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Third order analysis of the end-point mapping
Roberto Monti (Padova)
joint work with
F. Boarotto (Padova) and F. Palmurella (ETH Zurich)
Sub-Riemannian Geometry and InteractionsParis 6-11 September 2020
End-point mapping
(M,D) sub-Riemannian manifold
– M smooth manifold– D = span{f1, . . . , fk} ⊂ TM horiz. distribution (Hormander condition)
Fix a point q ∈ M and for u ∈ X := L2([0, 1];Rk) let γ = γq,u be thesolution to
γ =k∑
i=1
ui fi (γ) on [0, 1], γ(0) = q.
The end-point mapping is the smooth map F = Fq : X → M
F (u) = γq,u(1).
The control u ∈ X is singular if the differential duF : X → TF (u)M is notsurjective.
End-point mapping
(M,D) sub-Riemannian manifold
– M smooth manifold– D = span{f1, . . . , fk} ⊂ TM horiz. distribution (Hormander condition)
Fix a point q ∈ M and for u ∈ X := L2([0, 1];Rk) let γ = γq,u be thesolution to
γ =k∑
i=1
ui fi (γ) on [0, 1], γ(0) = q.
The end-point mapping is the smooth map F = Fq : X → M
F (u) = γq,u(1).
The control u ∈ X is singular if the differential duF : X → TF (u)M is notsurjective.
End-point mapping
(M,D) sub-Riemannian manifold
– M smooth manifold– D = span{f1, . . . , fk} ⊂ TM horiz. distribution (Hormander condition)
Fix a point q ∈ M and for u ∈ X := L2([0, 1];Rk) let γ = γq,u be thesolution to
γ =k∑
i=1
ui fi (γ) on [0, 1], γ(0) = q.
The end-point mapping is the smooth map F = Fq : X → M
F (u) = γq,u(1).
The control u ∈ X is singular if the differential duF : X → TF (u)M is notsurjective.
End-point mapping
(M,D) sub-Riemannian manifold
– M smooth manifold– D = span{f1, . . . , fk} ⊂ TM horiz. distribution (Hormander condition)
Fix a point q ∈ M and for u ∈ X := L2([0, 1];Rk) let γ = γq,u be thesolution to
γ =k∑
i=1
ui fi (γ) on [0, 1], γ(0) = q.
The end-point mapping is the smooth map F = Fq : X → M
F (u) = γq,u(1).
The control u ∈ X is singular if the differential duF : X → TF (u)M is notsurjective.
The end-point mapping is complicated
The main open problems in SR geometry:
–Regularity of singular length-minimizing curves–Size of the image of singular extremals (Sard problem)
are related to our limited understanding of the end-point mapping atsingular points.
NotationX =Banach spaceF : X → M × R smooth mappingu ∈ X singular, say u = 0
In our application
F = (F , J) : X → M × R ≡ Rn
is the extended end-point mapping, where J : X → R is the lengthfunctional. We can assume M ×R = Rn fixing a chart around F (0) ∈ M.
The cokernel of the differential of F is
coker(d0F ) = TF (0)M × R/Im(d0F ).
We denote by π : TF (0)M × R→ coker(d0F ) the projection.
NotationX =Banach spaceF : X → M × R smooth mappingu ∈ X singular, say u = 0
In our application
F = (F , J) : X → M × R ≡ Rn
is the extended end-point mapping, where J : X → R is the lengthfunctional. We can assume M ×R = Rn fixing a chart around F (0) ∈ M.
The cokernel of the differential of F is
coker(d0F ) = TF (0)M × R/Im(d0F ).
We denote by π : TF (0)M × R→ coker(d0F ) the projection.
NotationX =Banach spaceF : X → M × R smooth mappingu ∈ X singular, say u = 0
In our application
F = (F , J) : X → M × R ≡ Rn
is the extended end-point mapping, where J : X → R is the lengthfunctional. We can assume M ×R = Rn fixing a chart around F (0) ∈ M.
The cokernel of the differential of F is
coker(d0F ) = TF (0)M × R/Im(d0F ).
We denote by π : TF (0)M × R→ coker(d0F ) the projection.
Strictly singular case
The singular point u = 0 is strictly singular when
coker(d0F ) = coker(d0F ) = TF (0)M/Im(d0F ).
The J component is quotiented out by π.
In this case we may study the end-point map F and deduce properties forthe extended map F (e.g., being an open mapping).
Strictly singular case
The singular point u = 0 is strictly singular when
coker(d0F ) = coker(d0F ) = TF (0)M/Im(d0F ).
The J component is quotiented out by π.
In this case we may study the end-point map F and deduce properties forthe extended map F (e.g., being an open mapping).
