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Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems...
Transcript of Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems...
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Invariant Control Systems on Lie Groups
Rory Biggs∗ and Claudiu C. Remsing
Geometry and Geometric Control (GGC) Research GroupDepartment of Mathematics (Pure & Applied)Rhodes University, Grahamstown, South Africa
2nd International Conference“Lie Groups, Differential Equations and Geometry”
Universita di Palermo, Palermo, ItalyJune 30 – July 4, 2014
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 1 / 83
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Outline
1 Introduction
2 Equivalence of control systems
3 Invariant optimal control
4 Quadratic Hamilton-Poisson systems
5 Conclusion
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
![Page 3: Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems 3 Invariant optimal control 4 Quadratic Hamilton-Poisson systems 5 Conclusion R.](https://reader035.fdocuments.us/reader035/viewer/2022081611/5f0bd32f7e708231d43265a5/html5/thumbnails/3.jpg)
Outline
1 Introduction
2 Equivalence of control systems
3 Invariant optimal control
4 Quadratic Hamilton-Poisson systems
5 Conclusion
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
![Page 4: Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems 3 Invariant optimal control 4 Quadratic Hamilton-Poisson systems 5 Conclusion R.](https://reader035.fdocuments.us/reader035/viewer/2022081611/5f0bd32f7e708231d43265a5/html5/thumbnails/4.jpg)
Geometric control theory
Overview
began in the early 1970s
study control systems using methods from differential geometry
blend of differential equations, differential geometry, and analysis
R.W. Brockett, C. Lobry, A.J. Krener, H.J. Sussmann, V. Jurdjevic,B. Bonnnard, J.P. Gauthier, A.A. Agrachev, Y.L. Sachkov, U. Boscain
Smooth control systems
family of vector fields, parametrized by controls
state space, input space, control (or input), trajectories
characterize set of reachable points: controllability problem
reach in the best possible way: optimal control problem
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Invariant control systems on Lie groups
Overview
rich in symmetry
first considered in 1972 by Brockett and by Jurdjevic and Sussmann
natural geometric framework for various (variational) problems inmathematical physics, mechanics, elasticity, and dynamical systems
Last few decades: invariant control affine systems evolving on matrixLie groups of low dimension have received much attention.
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 3 / 83
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Outline
1 IntroductionInvariant control affine systemsExamples of invariant optimal control problems
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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Invariant control affine systems
Left-invariant control affine system
(Σ) g = Ξ(g , u) = g (A + u1B1 + · · ·+ u`B`), g ∈ G, u ∈ R`
state space: G is a connected (matrix) Lie group with Lie algebra g
input set: R`
dynamics: family of left-invariant vector fields Ξu = Ξ(·, u)
parametrization map: Ξ (1, ·) : R` → g, u 7→ A + u1B1 + · · ·+ u`B`is an injective (affine) map
trace: Γ = A + Γ0 = A + 〈B1, . . . ,B`〉 is an affine subspace of g
When the state space is fixed, we simply write
Σ : A + u1B1 + · · ·+ u`B`.
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 4 / 83
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Trajectories, controllability, and full rank
admissible controls: piecewise continuous curves u(·) : [0,T ]→ R`
trajectory: absolutely continuous curve s.t. g(t) = Ξ(g(t), u(t))
controlled trajectory: pair (g(·), u(·))
controllable: exists trajectory from any point to any other
full rank: Lie(Γ) = g; necessary condition for controllability
Σ : A + u1B1 + · · ·+ u`B`
trace: Γ = A + Γ0 = A + 〈B1, . . . ,B`〉 is an affine subspace of g
homogeneous: A ∈ Γ0
inhomogeneous: A /∈ Γ0
drift-free: A = 0
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 5 / 83
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Simple example (simplified model of a car)
Euclidean group SE (2)1 0 0x cos θ − sin θy sin θ cos θ
: x , y , θ ∈ R
System
Σ : u1E2 + u2E3
In coordinates
x = −u1 sin θ y = u1 cos θ θ = u2
se (2) : [E2,E3] = E1 [E3,E1] = E2 [E1,E2] = 0
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 6 / 83
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Outline
1 IntroductionInvariant control affine systemsExamples of invariant optimal control problems
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
![Page 11: Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems 3 Invariant optimal control 4 Quadratic Hamilton-Poisson systems 5 Conclusion R.](https://reader035.fdocuments.us/reader035/viewer/2022081611/5f0bd32f7e708231d43265a5/html5/thumbnails/11.jpg)
The plate-ball problem
Kinematic situation
ball rolls without slippingbetween two horizontal plates
through the horizontalmovement of the upper plate
Problem
transfer ball from initial positionand orientation to final positionand orientation
along a path which minimizes∫ T0 ‖v(t)‖dt
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 7 / 83
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The plate-ball problem
As invariant optimal control problem
Can be regarded as invariant optimal control problem on 5D group
R2 × SO (3) =
x1 0 0
0 x2 00 0 R
: x1, x2 ∈ R, R ∈ SO (3)
specified by
g = g
u1
1 0 0 0 00 0 0 0 00 0 0 0 −10 0 0 0 00 0 1 0 0
+ u2
0 0 0 0 00 1 0 0 00 0 0 0 00 0 0 0 −10 0 0 1 0
g(0) = g0, g(T ) = g1,
∫ T
0
(u2
1 + u22
)dt −→ min.
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Control of Serret-Frenet systems
Consider curve γ(t) in E2 with moving frame (v1(t), v2(t))
γ(t) = v1(t), v1(t) = κ(t)v2(t), v2(t) = −κ(t)v1(t).
Here κ(t) is the signed curvature of γ(t).
Lift to group of motions of E2
SE (2) =
1 0 0γ1
γ2R
: γ1, γ2 ∈ R,R ∈ SO (2)
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 9 / 83
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Control of Serret-Frenet systems
Interpreting the curvature κ(t) as a control function, we have:inhomogeneous invariant control affine system
g = g
0 0 01 0 00 0 0
+ κ(t)
0 0 00 0 −10 1 0
, g ∈ SE (2).
Many classic variational problems in geometry become problems inoptimal control.
Euler’s elastica: find curve γ(t) minimizing∫ T
0 κ2(t) dt such that
γ(0) = a, γ(0) = a, γ(T ) = b, γ(T ) = b.
