Extremal trajectories and Maxwell points in sub-Riemannian problem - PowerPoint PPT Presentation

Extremal trajectories and Maxwell points in sub-Riemannian problem on the Engel group A. A. Ardentov Program Systems Institute Russian Academy of Sciences Pereslavl-Zalessky, Russia aaa@pereslavl.ru Workshop on Nonlinear Control and Singularities Toulon, October 24 – 28, 2010

Problem Statement      0  x ˙ 1 1 y ˙ 0       q = ˙  = u 1  + u 2  , x    − y    z ˙    2 2 x 2 + y 2 v ˙ 0 2 q = ( x , y , z , v ) ∈ R 4 , u = ( u 1 , u 2 ) ∈ R 2 . q ( 0 ) = q 0 = ( 0 , 0 , 0 , 0 ) T , q ( t 1 ) = q 1 = ( x 1 , y 1 , z 1 , v 1 ) T , � t 1 � t 1 u 2 1 + u 2 � u 2 1 + u 2 2 2 dt → min ⇐ ⇒ dt → min . 2 0 0

Geometric formulation of the problem Given: a 0 , a 1 ∈ R 2 , a 1 a 1 γ 0 ⊂ R 2 connecting a 1 to a 0 Γ 0 Γ 0 line L ⊂ R 2 S ∈ R , D Find: γ 1 ⊂ R 2 connecting a 0 to a 1 , s. t. γ 1 ∪ γ 0 = ∂ D , Γ 1 Γ 1 area ( D ) = S , a 0 a 0 L center of mass of D ∈ L , length ( γ 1 ) → min.

Overview • Parameterization of extremal curves. • Symmetries of exponential mapping and construction of the Maxwell sets. • Global bound of the cut time and necessary optimality conditions for extremal curves. • Algorithm and software for numerical solution of the problem.

Known results for invariant sub-Riemannian problems on Lie groups 1. Three-dimensional Lie groups: • Heisenberg group (A. M. Vershik, V. Ya. Gershkovich 1986), • SL ( 2 ) , SO ( 3 ) , S 3 (U. Boscain, F. Rossi 2008), • SE ( 2 ) (Yu. L. Sachkov 2010) 2. 5-dimensional nilpotent Lie group with growth vector ( 2 , 3 , 5 ) (Yu.L.Sachkov 2006). 3. 6-dimensional nilpotent Lie group with growth vector ( 3 , 6 ) (O.M. Myasnichenko 2002).

Nilpotent sub-Riemannian problem on the Engel group 2 , x 2 + y 2 X 1 = ( 1 , 0 , − y X 2 = ( 0 , 1 , x 2 , 0 ) T , ) T . 2 Lie ( X 1 , X 2 ) = span ( X 1 , X 2 , X 3 , X 4 ) , dim Lie ( X 1 , X 2 )( q ) = 4 , [ X 1 , X 2 ] = X 3 , [ X 1 , X 3 ] = X 4 , [ X 1 , X 4 ] = [ X 2 , X 3 ] = [ X 2 , X 4 ] = 0 . Growth vector (2, 3, 4). Nilpotent approximation of nonholonomic control systems in four-dimensional space with two-dimensional control (e. g. car with trailer).

Controllability and existence of optimal curves 1. X 1 ( q ) , . . . , X 4 ( q ) are linearly independent ∀ q ∈ R 4 Rashevskii–Chow theorem − − − − − − − − − − − − − − − → complete controllability. 2. Existence of optimal solutions is implied by Filippov theorem.

Pontryagin’s maximum principle : Abnormal extremal trajectories v = ± t 3 y = ± t , x = 0 , z = 0 , 6 .

Normal Hamiltonian system ˙ θ ∈ S 1 , θ = c , c = − α sin θ, ˙ c ∈ R , α = 0 , ˙ α ∈ R , x = − sin θ, ˙ y = cos θ, ˙ z = x cos θ + y sin θ ˙ , 2 v = cos θ x 2 + y 2 ˙ . 2 E = c 2 2 − α cos θ ∈ [ −| α | , + ∞ ) .