Second order analysis
Notation for the kth-order directional derivative at u = 0
dk0 F (v) =
dk
dtkF (tv)
∣∣∣∣t=0
, v ∈ X .
The intrinsic Hessian of F at u = 0 is the quadratic mapD2
0F : ker(d0F )→ coker(d0F )
D20F (v) = π(d2
0F (v)), v ∈ ker(d0F ).
Definition. An element v ∈ ker(d0F ) is a regular zero for D20F if:
– D20F (v) = 0;
– the linear map w 7→ D20F (v ,w) is su from ker(d0F ) to coker(d0F ).
Second order analysis
Notation for the kth-order directional derivative at u = 0
dk0 F (v) =
dk
dtkF (tv)
∣∣∣∣t=0
, v ∈ X .
The intrinsic Hessian of F at u = 0 is the quadratic mapD2
0F : ker(d0F )→ coker(d0F )
D20F (v) = π(d2
0F (v)), v ∈ ker(d0F ).
Definition. An element v ∈ ker(d0F ) is a regular zero for D20F if:
– D20F (v) = 0;
– the linear map w 7→ D20F (v ,w) is su from ker(d0F ) to coker(d0F ).
Second order analysis
Notation for the kth-order directional derivative at u = 0
dk0 F (v) =
dk
dtkF (tv)
∣∣∣∣t=0
, v ∈ X .
The intrinsic Hessian of F at u = 0 is the quadratic mapD2
0F : ker(d0F )→ coker(d0F )
D20F (v) = π(d2
0F (v)), v ∈ ker(d0F ).
Definition. An element v ∈ ker(d0F ) is a regular zero for D20F if:
– D20F (v) = 0;
– the linear map w 7→ D20F (v ,w) is su from ker(d0F ) to coker(d0F ).
Second order analysis
Notation for the kth-order directional derivative at u = 0
dk0 F (v) =
dk
dtkF (tv)
∣∣∣∣t=0
, v ∈ X .
The intrinsic Hessian of F at u = 0 is the quadratic mapD2
0F : ker(d0F )→ coker(d0F )
D20F (v) = π(d2
0F (v)), v ∈ ker(d0F ).
Definition. An element v ∈ ker(d0F ) is a regular zero for D20F if:
– D20F (v) = 0;
– the linear map w 7→ D20F (v ,w) is su from ker(d0F ) to coker(d0F ).
Agracev-Sarychev theory
Theorem. Let u ∈ X be strictly singular. If D2uF has a regular zero
then the extended map F is open at u.
Thus u is not J-optimal (it is not length optimal).
Lemma. If for any λ 6= 0 orthogonal to Im(duF ) the scalarizationλD2
uF has index
ind(λD2uF ) ≥ dim(TF (u)/Im(duF )) =: corank(u)
then D2uF has a regular zero.
Corollary (Goh). Let (γ, u) be a strictly singular minimizingextremal. Then any adjoint curve λ : [0, 1]→ T ∗M satisfies
〈λ(t), [fi , fj ](γ(t))〉 = 0 t ∈ [0, 1], i , j = 1, . . . , k.
Comments. 1) If Goh’s condition is violated the index above is ∞.
2) The bracket [fi , fj ] appears in the computation for D2uF restricted
to its correct domain, i.e., ker(duF ).
Agracev-Sarychev theory
Theorem. Let u ∈ X be strictly singular. If D2uF has a regular zero
then the extended map F is open at u.
Thus u is not J-optimal (it is not length optimal).
Lemma. If for any λ 6= 0 orthogonal to Im(duF ) the scalarizationλD2
uF has index
ind(λD2uF ) ≥ dim(TF (u)/Im(duF )) =: corank(u)
then D2uF has a regular zero.
Corollary (Goh). Let (γ, u) be a strictly singular minimizingextremal. Then any adjoint curve λ : [0, 1]→ T ∗M satisfies
〈λ(t), [fi , fj ](γ(t))〉 = 0 t ∈ [0, 1], i , j = 1, . . . , k.
Comments. 1) If Goh’s condition is violated the index above is ∞.
2) The bracket [fi , fj ] appears in the computation for D2uF restricted
to its correct domain, i.e., ker(duF ).
Agracev-Sarychev theory
Theorem. Let u ∈ X be strictly singular. If D2uF has a regular zero
then the extended map F is open at u.
Thus u is not J-optimal (it is not length optimal).
Lemma. If for any λ 6= 0 orthogonal to Im(duF ) the scalarizationλD2
uF has index
ind(λD2uF ) ≥ dim(TF (u)/Im(duF )) =: corank(u)
then D2uF has a regular zero.