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 10 / 83
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Outline
1 Introduction
2 Equivalence of control systems
3 Invariant optimal control
4 Quadratic Hamilton-Poisson systems
5 Conclusion
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
![Page 16: Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems 3 Invariant optimal control 4 Quadratic Hamilton-Poisson systems 5 Conclusion R.](https://reader035.fdocuments.us/reader035/viewer/2022081611/5f0bd32f7e708231d43265a5/html5/thumbnails/16.jpg)
Equivalence of control systems
Overview
state space equivalence: equivalence up to coordinate changes in thestate space; well understood
establishes a one-to-one correspondence between the trajectories ofthe equivalent systems
feedback equivalence: (feedback) transformations of controls alsopermitted
extensively studied; much weaker than state space equivalence
Note
we specialize to left-invariant systems on Lie groups
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 11 / 83
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Outline
2 Equivalence of control systemsState space equivalenceDetached feedback equivalenceClassification in three dimensions
Solvable caseSemisimple case
Controllability characterizationsClassification beyond three dimensions
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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State space equivalence
Definition
Σ and Σ′ are state space equivalent if
there exists a diffeomorphism φ : G→ G′ such that φ∗Ξu = Ξ′u.
Theorem
Full-rank systems Σ and Σ′ are state space equivalent if and only if thereexists a Lie group isomorphism φ : G→ G′ such that
T1φ · Ξ(1, ·) = Ξ′(1, ·).
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 12 / 83
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Example: classification on the Euclidean group
Result
On the Euclidean group SE (2), any inhomogeneous full-rank system
Σ : A + u1B1 + u2B2
is state space equivalent to exactly one of the following systems
Σ1,αβγ : αE3 + u1(E1 + γ1E2) + u2(βE2), α>0, β≥0, γi∈R
Σ2,αβγ : βE1 + γ1E2 + γ2E3 + u1(αE3) + u2E2, α>0, β≥0, γi∈R
Σ3,αβγ : βE1 + γ1E2 + γ2E3 + u1(E2 + γ3E3) + u2(αE3), α>0, β≥0, γi∈R.
d Aut (SE (2)) :
x y v−σy σx w
0 0 1
, σ = ±1, x2 + y2 6= 0
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 13 / 83
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Concrete cases covered (state space equivalence)
Classifications on
Euclidean group SE (2)
semi-Euclidean group SE (1, 1)
pseudo-orthogonal group SO (2, 1)0 (resp. SL (2,R))
Remarks
many equivalence classes
limited use
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 14 / 83
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Outline
2 Equivalence of control systemsState space equivalenceDetached feedback equivalenceClassification in three dimensions
Solvable caseSemisimple case
Controllability characterizationsClassification beyond three dimensions
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
![Page 22: Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems 3 Invariant optimal control 4 Quadratic Hamilton-Poisson systems 5 Conclusion R.](https://reader035.fdocuments.us/reader035/viewer/2022081611/5f0bd32f7e708231d43265a5/html5/thumbnails/22.jpg)
Detached feedback equivalence
Definition
Σ and Σ′ are detached feedback equivalent if there
exist diffeomorphisms φ : G→ G′, ϕ : R` → R` such that φ∗Ξu = Ξ′ϕ(u).
one-to-one correspondence between trajectories
specialized feedback transformations
φ preserves left-invariant vector fields
Theorem
Full-rank systems Σ and Σ′ are detached feedback equivalent if and onlyif there exists a Lie group isomorphism φ : G→ G′ such that
T1φ · Γ = Γ′
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 15 / 83
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Detached feedback equivalence
Proof sketch (equivalent ⇐⇒ T1φ · Γ = Γ′)
Suppose Σ and Σ′ equivalent.
We may assume φ(1) = 1′; hence T1φ · Ξ(1, u) = Ξ′(1′, ϕ(u)) andso T1φ · Γ = Γ′.
Full-rank implies elements Ξ(1, u) ∈ g, u ∈ R` generate g.
Also push-forward of left-invariant vector fields Ξu = Ξ(·, u) areleft-invariant.
It follows that φ is a group homomorphism.
Conversely, suppose φ · Γ = Γ′.
There exists a unique affine isomorphism ϕ : R` → R` such thatT1φ · Ξ(1, u) = Ξ′(1′, ϕ(u)).
By left-invariance (and the fact that φ is a homomorphism) it thenfollows that Tgφ · Ξ(g , u) = Ξ′(φ(g), ϕ(u)).
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 16 / 83
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Example: classification on the Euclidean group
Result
On the Euclidean group SE (2), any inhomogeneous full-rank system
Σ : A + u1B1 + u2B2
is detached feedback equivalent to exactly one of the following systems
Σ1 : E1 + u1E2 + u2E3
Σ2,α : αE3 + u1E1 + u2E2, α > 0.