Equation of pendulum and physical meaning of α α = g ¨ θ = − α sin θ, L = const ∈ R s ❙ ❙ L θ ❙ s ❙ ❙ mg m ❙ s θ ❙ ✻ ❄ L ❙ mg m s Figure: Mathematical pendulum Figure: Mathematical pendulum with α > 0 with α < 0

Stratification of phase cylinder of pendulum q 0 M ∩ { H = 1 / 2 } = { λ = ( θ, c , α ) | θ ∈ S 1 , c , α ∈ R } . C = T ∗ C = ∪ 7 i = 1 C i , C i ∩ C j = ∅ , i � = j . C − C + = C i ∩ { α > 0 } , = C i ∩ { α < 0 } , i ∈ { 1 , . . . , 5 } , i i C ± i + = C ± C ± i − = C ± ∩ { c > 0 } , ∩ { c < 0 } , i ∈ { 2 , 3 } . i i c c C 3 � C 3 � C 3 � C 3 � � � � � C 2 � C 2 � � � C 2 � C 2 � � � C 1 C 1 C 1 C 1 � � � � 0 0 0 2 Π 2 Π �Π �Π Π Π Θ Θ C 4 C 4 � � C 4 C 4 � � C 5 C 5 � � C 5 C 5 � � C 2 � C 2 � C 2 � C 2 � � � � � C 3 � C 3 � C 3 � C 3 � � � � � Figure: Stratification for α > 0 Figure: Stratification for α < 0

Elliptic coordinates in C + λ ∈ C + 1 , � � c 2 E + α 4 α + sin 2 θ k = = 2 ∈ ( 0 , 1 ) , 2 α 2 = k sn ( √ αϕ ) , 2 = dn ( √ αϕ ) , sin θ cos θ 2 = k √ α cn ( √ αϕ ) , c ϕ ∈ [ 0 , 4 K ] , where sn , cn , dn , E are elliptic Jacobi’s functions. ˙ Equation of pendulum: ˙ ϕ = 1 , k = ˙ α = 0 .

Elliptic coordinates ( ϕ, k ) in the phase cylinder of pendulum c c � � � � � � 0 2 Π 2 Π 0 0 �Π �Π Π Π Θ Θ � �

Elliptic coordinates in C − Coordinates in the sets C − 1 , C − 2 , C − 3 : ϕ ( θ, c , α ) = ϕ ( θ − π, c , − α ) , k ( θ, c , α ) = k ( θ − π, c , − α ) .

Parametrization of extremal curves in the case α = 1 λ ∈ C + 1 (oscillations of pendulum) ⇒ x t = 2 k ( cn ϕ t − cn ϕ ) , y t = 2 ( E ( ϕ t ) − E ( ϕ )) − t , z t = 2 k ( sn ϕ t dn ϕ t − sn ϕ dn ϕ − y t 2 ( cn ϕ t + cn ϕ )) , v t = y 3 6 + 2 k 2 cn 2 ϕ y t − 4 k 2 cn ϕ ( sn ϕ t dn ϕ t − sn ϕ dn ϕ )+ t 3 cn ϕ dn ϕ sn ϕ + 1 − k 2 � 2 3 cn ϕ t dn ϕ t sn ϕ t − 2 + 2 k 2 3 k 2 t + 2 k 2 − 1 � ( E ( ϕ t ) − E ( ϕ )) . 3 k 2

Symmetries of Hamiltonian system Dilation of α : √ α, 1 , √ α x , √ α y , α z , α 2 v , √ α t ) , ( θ, c , α, x , y , z , v , t ) �→ ( θ, c 3 ( ϕ, k , α ) �→ ( √ αϕ, k , 1 ) . Inversion of α : ( θ, c , α, x , y , z , v , t ) �→ ( θ − π, c , − α, − x , − y , z , − v , t ) , ( ϕ, k , α ) �→ ( ϕ, k , − α ) .

Parametrization of extremal trajectories in general case with λ ∈ ∪ 3 i = 1 C i ( x t , y t , z t , v t )( ϕ, k , α ) = ( s 1 σ x σ t , s 1 σ y σ t , 1 σ 2 z σ t , s 1 σ 3 v σ t )( σϕ, k , 1 ) , � where σ = | α | , s 1 = sgn α .

General case with α � = 0 λ ∈ C 1 ⇒ x t = 2 k σ α ( cn ( σϕ t ) − cn ( σϕ )) , y t = 2 σ α ( E ( σϕ t ) − E ( σϕ )) − sgn α t , z t = 2 k | α | ( sn ( σϕ t ) dn ( σϕ t ) − sn ( σϕ ) dn ( σϕ ) − − σ ky t 2 α ( cn ( σϕ t ) + cn ( σϕ ))) , v t = . . .