Corollary (Goh). Let (γ, u) be a strictly singular minimizingextremal. Then any adjoint curve λ : [0, 1]→ T ∗M satisfies
〈λ(t), [fi , fj ](γ(t))〉 = 0 t ∈ [0, 1], i , j = 1, . . . , k.
Comments. 1) If Goh’s condition is violated the index above is ∞.
2) The bracket [fi , fj ] appears in the computation for D2uF restricted
to its correct domain, i.e., ker(duF ).
Agracev-Sarychev theory
Theorem. Let u ∈ X be strictly singular. If D2uF has a regular zero
then the extended map F is open at u.
Thus u is not J-optimal (it is not length optimal).
Lemma. If for any λ 6= 0 orthogonal to Im(duF ) the scalarizationλD2
uF has index
ind(λD2uF ) ≥ dim(TF (u)/Im(duF )) =: corank(u)
then D2uF has a regular zero.
Corollary (Goh). Let (γ, u) be a strictly singular minimizingextremal. Then any adjoint curve λ : [0, 1]→ T ∗M satisfies
〈λ(t), [fi , fj ](γ(t))〉 = 0 t ∈ [0, 1], i , j = 1, . . . , k.
Comments. 1) If Goh’s condition is violated the index above is ∞.
2) The bracket [fi , fj ] appears in the computation for D2uF restricted
to its correct domain, i.e., ker(duF ).
Agracev-Sarychev theory
Theorem. Let u ∈ X be strictly singular. If D2uF has a regular zero
then the extended map F is open at u.
Thus u is not J-optimal (it is not length optimal).
Lemma. If for any λ 6= 0 orthogonal to Im(duF ) the scalarizationλD2
uF has index
ind(λD2uF ) ≥ dim(TF (u)/Im(duF )) =: corank(u)
then D2uF has a regular zero.
Corollary (Goh). Let (γ, u) be a strictly singular minimizingextremal. Then any adjoint curve λ : [0, 1]→ T ∗M satisfies
〈λ(t), [fi , fj ](γ(t))〉 = 0 t ∈ [0, 1], i , j = 1, . . . , k.
Comments. 1) If Goh’s condition is violated the index above is ∞.
2) The bracket [fi , fj ] appears in the computation for D2uF restricted
to its correct domain, i.e., ker(duF ).
Agracev-Sarychev theory
Theorem. Let u ∈ X be strictly singular. If D2uF has a regular zero
then the extended map F is open at u.
Thus u is not J-optimal (it is not length optimal).
Lemma. If for any λ 6= 0 orthogonal to Im(duF ) the scalarizationλD2
uF has index
ind(λD2uF ) ≥ dim(TF (u)/Im(duF )) =: corank(u)
then D2uF has a regular zero.
Corollary (Goh). Let (γ, u) be a strictly singular minimizingextremal. Then any adjoint curve λ : [0, 1]→ T ∗M satisfies
〈λ(t), [fi , fj ](γ(t))〉 = 0 t ∈ [0, 1], i , j = 1, . . . , k.
Comments. 1) If Goh’s condition is violated the index above is ∞.
2) The bracket [fi , fj ] appears in the computation for D2uF restricted
to its correct domain, i.e., ker(duF ).
Third intrinsic differential
Consider a smooth map F : X → M. The subspace
dom(D30F ) := {v ∈ ker(d0F ) : π(d2
0F (v , x)) = 0 for all x ∈ X}
is chart-independent.
We define D30F : dom(D3
0F )→ coker(d0F ) in the natural way:
D30F (v) = π(d3
0F (v)).
Definition. Let w ∈ ker(d0F ) be such that D20F (w) = 0.
An element v ∈ dom(D30F ) is a w -regular zero for the third
differential if:– D3
0F (v) = 0;– the linear map u 7→ D3
0F (v , v , u) is surjective from dom(D30F ) to
coker(d0F )/Im(D20F (w , ·)).
Third intrinsic differential
Consider a smooth map F : X → M. The subspace
dom(D30F ) := {v ∈ ker(d0F ) : π(d2
0F (v , x)) = 0 for all x ∈ X}
is chart-independent.
We define D30F : dom(D3
0F )→ coker(d0F ) in the natural way:
D30F (v) = π(d3
0F (v)).
Definition. Let w ∈ ker(d0F ) be such that D20F (w) = 0.
An element v ∈ dom(D30F ) is a w -regular zero for the third
differential if:– D3
0F (v) = 0;– the linear map u 7→ D3
0F (v , v , u) is surjective from dom(D30F ) to
coker(d0F )/Im(D20F (w , ·)).
Third intrinsic differential
Consider a smooth map F : X → M. The subspace
dom(D30F ) := {v ∈ ker(d0F ) : π(d2
0F (v , x)) = 0 for all x ∈ X}
is chart-independent.