d Aut (SE (2)) :
x y v−σy σx w
0 0 1
, σ = ±1, x2 + y2 6= 0
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 17 / 83
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Outline
2 Equivalence of control systemsState space equivalenceDetached feedback equivalenceClassification in three dimensions
Solvable caseSemisimple case
Controllability characterizationsClassification beyond three dimensions
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
![Page 26: Invariant Control Systems on Lie Groups · Outline 1 Introduction 2 Equivalence of control systems 3 Invariant optimal control 4 Quadratic Hamilton-Poisson systems 5 Conclusion R.](https://reader035.fdocuments.us/reader035/viewer/2022081611/5f0bd32f7e708231d43265a5/html5/thumbnails/26.jpg)
Classification of 3D Lie algebras
Eleven types of real 3D Lie algebras
3g — R3 Abelian
g2.1 ⊕ g1 — aff (R)⊕ R cmpl. solvable
g3.1 — Heisenberg h3 nilpotent
g3.2 cmpl. solvable
g3.3 — book Lie algebra cmpl. solvable
g03.4 — semi-Euclidean se (1, 1) cmpl. solvable
ga3.4, a > 0, a 6= 1 cmpl. solvable
g03.5 — Euclidean se (2) solvable
ga3.5, a > 0 exponential
g03.6 — pseudo-orthogonal so (2, 1), sl (2,R) simple
g03.7 — orthogonal so (3), su (2) simple
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 18 / 83
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Classification of 3D Lie groups
3D Lie groups
3g1 — R3, R2 × T, R× T, T3 Abelian
g2.1 ⊕ g1 — Aff (R)0 × R, Aff (R)0 × T cmpl. solvable
g3.1 — H3, H∗3 = H3/Z(H3(Z)) nilpotent
g3.2 — G3.2 cmpl. solvable
g3.3 — G3.3 cmpl. solvable
g03.4 — SE (1, 1) cmpl. solvable
ga3.4 — Ga3.4 cmpl. solvable
g03.5 — SE (2), n-fold cov. SEn(2), univ. cov. SE (2) solvable
ga3.5 — Ga3.5 exponential
g3.6 — SO (2, 1)0, n-fold cov. A(n), univ. cov. A simple
g3.7 — SO (3), SU (2) simple
Only H∗3, An, n ≥ 3, and A are not matrix Lie groups.R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 19 / 83
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Solvable case study: Heisenberg group H3
H3 :
1 y x0 1 z0 0 1
h3 :
0 y x0 0 z0 0 0
= xE1 + yE2 + zE3
Theorem
On the Heisenberg group H3, any full-rank system is detached feedbackequivalent to exactly one of the following systems
Σ(1,1) : E2 + uE3
Σ(2,0) : u1E2 + u2E3
Σ(2,1)1 : E1 + u1E2 + u2E3
Σ(2,1)2 : E3 + u1E1 + u2E2
Σ(3,0) : u1E1 + u2E2 + u3E3.
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 20 / 83
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Solvable case study: Heisenberg group H3
Proof sketch (1/2)
d Aut(H3) = Aut(h3) =
yw − vz x u
0 y v0 z w
:x , y , z , u, v ,w ∈ R
yw − vz 6= 0
single-input system Σ with trace Γ =
∑3i=1 aiEi +
⟨∑3i=1 biEi
⟩;
ψ =
a2b3 − a3b2 a1 b1
0 a2 b2
0 a3 b3
∈ Aut(h3), ψ · (E2 + 〈E3〉) = Γ;
so Σ is equivalent to Σ(1,1)
two-input homogeneous system with trace Γ = 〈A,B〉; similarargument holds
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 21 / 83
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Solvable case study: Heisenberg group H3
Proof sketch (2/2)
two-input inhomogeneous system Σ with trace Γ = A + 〈B1,B2〉
if E1 ∈ 〈B1,B2〉, then Γ = A + 〈E1,B′2〉; like single-input case there
exists automorphism ψ such that ψ · Γ = E3 + 〈E1,E2〉
if E1 /∈ 〈B1,B2〉, construct automorphism ψ such thatψ · Γ = E1 + 〈E2,E3〉
Σ(2,1)1 and Σ
(2,1)2 are distinct as E1 is eigenvector of every
automorphism
three-input system: trivial
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 22 / 83
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Semisimple case study: orthogonal group SO (3)
SO (3) = g ∈ R3×3 : gg> = 1, det g = 1
so (3) :
0 −z yz 0 −x−y x 0
= xE1 + yE2 + zE3
Theorem
On the orthogonal group SO (3), any full-rank system is detachedfeedback equivalent to exactly one of the following systems
Σ(1,1)α : αE1 + uE2, α > 0
Σ(2,0) : u1E1 + u2E2
Σ(2,1)α : αE1 + u1E2 + u2E3, α > 0
Σ(3,0) : u1E1 + u2E2 + u3E3.
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 23 / 83
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Semisimple case study: orthogonal group SO (3)
Proof sketch
d Aut(SO (3)) = Aut(so (3)) = SO (3)
product A • B = a1b1 + a2b2 + a3b3 is preserved by automorphisms
critical point C•(Γ) at which an inhomogeneous affine subspace istangent to a sphere Sα = A ∈ so (3) : A • A = α is given by
C•(Γ) = A− A • BB • B
B
C•(Γ) = A−[B1 B2
] [B1 • B1 B1 • B2
B1 • B2 B2 • B2
]−1 [A • B1
A • B2
]ψ · C•(Γ) = C•(ψ · Γ) for any automorphism ψ ∈ SO (3)
scalar α2 = C•(Γ) • C•(Γ) invariant under automorphisms
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Outline
2 Equivalence of control systemsState space equivalenceDetached feedback equivalenceClassification in three dimensions
Solvable caseSemisimple case
Controllability characterizationsClassification beyond three dimensions
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Some controllability criteria for invariant systems
Sufficient conditions for full-rank system to be controllable
system is homogeneous
state space is compact
the direction space Γ0 generates g, i.e., Lie (Γ0) = g
there exists C ∈ Γ such that t 7→ exp(tC ) is periodic
the identity element 1 is in the interior of the attainable setA = g(t1) : g(·) is a trajectory such that g(0) = 1, t1 ≥ 0
[Jurdjevic and Sussmann 1972]
Systems on simply connected completely solvable groups
condition Lie (Γ0) = g is necessary and sufficient [Sachkov 2004]
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Characterization for 3D groups
Theorem
1 On Aff (R)0 × R, H3, G3.2, G3.3, SE (1, 1), and Ga3.4,
a full-rank system is controllable if and only if Lie (Γ0) = g.
2 On SEn(2), SO (3), and SU (2),all full-rank systems are controllable.
3 On Aff (R)× T, SL (2,R), and SO (2, 1)0,a full-rank system is controllable if and only if it is homogeneousor there exists A ∈ Γ such that t 7→ exp(tA) is periodic.
4 On SE (2) and Ga3.5,
a full-rank system is controllable if and only if E ∗3 (Γ0) 6= 0.
