cartan involution θ. In addition, the Killing form is
positivedefiniteonkandnegativedefiniteonp. So,connectedsemisimpleLiegroupsarepotential
candidates that satisfy the assumption of Proposition 1. Conversely, a Cartan decomposition
determines a Cartaninvolutionθ (see, e.g.,[18]). For more details, see [10], [15], [18]. Also, note
that the roles of k and p can be reversed in Proposition 1. ♦
OPTIMAL CONTROL PROBLEMS WITH SYMMETRY BREAKING COST FUNCTIONS 7
3.2. LegendreTransformation. IfweassumethatthereducedLagrangianℓ ishyper-regular,
a
then we can obtain the reduced Hamiltonian h : g∗ × p∗ × X∗ → R (by using a Legendre
a
transformation) given by
h (µ,λ,α)=hµ,ui−ℓ (u,λ,α),
a a
where µ = ∂ℓa =(∂C +λ), with µ(·)∈C1([0,T],g∗). The Euler–Poincar´eequations (3.7)–(3.8)
∂u ∂u
can now be written as the Lie–Poissonequations (see, e.g., [12], [16]), which are given by
∂V
µ˙ =ad∗µ+J ext,α , (3.10)
u X(cid:18) ∂α (cid:19)
α˙ =ρ′∗(α), α(0)=ρ∗ (α ). (3.11)
u g0 0
Remark. TheLie–Poissonequations(3.10)–(3.11)arealsoobtainedin[5],[6]usingPontryagin’s
maximum principle. For more details, see [5], [6]. ♦
3.3. Discrete Lagrange–Pontryagin Principle. Consider the discrete-time version of
(P1) given by
N−1
min J := min [C (g ,u )+V (g )]
d d k k d k
{uk}Nk=−01 {uk}Nk=−01 kX=0
subject to (P2)
g =g τ(hu ), with given boundary conditions g and g ,
k+1 k k 0 N
where h∈R is the time-step, τ :g→G is the retractionmap(see, e.g.,[7],[8], [20], [19],[21])
>0
and the functions C and V satisfy assumptions (i)–(vi).
d d
Wecansolve(P2)usingadiscreteanalogueoftheLagrange–Pontryaginvariationalprinciple.
Define the augmented cost function as follows
N−1
J = L (g ,g ,u ,µ ,α),
d,a d,a k k+1 k k
Xk=0
where the augmented LagrangianL :G×E×g∗×X∗ →R is defined as follows
d,a
L (g ,g ,u ,µ ,α)=C (g ,u −e )+V (g ,α) (3.12)
d,a k k+1 k k d k k 0 d,ext k
1
+h µ , τ−1(g−1g )−(u −e ) .
(cid:28) k h k k+1 k 0 (cid:29)
Define the left action of G on G×E×g∗×X∗ as follows
Γ:G×(G×E×g∗×X∗)−→G×E×g∗×X∗,
(g,(h ,h ,u,µ,α))7−→(L (h ),L (h ),u,µ,ρ∗ (α)). (3.13)
1 2 g 1 g 2 g−1
Undertheassumption(v),itfollowsthatL :G×E×g∗×X∗ →RisG-invariantunder(3.13),
d,a
i.e., L ◦Γ = L , for any g ∈ G. In particular, under the assumption (vi), it follows that
d,a g d,a
the augmented Lagrangian L (·,·,·,·,α ) =: L : G×E×g∗ → R is G -invariant under
d,a 0 d,a,α0 α0
(3.13), i.e., L ◦Γ = L , for any g ∈ G . We can now obtain the reduced augmented
d,a,α0 g d,a,α0 α0
Lagrangianℓ :E×g∗×X∗ →R, which is given by
d,a
ℓ (π(g ,g ),u ,µ ,α¯ ):=L (e,π(g ,g ),u ,µ ,α¯ ),
d,a k k+1 k k k d,a k k+1 k k k
=C (u −e )+V (α¯ )
d k 0 d,ext k
1
+h µ , τ−1(π(g ,g ))−(u −e ) ,
(cid:28) k h k k+1 k 0 (cid:29)
where π(g ,g ) := g−1g ∈ G, α¯ = ρ∗ (α ) and with a slight abuse of notation, we write
k k+1 k k+1 k gk 0
C (e,u −e )=C (u −e )andV (α¯ )=V (e,α¯ ). Anormalextremalfor(P2) satisfies
d k 0 d k 0 d,ext k d,ext k
the following discrete-time Lie–Poissonequations (see Theorem 2 below for a proof)
∂V
(dτ−1)∗µ =(dτ−1 )∗µ +J d,ext,α¯ , (3.14)
huk k −huk−1 k−1 X(cid:18) ∂α¯k k(cid:19)
8 A.M.BLOCH,L.J.COLOMBO,R.GUPTA,ANDT.OHSAWA
α¯ =ρ∗ (α¯ ), α¯ =ρ∗ (α ), (3.15)
k+1 τ(huk) k 0 g0 0
where µ = 1∂Cd ∈ g∗. To summarize, we have the following theorem but before proceeding
k h∂uk
further, we recall a lemma, which will be used in its proof.