Parametrization of extremal curves for degenerate cases v t = t 3 λ ∈ C 4 ⇒ x t = 0 , y t = t sgn α, z t = 0 , 6 sgn α. v t = − t 3 λ ∈ C 5 ⇒ x t = 0 , y t = − t sgn α, z t = 0 , 6 sgn α. λ ∈ C 6 ⇒ x t = cos ( ct + θ ) − cos θ y t = sin ( ct + θ ) − sin θ , , c c z t = ct − sin ( ct ) v t = − 2 c cos θ t − 4 sin ( ct + θ ) + sin ( 2 ct + θ ) , . 2 c 2 4 c 3 v t = cos θ t 3 . λ ∈ C 7 ⇒ x t = − t sin θ, y t = t cos θ, z t = 0 , 6

Euler elasticae 3.5 3 3.5 3 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5 0 0 0 0 1 2 3 4 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -3 -2 -1 0 Figure: Inflectional elasticae 2 1.2 1 1.5 0.8 1 0.6 0.4 0.5 0.2 0 0 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 Figure: Critical elastica Figure: Non-inflectional elastica

Exponential mapping, Maxwell points and cut time Exp : C × R + → M = R 4 , Exp ( λ, t ) = q t , λ = ( θ, c , α ) ∈ C , t ∈ R + , q t ∈ M . MAX = { ( λ, t ) | ∃ ˜ λ � = λ, Exp ( λ, t ) = Exp (˜ λ, t ) } , t cut ( λ ) = sup { t > 0 | Exp ( λ, s ) is optimal for s ∈ [ 0 , t ] } , t cut ( λ ) ≤ t for any ( λ, t ) ∈ MAX .

Group of symmetries of exponential mapping ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ε 7 G ε 1 ε 3 ε 2 ε 5 ε 4 ε 7 ε 6 Id ε 2 ε 1 ε 6 ε 7 ε 4 ε 5 Id ε 3 ε 7 ε 6 ε 5 ε 4 Id ε 4 ε 1 ε 2 ε 3 Id ε 5 ε 3 ε 2 Id ε 6 ε 1 Id ε 7 Id Table: Multiplication in G = { Id , ε 1 , ε 2 , ε 3 , ε 4 , ε 5 , ε 6 , ε 7 }

Reflections of trajectories of pendulum c c Γ 2 Γ 2 Γ 6 Γ 6 Γ 4 Γ 4 Γ 0 0 2 Π 2 Π �Π �Π Π Π Θ Θ Γ 3 Γ 3 Γ 1 Γ 1 Γ 7 Γ 7 Γ 5 Γ 5

Reflections of trajectories of pendulum ε 1 : γ �→ γ 1 = { ( θ 1 s , c 1 s , α 1 ) } = { ( θ t − s , − c t − s , α ) | s ∈ [ 0 , t ] } , ε 2 : γ �→ γ 2 = { ( θ 2 s , c 2 s , α 2 ) } = { ( − θ t − s , c t − s , α ) | s ∈ [ 0 , t ] } , ε 3 : γ �→ γ 3 = { ( θ 3 s , c 3 s , α 3 ) } = { ( − θ s , − c s , α ) | s ∈ [ 0 , t ] } , ε 4 : γ �→ γ 4 = { ( θ 4 s , c 4 s , α 4 ) } = { ( θ s + π, c s , − α ) | s ∈ [ 0 , t ] } , ε 5 : γ �→ γ 5 = { ( θ 5 s , c 5 s , α 5 ) } = { ( θ t − s + π, − c t − s , − α ) | s ∈ [ 0 , t ] } , ε 6 : γ �→ γ 6 = { ( θ 6 s , c 6 s , α 6 ) } = { ( − θ t − s + π, c t − s , − α ) | s ∈ [ 0 , t ] } , ε 7 : γ �→ γ 7 = { ( θ 7 s , c 7 s , α 7 ) } = { ( − θ s + π, − c s , − α ) | s ∈ [ 0 , t ] } , where γ = { ( θ s , c s , α ) | s ∈ [ 0 , t ] } .

Reflections of Euler elasticae � 6 � 6 � 5 � 5 � 3 � 3 id id � 7 � 7 � 4 � 4 � 1 � 1 � 2 � 2

Action of ε i in the preimage of exponential mapping ε i : C × R → C × R , ε i ( θ, c , α, t ) = ( θ i , c i , α i , t ) , ( θ 1 , c 1 , α 1 ) = ( θ t , − c t , α ) , ( θ 2 , c 2 , α 2 ) = ( − θ t , c t , α ) , ( θ 3 , c 3 , α 3 ) = ( − θ t , − c t , α ) , ( θ 4 , c 4 , α 4 ) = ( θ t + π, c t , − α ) , ( θ 5 , c 5 , α 5 ) = ( θ t + π, − c t , − α ) , ( θ 6 , c 6 , α 6 ) = ( − θ t + π, c t , − α ) , ( θ 7 , c 7 , α 7 ) = ( − θ t + π, − c t , − α ) .