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Characterization for 3D groups
Proof sketch (1/2)
1 completely solvable simply connected groups; characterization known
2 the groups SO (3) and SU (2) are compact, hence all full-ranksystems are controllable
SEn(2) decomposes as semidirect product of vector space andcompact subgroup; hence result follows from [Bonnard et al. 1982]
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Characterization for 3D groups
Proof sketch (2/2)
3 study normal forms of these systems obtained in classification
full-rank homogeneous systems are controllable
for each full-rank inhomogeneous system we either explicitly findA ∈ Γ such that t 7→ exp(tA) is periodic
or prove that some states are not attainable by inspection ofcoordinates of g = Ξ(g , u)
as properties are invariant under equivalence, characterization holds
4 study normal forms of these systems obtained in classification
condition invariant under equivalence
similar techniques with extensions; however for one system on Ga3.5 we
could only prove controllability by showing 1 ∈ intA
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Outline
2 Equivalence of control systemsState space equivalenceDetached feedback equivalenceClassification in three dimensions
Solvable caseSemisimple case
Controllability characterizationsClassification beyond three dimensions
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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Classification beyond three dimensions
Homogeneous systems
four-dimensional central extension of SE (2) — oscillator group
four-dimensional central extension of SE (1, 1)
all simply connected four-dimensional groups
All systems
six-dimensional orthogonal group SO (4)
Controllable systems
(2n + 1)-dimensional Heisenberg groups
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Example: controllable systems on H2n+1
hn :
0 x1 x2 · · · xn z0 0 0 0 y1
0 0 0 0 y2...
. . ....
0 · · · 0 yn0 · · · 0 0
= zZ +
n∑i=1
(xiXi + yiYi ) , z , xi , yi ∈ R
Theorem
Every controllable system on the Heisenberg group H2n+1 is detachedfeedback equivalent to exactly one of the following three systems
Σ(2n,0) : u1X1 + · · ·+ unXn + un+1Y1 + · · ·+ u2nYn
Σ(2n,1) : Z + u1X1 + · · ·+ unXn + un+1Y1 + · · ·+ u2nYn
Σ(2n+1,0) : u1X1 + · · ·+ unXn + un+1Y1 + · · ·+ u2nYn + u2n+1Z .
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Outline
1 Introduction
2 Equivalence of control systems
3 Invariant optimal control
4 Quadratic Hamilton-Poisson systems
5 Conclusion
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Invariant optimal control problems
Problem
Minimize cost functional J =∫ T
0 χ(u(t)) dtover controlled trajectories of a system Σ
subject to boundary data.
Formal statement LiCP
g = g (A + u1B1 + · · ·+ u`B`) , g ∈ G, u ∈ R`
g(0) = g0, g(T ) = g1
J =
∫ T
0(u(t)− µ)>Q (u(t)− µ) dt −→ min.
µ ∈ R`, Q ∈ R`×` is positive definite.
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Invariant optimal control problems
Examples
optimal path planning for airplanes
motion planning for wheeled mobile robots
spacecraft attitude control
control of underactuated underwater vehicles
control of quantum systems
dynamic formation of DNA
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Outline
3 Invariant optimal controlPontryagin Maximum PrincipleEquivalence of cost-extended systems
Classification
Pontryagin liftSub-Riemannian structures
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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Pontryagin Maximum Principle
Associate Hamiltonian function on T ∗G = G× g∗:
Hλu (ξ) = λχ(u) + ξ(Ξ(g , u))
= λχ(u) + p (Ξ(1, u)) , ξ = (g , p) ∈ G× g∗.
Maximum Principle Pontryagin et al. 1964
If (g(·), u(·)) is a solution, then there exists a curve
ξ(·) : [0,T ]→ T ∗G, ξ(t) ∈ T ∗g(t)G, t ∈ [0,T ]
and λ ≤ 0, such that (for almost every t ∈ [0,T ]):
(λ, ξ(t)) 6≡ (0, 0)
ξ(t) = ~Hλu(t)(ξ(t))
Hλu(t) (ξ(t)) = max
uHλu (ξ(t)) = constant.
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Pontryagin Maximum Principle
Definition
A pair (ξ(·), u(·)) is said to be an extremal pair if, for some λ ≤ 0,
(λ, ξ(t)) 6≡ (0, 0)
ξ(t) = ~Hλu(t)(ξ(t))
Hλu(t) (ξ(t)) = max
uHλu (ξ(t)) = constant
extremal trajectory: projection to G of curve ξ(·) on T ∗G
extremal control: component u(·) of extremal pair (ξ(·), u(·))
An extremal is said to be
normal if λ < 0
abnormal if λ = 0
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Outline
3 Invariant optimal controlPontryagin Maximum PrincipleEquivalence of cost-extended systems
Classification
Pontryagin liftSub-Riemannian structures
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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Cost-extended system
Definition
Cost-extended system is a pair (Σ, χ) where
Σ : A + u1B1 + · · ·+ u`B`
χ(u) = (u(t)− µ)>Q (u(t)− µ).
(Σ, χ) + boundary data = optimal control problem
VOCT — virtually optimal controlled-trajectory (g(·), u(·)):solution of some associated optimal control problem
ECT — extremal controlled-trajectory (g(·), u(·)):extremal CT for some associated optimal control problem
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Cost equivalence
Definition
(Σ, χ) and (Σ′, χ′) are cost equivalent if there exist
a Lie group isomorphism φ : G→ G′
an affine isomorphism ϕ : R` → R`
such thatφ∗Ξu = Ξ′ϕ(u) and ∃r>0 χ′ ϕ = rχ.
G× R`
Ξ
φ×ϕ// G′ × R`
Ξ′
TGTφ
// TG′
R`
χ
ϕ// R`
χ′
Rδr
// R
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Relation of equivalences
Proposition
(Σ, χ) and (Σ′, χ′)cost equivalent
=⇒ Σ and Σ′
detached feedback equivalent
Proposition
Σ and Σ′
state space equivalent=⇒ (Σ, χ) and (Σ′, χ)
cost equivalent for any χ
Σ and Σ′
detached feedback equivalentw.r.t. ϕ ∈ Aff (R`)
=⇒ (Σ, χ ϕ) and (Σ′, χ)cost equivalent for any χ
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Preservation of VOCTs and ECTs
Theorem
If (Σ, χ) and (Σ′, χ′) are cost equivalent w.r.t. φ× ϕ, then
(g(·), u(·)) is a VOCT if and only if (φ g(·), ϕ u(·)) is a VOCT;
(g(·), u(·)) is an ECT if and only if (φ g(·), ϕ u(·)) is an ECT.