Lemma 1 ( [7], [8], [21]). The following properties hold:
1 1 1
δτ−1(g−1g )= δτ−1(π(g ,g ))= dτ−1(−η +Ad η ), (3.16)
h k k+1 h k k+1 h huk k τ(huk) k+1
where ηk =TgkLgk−1(δgk)∈g.
(dτ−1 )∗µ =Ad∗ (dτ−1)∗µ , (3.17)
−huk k τ(huk) huk k
where µ ∈g∗ and dτ−1 is the inverse right trivialized tangent of τ.
k
Theorem 2. Anormalextremalfor (P2), undertheassumptions (i)–(vi), satisfiesthefollowing
discrete-time Lie–Poisson equations
∂V
(dτ−1)∗µ =(dτ−1 )∗µ +J d,ext,α¯ , (3.18)
huk k −huk−1 k−1 X(cid:18) ∂α¯k k(cid:19)
α¯ =ρ∗ (α¯ ), α¯ =ρ∗ (α ), (3.19)
k+1 τ(huk) k 0 g0 0
where µ = 1∂Cd ∈g∗.
k h∂uk
Proof. Consider the following variational principles:
(a) The variational principle
N−1
δ L (g ,g ,u ,µ )=0,
d,a,α0 k k+1 k k
Xk=0
holds for all variations of g (vanishing at the endpoints), τ−1(g−1g ) (induced by the
k k k+1
variations of g ) and u .
k k
(b) The variational principle
N−1
δ ℓ (π(g ,g ),u ,µ ,α¯ )=0,
d,a k k+1 k k k
Xk=0
holds for all variations of τ−1(π(g ,g )) (induced by the variations of g vanishing at the
k k+1 k
endpoints), u and α¯ of the form
k k
δα¯ =ρ′∗(α¯ ),
k ηk k
where η ∈g vanishes at the endpoints.
k
Wewillfirstshowthatthevariationalprinciple(a)impliesthevariationalprinciple(b). Webegin
by noting that L : G×E×g∗ ×X∗ → R is G-invariant under (3.13), i.e., L ◦Γ = L ,
d,a d,a g d,a
for any g ∈ G and α¯ = ρ∗ (α ), so the summand in the variational principle (a) is equal to
k gk 0
the summand in the variational principle (b). In addition, we have δα¯ = ρ′∗(α¯ ). Hence, the
k ηk k
variational principle (a) implies the variational principle (b).
A normal extremal for (P2), satisfies the variational principle (a) and hence, the variational
principle (b). So, we have
N−1
0=δ ℓ (π(g ,g ),u ,µ ,α¯ )
d,a k k+1 k k k
kX=0
N−1
∂C ∂V
= d,δu −hhµ ,δu i+ δα¯ , d,ext
(cid:20)(cid:28)∂u k(cid:29) k k (cid:28) k ∂α¯ (cid:29)
kX=0 k k
1
+h µ , dτ−1(−η +Ad η )
(cid:28) k h huk k τ(huk) k+1 (cid:29)(cid:21)
OPTIMAL CONTROL PROBLEMS WITH SYMMETRY BREAKING COST FUNCTIONS 9
N−1
∂C ∂V
= d −hµ ,δu + δα¯ , d,ext
(cid:20)(cid:28)∂u k k(cid:29) (cid:28) k ∂α¯ (cid:29)
kX=0 k k
1
+h µ , dτ−1(−η +Ad η )
(cid:28) k h huk k τ(huk) k+1 (cid:29)(cid:21)
N−1
∂C ∂V
= d −hµ ,δu + ρ′∗(α¯ ), d,ext
(cid:20)(cid:28)∂u k k(cid:29) (cid:28) ηk k ∂α¯ (cid:29)
kX=1 k k
+ (dτ−1 )∗µ −(dτ−1)∗µ ,η
D −huk−1 k−1 huk k kE(cid:21)
N−1
∂C
= d −hµ ,δu
(cid:20)(cid:28)∂u k k(cid:29)
kX=1 k
∂V
+ (dτ−1 )∗µ +J d,ext,α¯ −(dτ−1)∗µ ,η ,
(cid:28) −huk−1 k−1 X(cid:18) ∂α¯k k(cid:19) huk k k(cid:29)(cid:21)
where we have used Lemma 1, the analogue of integration by parts in the discrete-time setting
along with the fact that η =η =0 and so, we have
0 N
∂V
(dτ−1)∗µ =(dτ−1 )∗µ +J d,ext,α¯ .
huk k −huk−1 k−1 X(cid:18) ∂α¯k k(cid:19)
Finally, after a simple calculation, we have
α¯ =ρ∗ (α¯ ), α¯ =ρ∗ (α ).
k+1 τ(huk) k 0 g0 0
(cid:3)
As in the continuous-time case, there are are two special cases of Theorem 2 that are of
particular interest and we state them as corollaries.