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Classification under cost equivalence
Algorithm
1 classify underlying systems under detached feedback equivalence2 for each normal form Σi ,
determine transformations TΣi preserving system Σi
normalize (admissible) associated cost χ by dilating by r > 0 andcomposing with ϕ ∈ TΣi
TΣ =
ϕ ∈ Aff (R`) :
∃ψ ∈ d Aut (G), ψ · Γ = Γψ · Ξ(1, u) = Ξ(1, ϕ(u))
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Example: structures on SE (2)
SE (2) :
1 0 0x cos θ − sin θy sin θ cos θ
se (2) :
0 0 0x 0 −θy θ 0
= xE1+yE2+θE3
Result
On the Euclidean group SE (2) , any full-rank cost-extended system
Σ : u1B1 + u2B2 χ(u) = u>Qu
is cost equivalent to
(Σ(2,0), χ(2,0)) :
Σ : u1E2 + u2E3
χ(u) = u21 + u2
2
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Example: structures on SE (2)
Proof sketch
Σ is detached feedback equivalent to Σ(2,0) : u1E2 + u2E3
thus (Σ, χ) is cost equivalent to (Σ(2,0), χ′) for someχ′ : u 7→ u>Q ′ u (feedback transformation linear)
TΣ(2,0) =
u 7→
[ςx w0 ς
]u : x 6= 0, w ∈ R, ς = ±1
let Q ′ =
[a1 bb a2
]ϕ1 =
[1 − b
a1
0 1
]∈ TΣ1 and (χ′ ϕ1)(u) = u> diag(a1, a
′2) u
ϕ2 = diag(√
a′2a1, 1) ∈ TΣ1 and
(χ′ (ϕ1 ϕ2))(u) = a′2 u> u = a′2 χ
(2,0)(u)
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Example: structures on Heisenberg group H2n+1
Result
On the Heisenberg group H2n+1, any controllable cost-extended system
Σ : u1B1 + · · ·+ u`B` χ(u) = u>Qu
is cost-equivalent to exactly one of the following systems:
(Σ(2n,0), χ(2n,0)λ ) :
Σ(2n,0) :
∑ni=1 (uiXi + un+iYi )
χ(2n,0)λ (u) =
∑ni=1λi
(u2i + u2
n+i
)(Σ(2n+1,0), χ
(2n+1,0)λ ) :
Σ(2n+1,0) :
∑ni=1 (uiXi + un+iYi ) + u2n+1Z
χ(2n+1,0)λ (u) =
∑ni=1λi
(u2i + u2
n+i
)+ u2
2n+1.
1 = λ1 ≥ λ2 ≥ · · · ≥ λn > 0
Commutators: [Xi ,Yj ] = δijZ
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Outline
3 Invariant optimal controlPontryagin Maximum PrincipleEquivalence of cost-extended systems
Classification
Pontryagin liftSub-Riemannian structures
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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Pontryagin lift
Reduction of LiCP (normal case, i.e., λ < 0)
maximal condition
Hλu(t) (ξ(t)) = max
uHλu (ξ(t)) = constant
eliminates the parameter u
obtain a smooth G-invariant function H on T ∗G = G× g∗
reduced to Hamilton-Poisson system on Lie-Poisson space g∗−:
F ,G = −p([dF (p), dG (p)])
(here F ,G ∈ C∞(g∗) and dF (p), dG (p) ∈ g∗∗ ∼= g)
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Hamiltonian system and normal extremals
Coordinate representation
a1E1 + · · ·+ anEn ∈ g ←→[a1 · · · an
]>p1E
∗1 + · · ·+ pnE
∗n ∈ g∗ ←→
[p1 · · · pn
]Let (Σ, χ) be a cost-extended system with
Ξu(1) = A + B u, B =[B1 · · · B`
], χ(u) = (u − µ)>Q(u − µ).
Theorem
Any normal ECT (g(·), u(·)) of (Σ, χ) is given by
g(t) = Ξ(g(t), u(t)), u(t) = Q−1 B> p(t)> + µ
where p(·) : [0,T ]→ g∗ is an integral curve for the Hamilton-Poissonsystem on g∗− specified by
H(p) = p (A + Bµ) + 12 p B Q−1 B> p>.
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Cost equivalence and linear equivalence
Definition
Hamilton-Poisson systems (g∗−,G ) and (h∗−,H)are linearly equivalent if there
exists linear isomorphism ψ : g∗ → h∗ such that ψ∗ ~G = ~H.
Theorem
If two cost-extended systems are cost equivalent, then their associatedHamilton-Poisson systems are linearly equivalent.
one shows that r(T1φ)∗ is the required linear isomorphism
converse of theorem does not hold
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Outline
3 Invariant optimal controlPontryagin Maximum PrincipleEquivalence of cost-extended systems
Classification
Pontryagin liftSub-Riemannian structures
R. Biggs, C.C. Remsing (Rhodes) Invariant Control Systems on Lie Groups Palermo 2014 * / 83
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Invariant sub-Riemannian manifolds
Left-invariant sub-Riemannian manifold (G,D, g)
Lie group G with Lie algebra g
left-invariant bracket-generating distribution DD(g) = T1Lg · D(1)Lie(D(1)) = g
left-invariant Riemannian metric g on Dgg is a symmetric positive definite bilinear form on D(g)gg (T1Lg · A,T1Lg · B) = g1(A,B) for A,B ∈ g
horizontal curve: a.c. curve g(·) s.t. g(t) ∈ D(g(t))
Remark
Structure (D, g) on G is fully specified by
subspace D(1) of Lie algebra g
inner product g1 on D(1).
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Geodesics
Relation to cost-extended systems
Length minimization problem
g(t) ∈ D(g(t)), g(0) = g0, g(T ) = g1,
∫ T
0
√g(g(t), g(t)) −→ min
is equivalent to the energy minimization problem
g = Ξu(g), g(0) = g0, g(T ) = g1,
∫ T
0χ(u(t)) dt −→ min
with
Ξu(1) = u1B1 + · · ·+ u`B` such that 〈B1, . . . ,B`〉 = D(1);
χ(u(t)) = u(t)>Q u(t) = g1(Ξu(t)(1),Ξu(t)(1))
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SR structures and cost-extended systems
VOCTs ←→ minimizing geodesics
ECTs ←→ geodesics
SR structure + parametrizarion map = cost-extended syst.
To a cost-extended system (Σ, χ) on G
Σ : u1B1 + · · ·+ u`B`, χ(u) = u>Qu
we associate the SR structure (G,D, g)
D(1) = 〈B1, . . . ,B`〉g1(u1B1 + · · ·+ u`B`, u1B1 + · · ·+ u`B`) = χ(u).