Corollary 3. Let X =g and ρ be the adjoint representation of G on X, i.e., ρ =Ad , for any
g g
g ∈G. Then, the discrete-time Lie–Poisson equations (3.18)–(3.19) give the following equations
(dτ−1)∗µ =(dτ−1 )∗µ −ad∗ α¯ ,
huk k −huk−1 k−1 ∂V∂dα¯,ekxt k
α¯ =Ad∗ α¯ , α¯ =Ad∗ α ,
k+1 τ(huk) k 0 g0 0
where µ = 1∂Cd ∈g∗.
k h∂uk
Proof. See Corollary 1. (cid:3)
Corollary 4. Let X = g∗ and ρ be the coadjoint representation of G on X, i.e., ρg = Ad∗g−1,
for any g ∈ G. Then, the discrete-time Lie–Poisson equations (3.18)–(3.19) give the following
equations
∂V
(dτ−1)∗µ =(dτ−1 )∗µ +ad∗ d,ext,
huk k −huk−1 k−1 α¯k ∂α¯k
α¯k+1 =Adτ(huk)−1α¯k, α¯0 =Adg0−1α0,
where µ = 1∂Cd ∈g∗.
k h∂uk
Proof. See Corollary 2. (cid:3)
4. Examples
We will now consider two examples.
10 A.M.BLOCH,L.J.COLOMBO,R.GUPTA,ANDT.OHSAWA
4.1. The Heavy Top as an Optimal Control Problem. We will now show that the heavy
top equations can also be obtained from OCPs for left-invariant control systems on Lie groups,
with partial symmetry breaking cost functions.
We now consider (P1), with G=SO(3), e =0 and with the cost function given by
0 3×3
1
C(g,u)= hu,Iui,
2
V(g)=−mglhe ,gχi,
3
where hξ,ξi:=tr(ξTξ), for any ξ ∈g=so(3), u(·)= 3 ui(·)e , with the elements of the basis
i=1 i
of g (which is semisimple) given by P
0 0 0 0 0 1 0 −1 0
e1 =0 0 −1, e2 = 0 0 0, e3 =1 0 0,
0 1 0 −1 0 0 0 0 0
     
which satisfy
[e ,e ]=e , [e ,e ]=e , [e ,e ]=e ,
1 2 3 2 3 1 3 1 2
I:g→g∗ =so(3)∗istheinertiatensorofthetop2,misthemassofthebody,gistheacceleration
due to gravity, e is the verticalunit vector, χ∈R3 is the unit vector from the point of support
3
to the direction of the body’s center of mass (constant) in body coordinates and l is the length
of the line segment between these two points.
Under the dual pairing, where hα,ξi :=tr(αξ), for any ξ ∈g and α∈g∗, the elements of the
basis of g∗ are given by
0 0 0 0 0 −1 0 1 0
 1  2  2 
e1 =00 −021 02, e2 = 201 00 00, e3 =−012 00 00.
It is easy to verify that the potential function is SO(2)-invariantbut not G-invariantand so,the
potential function breaks the symmetry partially. We will now show how the theory developed
in the paper can be applied to the OCP under consideration.
Let X = g and ρ be the adjoint representation of G on X, i.e., ρ = Ad , for any g ∈ G.
g g
So, ρ∗ is the coadjoint representation of G on X∗ = g∗, i.e., ρ∗ = Ad∗, for any g ∈ G. Let the
g g
extended potential function V :G×X∗ →R be given as follows
ext
V (g,α˘)=−mglhg−1α,χi,
ext
where α ∈ R3 is identified with α˘ ∈ g∗ (see, e.g., [17, Section 5.3]). If we set α˘ = −2e3 in the
0
extended potential function, we recover our original potential function. It is easy to verify that
theextendedpotentialfunctionisG-invariantunder(3.1),i.e., V ◦Φ =V ,foranyg ∈G. It
ext g ext
is also easy to verify that the potential function is invariant under the left action of the isotropy
group
G :={g ∈G|Ad∗α =α }∼=SO(2),
α0 g 0 0
i.e., rotations about the vertical axis e . We can now see that the assumptions (i)–(vi) are
3
satisfied. By Corollary 1, the Euler–Poincar´e equations (3.7)–(3.8) (under the identifications
g∼=R3 and g∗ ∼=R3) give the following equations
Iu˙ =ad∗Iu−ad∗ α, (4.1)
u ∂Vext
∂α
α˙ =ad∗α, α(0)=Ad∗ α , (4.2)
u g0 0
where
ad∗Iu=Iu×u, ad∗ α=−mglα×χ, ad∗α=α×u, Ad∗ α =g−1α
u ∂Vext u g0 0 0 0
∂α
2Itiscalculatedwithrespecttothepivot,whichisnotingeneralthecenter ofmass.