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Cost equivalence reinterpreted
Let (G,D, g), (G′,D′, g′) be SR structures associated to (Σ, χ), (Σ′, χ′).
Theorem
(Σ, χ) and (Σ′, χ′) are cost equivalent if and only if there exists a Liegroup isomorphism φ : G→ G′ and r > 0 such that
φ∗D = D′ and g = r φ∗g′.
cost equivalence =⇒ isometric up to rescaling
Remark
At least for
invariant Riemannian structures on nilpotent groups [Wilson 1982]
sub-Riemannian Carnot groups [Capogna et al. 2014]
isometric =⇒ cost equivalence.
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Example: sub-Riemannian structures on SE (2)
Normal form for drift-free systems on SE (2) (recalled)
(Σ(2,0), χ(2,0)) :
Σ : u1E2 + u2E3
χ(u) = u21 + u2
2
Result
On the Euclidean group SE (2), any left-invariant sub-Riemannianstructure (D, g) isometric (up to rescaling) to the structure (D, g)specified by
D(1) = 〈E2,E3〉
g1 =
[1 00 1
]i.e., with orthonormal basis (E2,E3).
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Example: sub-Riemannian structures on H2n+1
Result
On the Heisenberg group H2n+1, any left-invariant sub-Riemannianstructure (D, g) is isometric to exactly one of the structures (D, gλ)specified by
D(1) = 〈X1,Y1, . . . ,Xn,Yn〉gλ1 = diag(λ1, λ1, λ2, λ2, . . . , λn, λn)
i.e., with orthonormal basis
( 1√λ1X1,
1√λ1Y1,
1√λ2X2,
1√λ2Y2, . . . ,
1√λnXn,
1√λnYn).
Here 1 = λ1 ≥ λ2 ≥ · · · ≥ λn > 0 parametrize a family of classrepresentatives.
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Example: Riemannian structures on H2n+1
Result
On the Heisenberg group H2n+1, any left-invariant Riemannian structureg is isometric to exactly one of the structures
gλ1 =
[1 00 Λ
], Λ = diag(λ1, λ1, λ2, λ2, . . . , λn, λn)
i.e., with orthonormal basis
(Z , 1√λ1X1,
1√λ1Y1,
1√λ2X2,
1√λ2Y2, . . . ,
1√λnXn,
1√λnYn).
Here λ1 ≥ λ2 ≥ · · · ≥ λn > 0 parametrize a family of classrepresentatives.
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Outline
1 Introduction
2 Equivalence of control systems
3 Invariant optimal control
4 Quadratic Hamilton-Poisson systems
5 Conclusion
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Lie-Poisson spaces
Overview
dual space of a Lie algebra admits a natural Poisson structure
one-to-one correspondence with linear Poisson structures
many dynamical systems are naturally expressed as quadraticHamilton-Poisson systems on Lie-Poisson spaces
prevalent examples are Euler’s classic equations for the rigid body, itsextensions and its generalizations
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Lie-Poisson structure
(Minus) Lie-Poisson structure
F ,G (p) = −p ([dF (p), dG (p)]) , p ∈ g∗, F ,G ∈ C∞(g∗)
Hamiltonian vector field: ~H[F ] = F ,H
Casimir function: C ,F = 0
quadratic system: HA,Q(p) = p A + p Q p>
Equivalence
Systems (g∗−,G ) and (h∗−,H) are linearly equivalent if
there exists a linear isomorphism ψ : g∗ → h∗ such that ψ∗ ~G = ~H.
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Outline
4 Quadratic Hamilton-Poisson systemsClassification in three dimensions
Homogeneous systemsInhomogeneous systems
On integration and stability
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Classification algorithm
Proposition
The following systems are linearly equivalent to HA,Q :
1 HA,Q ψ, where ψ : g∗ → g∗ is a linear Poisson automorphism;
2 HA,Q + C , where C is a Casimir function;
3 HA,rQ , where r 6= 0.
Algorithm
1 Normalize as much as possible at level of Hamiltonians (as above).
2 Normalize at level of vector fields, i.e., solve ψ∗ ~Hi = ~Hj .
In some cases, only step 1 is required to obtain normal forms.
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Classification of Lie-Poisson spaces
Lie-Poisson spaces admitting global Casimirs [Patera et al. 1976]
R3 C∞(R3)
(h3)∗− C (p) = p1
(aff (R)⊕ R)∗− C (p) = p3
se (1, 1)∗− C (p) = p21 − p2
2
se (2)∗− C (p) = p21 + p2
2
so (2, 1)∗− C (p) = p21 + p2
2 − p23
so (3)∗− C (p) = p21 + p2
2 + p23
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Coadjoint orbits (spaces admitting global Casimirs)
aff(R) ⊕ R h3 se (1, 1)
se (2) so (2, 1) so (3)
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Classification by Lie-Poisson space
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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General classification
Consider equivalence of systems on different spaces— direct computation with Mathematica
Types of systems
linear: integral curves contained in lines(sufficient: has two linear constants of motion)
planar: integral curves contained in planes, not linear(sufficient: has one linear constant of motion)
otherwise: non-planar
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Classification by Lie-Poisson space
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Linear systems
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Linear systems (3 classes)
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Linear systems L (1), L (2), L(3)
(aff (R)⊕ R)∗−
p21
1 : p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
2 : (p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
3 : p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Planar systems
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Planar systems (5 classes)
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Planar systems P(1), . . . ,P(5)
(aff (R)⊕ R)∗−
p21
p22
1 : p21 + p2
2
(p1 + p3)2
2 : p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
3 : p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
4 : p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
5 : (p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Non-planar systems
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Non-planar systems (2 classes)
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
(p1 + p2)2 + p23
se (2)∗−
p22
p23
p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Non-planar systems Np(1), Np(2)
(aff (R)⊕ R)∗−
p21
p22
p21 + p2
2
(p1 + p3)2
p22 + (p1 + p3)2
(h3)∗−
p23
p22 + p2
3
se (1, 1)∗−
p21
p23
p21 + p2
3
(p1 + p2)2
1 : (p1 + p2)2 + p23
se (2)∗−
p22
p23
2 : p22 + p2
3
so (2, 1)∗−
p21
p23
p21 + p2
3
(p2 + p3)2
p22 + (p1 + p3)2
so (3)∗−
p21
p21 + 1
2p22
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Remarks
Interesting features
systems on (h3)∗− or so (3)∗−
— equivalent to ones on se (2)∗−
systems on (aff(R)⊕ R)∗− or (h3)∗−
— planar or linear
systems on (h3)∗−, se (1, 1)∗−, se (2)∗− and so (3)∗−
— may be realized on multiple spaces
(for so (2, 1)∗− exception is P (5))
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Inhomogeneous systems
Theorem
Any (strictly) inhomogeneous quadratic system (so (3)∗−,H) is affinelyequivalent to exactly one of the systems:
H01,α(p) = αp1, α > 0
H10 (p) = 1
2p21
H11 (p) = p2 + 1
2p21
H12,α(p) = p1 + αp2 + 1
2p21 , α > 0
H21,α(p) = αp1 + p2
1 + 12p
22 , α > 0
H22,α(p) = αp2 + p2
1 + 12p
22 , α > 0
H23,α(p) = α1p1 + α2p2 + p2
1 + 12p
22 , α1, α2 > 0
H24,α(p) = α1p1 + α3p3 + p2
1 + 12p
22 , α1 ≥ α3 > 0
H25,α(p) = α1p1 + α2p2 + α3p3 + p2
1 + 12p
22 , α2>0, α1>|α3|>0 or α2>0, α1=α3>0
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Outline
4 Quadratic Hamilton-Poisson systemsClassification in three dimensions
Homogeneous systemsInhomogeneous systems
On integration and stability
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Overview: 3D Lie-Poisson spaces
Integration
homogeneous systems admitting global Casimirs — integrable byelementary functions; exception Np(2) : (se(2)∗−, p
22 + p2
3) which isintegrable by Jacobi elliptic functions
inhomogeneous systems — integrable by Jacobi elliptic functions (atleast some)
Stability of equilibria
instability — usually follows from spectral instability
stability — usually follows from the energy Casimir method or one ofits extensions [Ortega et al. 2005]
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Example: inhomogeneous system on se (2)∗−
Hα(p) = p1 + 12
(αp22 + p2
3)
Equations of motion: p1 = p2p3
p2 = −p1p3
p3 = (αp1 − 1) p2.
Equilibria:
eµ1 = (µ, 0, 0), eν2 = ( 1α , ν, 0), eν3 = (0, 0, ν)
where µ, ν ∈ R, ν 6= 0.
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Stability
eµ1 = (µ, 0, 0)
eν2 = ( 1α , ν, 0)
eν3 = (0, 0, ν)
Theorem
1 The states eµ1 , 0 < µ < 1α are spectrally unstable.
2 The state eµ1 , µ = 1α is unstable.
3 The states eµ1 , µ ∈ (−∞, 0] ∪ ( 1α ,∞) are stable.
4 The states eν2 are spectrally unstable.
5 The states eν3 are stable.
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Stability
Proof (1/2)
States eµ1 , 0 < µ < 1α are spectrally unstable.
Linearization of ~Hα at eµ1 has eigenvalues λ1 = 0,λ2,3 = ±
√µ(1− αµ). Hence for 0 < µ < 1
α spectrally unstable.
States eν2 are spectrally unstable — follows similarly.
State eµ1 , µ = 1α is unstable. We have an integral curve
p(t) =(
t2−αα(t2+α)
, 2t√α(t2+α)
, 2√α
t2+α
)such that limt→−∞ p(t) = e
1/α1 and ‖e1/α
1 − p(0)‖ = 2√
1+αα2 > 0.
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Stability
Proof (2/2)
States eµ1 , µ ∈ (−∞, 0] ∪ ( 1α ,∞) are stable.
For energy function G = λ1H + λ2C , λ1 = −µ, λ2 = 2µ2
dG (eµ1 ) = 0 and d2G (eµ1 ) =
−2µ 0 00 2µ(−1 + αµ) 00 0 2µ2
.Restriction of d2G (eµ1 ) to
ker dH(eµ1 ) ∩ ker dC (eµ1 ) = (p1, 0, 0) : p1 ∈ R
is PD for µ < 0 and µ > 1α — stable.
Intersection C−1(0) ∩ H−1α (0) is a singleton e0
1; stable.
States eν3 are stable — follows similarly.
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Integration
Basic Jacobi elliptic functions
Given a modulus k ∈ [0, 1],
sn(x , k) = sin am(x , k)
cn(x , k) = cos am(x , k)
dn(x , k) =
√1− k2 sin2 am(x , k)
where am(·, k) = F (·, k)−1 and F (ϕ, k) =∫ ϕ
0dt
1−k2 sin2 t.
k = 0 / 1 ←→ circular / hyperbolic functions.
K = F (π2 , k);sn(·, k), cn(·, k) are 4K periodic; dn(·, k) is 2K periodic.
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Integration
First problem
Several cases (usually corresponding to qualitatively different cases)
Let p(·) be integral curve, c0 = C (p(0)), and h0 = Hα(p(0)).
We consider case c0 >1α2 and h0 >
1+α2c02α .
Figure: Intersection of C−1(c0) and H−1α (h0).
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Integration
Theorem
Let p(·) : (−ε, ε)→ se (2)∗ be an integral curve of ~Hα and let
h0 = H(p(0)), c0 = C (p(0)). If c0 >1α2 and h0 >
1+α2c02α , then there
exists t0 ∈ R and σ ∈ −1, 1 such that p(t) = p(t + t0) fort ∈ (−ε, ε), where
p1(t) =√c0
√h0 − δ −
√h0 + δ cn (Ω t, k)√
h0 + δ −√h0 − δ cn (Ω t, k)
p2(t) = σ√
2c0δsn (Ω t, k)√
h0 + δ −√h0 − δ cn (Ω t, k)
p3(t) = 2σδdn (Ω t, k)√
h0 + δ −√h0 − δ cn (Ω t, k)
·
Here δ =√
h20 − c0, Ω =
√2δ and k = 1√
2δ
√(h0 − δ) (αh0 + αδ − 1).
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Integration
Proof sketch (1/3)
Equations of motion: p1 = p2p3
p2 = −p1p3
p3 = (αp1 − 1) p2.
By squaring first equation, we obtain p21 = p2
2p23 .
By constants of motion c0 = p21 + p2
2 and h0 = p1 + 12
(αp2
2 + p23
),
eliminate p22 and p2
3 .
Obtain separable differential equation
dp1
dt= ±
√(c0 − p1
2) (h0 − 2p1 − (c0 − p12)α).
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Integration
Proof sketch (2/3)
Integral ∫dp1√
(c0 − p12) (h0 − 2p1 − (c0 − p1
2)α)
transformed into a standard form; elliptic integral formula applied.
Above is nontrivial; also, depends on c0 >1α2 and h0 >
1+α2c02α .
After simplification, we obtain
p1(t) =√c0
√h0 − δ −
√h0 + δ cn (Ω t, k)√
h0 + δ −√h0 − δ cn (Ω t, k)
.
Coordinates p2(t) and p3(t) are recovered by means of the identitiesc0 = p2
1 + p22 and h0 = p1 + 1
2
(αp2
2 + p23
).
Verify ˙p(t) = ~Hα(p(t)) (& determine choices of sign).
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Integration
Proof sketch (3/3)
Remains to be shown: p(·) is p(·) up to a translation in t.
We have p1(0) = −√c0 and p1( 2KΩ ) =
√c0.
Also, as p21 + p2
2 = c0, we have that −√c0 ≤ p1(0) ≤ √c0.
Exists t1 ∈ [0, 2KΩ ] such that p1(t1) = p1(0).
Choosing σ ∈ −1, 1 appropriately and using constant of motions,we get p(0) = p(t0) where t0 = t1 or t0 = −t1.
Curves t 7→ p(t) and t 7→ p(t + t0) solve the same Cauchy problemand therefore are identical.
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Outline
1 Introduction
2 Equivalence of control systems
3 Invariant optimal control
4 Quadratic Hamilton-Poisson systems
5 Conclusion
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Conclusion
Summary
effective means of classifying systems (at least in lower dimensions)
natural extension to optimal control problems
relates to equivalence of Hamilton-Poisson systems
Outlook
point affine distributions and strong detached feedback equivalence
systematic study of homogeneous cost-extended systems in lowdimensions (i.e., Riemannian and sub-Riemannian structures)
(invariant) nonholonomic structures
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References (standard: geometric control theory)
V. Jurdjevic
Geometric Control Theory
Cambridge University Press, 1997.
A.A. Agrachev and Yu.L. Sachkov
Control Theory from the Geometric Viewpoint
Springer, Berlin, 2004.
R.W. Brockett
System theory on group manifolds and coset spaces
SIAM J. Control 10 (1972), 265–284.
V. Jurdjevic and H.J. Sussmann
Control systems on Lie groups
J. Diff. Equations 12 (1972), 313–329.
Yu.L. Sachkov
Control theory on Lie groups
J. Math. Sci. 156 (2009), 381–439.
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GGC References I
Invariant control systems
C.C. Remsing
Optimal control and Hamilton-Poisson formalism
Int. J. Pure Appl. Math. 59 (2010), 11–17.
R. Biggs and C.C. Remsing
A category of control systems
An. St. Univ. “Ovidius” Constanta, Ser. Mat. 20 (2012), 355–368.
R. Biggs and C.C. Remsing
Cost-extended control systems on Lie groups
Mediterr. J. Math. 11 (2014), 193–215.
R. Biggs and C.C. Remsing,
On the equivalence of control systems on Lie groups
submitted.
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GGC References II
State space equivalence
R.M. Adams, R. Biggs, and C.C. Remsing
Equivalence of control systems on the Euclidean group SE (2)
Control Cybernet. 41 (2012), 513–524.
D.I. Barrett, R. Biggs, and C.C. Remsing
Affine subspaces of the Lie algebra se (1, 1)
Eur. J. Pure Appl. Math. 7 (2014), 140–155.
R. Biggs, C.C. Remsing
Equivalence of control systems on the pseudo-orthogonal group SO (2, 1)
submitted.
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GGC References III
Detached feedback equivalence
R.M. Adams, R. Biggs, and C.C. Remsing
Control systems on the orthogonal group SO (4)
Commun. Math. 21 (2013), 107–128.
R. Biggs and C.C. Remsing
Control systems on three-dimensional Lie groups: equivalence and controllability
J. Dyn. Control Syst. (2014), doi: 10.1007/s10883-014-9212-0.
R. Biggs and C.C. Remsing
Some remarks on the oscillator group
Differential Geom. Appl. (2014), in press.
Invariant sub-Riemannian structures
R. Biggs and P.T. Nagy
A classification of sub-Riemannian structures on the Heisenberg groups
Acta Polytech. Hungar. 10 (2013), 41–52.
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GGC References IV
Optimal control and Hamilton-Poisson systems
R.M. Adams, R. Biggs, C.C. Remsing
Two-input control systems on the Euclidean group SE (2)
ESAIM Control Optim. Calc. Var. 19 (2013), 947–975.
R. Biggs and C.C. Remsing
A classification of quadratic Hamilton-Poisson systems in three dimensions
Fifteenth International Conference on Geometry, Integrability and Quantization,Varna, Bulgaria, 2013, 67–78.
R. Biggs and C.C. Remsing
Quadratic Hamilton-Poisson systems in three dimensions: equivalence, stability,and integration
preprint.
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References (other)
B. Bonnard, V. Jurdjevic, I. Kupka, G. Sallet
Transitivity of families of invariant vector fields on the semidirect products of Liegroups
Trans. Amer. Math. Soc. 271 (1982), 525–535.
L. Capogna, E. Le Donne
Smoothness of subRiemannian isometries
2014 preprint, arXiv:1305.5286.
J.P. Ortega, V. Planas-Bielsa, T.S. Ratiu
Asymptotic and Lyapunov stability of constrained and Poisson equilibria
J. Differential Equations 214 (2005), 92?127
J. Patera, R.T. Sharp, P. Winternitz, H. Zassenhaus
Invariants of real low dimension Lie algebras
J. Mathematical Phys. 17 (1976), 986–994.
E.N. Wilson
Isometry groups on homogeneous nilmanifolds
Geom. Dedicata 12 (1982), 337?346.
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