We define D30F : dom(D3
0F )→ coker(d0F ) in the natural way:
D30F (v) = π(d3
0F (v)).
Definition. Let w ∈ ker(d0F ) be such that D20F (w) = 0.
An element v ∈ dom(D30F ) is a w -regular zero for the third
differential if:– D3
0F (v) = 0;– the linear map u 7→ D3
0F (v , v , u) is surjective from dom(D30F ) to
coker(d0F )/Im(D20F (w , ·)).
Third intrinsic differential
Consider a smooth map F : X → M. The subspace
dom(D30F ) := {v ∈ ker(d0F ) : π(d2
0F (v , x)) = 0 for all x ∈ X}
is chart-independent.
We define D30F : dom(D3
0F )→ coker(d0F ) in the natural way:
D30F (v) = π(d3
0F (v)).
Definition. Let w ∈ ker(d0F ) be such that D20F (w) = 0.
An element v ∈ dom(D30F ) is a w -regular zero for the third
differential if:– D3
0F (v) = 0;– the linear map u 7→ D3
0F (v , v , u) is surjective from dom(D30F ) to
coker(d0F )/Im(D20F (w , ·)).
Third order open mapping theorem
Theorem. Let F : X → M be a smooth mapping and let u = 0 be asingular point.
1) Case corank(u) = 1. If there exists v ∈ dom(D30F ) such that
D30F (v) 6= 0 then F is open at u = 0.
2) Case corank(u) ≥ 1. If there exist w and v such that v is aw -regular zero for D3
0F then F is open at u = 0.
Comments.
1) In our application the extended map F will be open at u.
2) We could not find any condition of the algebraic type (extension ofthe notion of index for a quadratic form) ensuring the existence of aw -regular zero for a cubic map.
Third order open mapping theorem
Theorem. Let F : X → M be a smooth mapping and let u = 0 be asingular point.
1) Case corank(u) = 1. If there exists v ∈ dom(D30F ) such that
D30F (v) 6= 0 then F is open at u = 0.
2) Case corank(u) ≥ 1. If there exist w and v such that v is aw -regular zero for D3
0F then F is open at u = 0.
Comments.
1) In our application the extended map F will be open at u.
2) We could not find any condition of the algebraic type (extension ofthe notion of index for a quadratic form) ensuring the existence of aw -regular zero for a cubic map.
Third order open mapping theorem
Theorem. Let F : X → M be a smooth mapping and let u = 0 be asingular point.
1) Case corank(u) = 1. If there exists v ∈ dom(D30F ) such that
D30F (v) 6= 0 then F is open at u = 0.
2) Case corank(u) ≥ 1. If there exist w and v such that v is aw -regular zero for D3
0F then F is open at u = 0.
Comments.
1) In our application the extended map F will be open at u.
2) We could not find any condition of the algebraic type (extension ofthe notion of index for a quadratic form) ensuring the existence of aw -regular zero for a cubic map.
Third order open mapping theorem
Theorem. Let F : X → M be a smooth mapping and let u = 0 be asingular point.
1) Case corank(u) = 1. If there exists v ∈ dom(D30F ) such that
D30F (v) 6= 0 then F is open at u = 0.
2) Case corank(u) ≥ 1. If there exist w and v such that v is aw -regular zero for D3
0F then F is open at u = 0.
Comments.
1) In our application the extended map F will be open at u.
2) We could not find any condition of the algebraic type (extension ofthe notion of index for a quadratic form) ensuring the existence of aw -regular zero for a cubic map.
Third order open mapping theorem
Theorem. Let F : X → M be a smooth mapping and let u = 0 be asingular point.
1) Case corank(u) = 1. If there exists v ∈ dom(D30F ) such that
D30F (v) 6= 0 then F is open at u = 0.
2) Case corank(u) ≥ 1. If there exist w and v such that v is aw -regular zero for D3
0F then F is open at u = 0.
Comments.
1) In our application the extended map F will be open at u.
2) We could not find any condition of the algebraic type (extension ofthe notion of index for a quadratic form) ensuring the existence of aw -regular zero for a cubic map.
Computation of D3uF
New notation G (v) = F (u + v).
The end-point mapping is defined via a flow that can be expressed as aright (or left) chronological exponential and can be interpreted as anoperator acting on C∞(M).
In this sense we have the expansion
G (v) = Id + d0G (v) +1
2d2
0G (v) +1
6d3
0G (v) + error
The third order term is (we do not explain the notation)
d30G (v) =
∫Σ3
[gτ3
v(τ3), [gτ2
v(τ2), gτ1
v(τ1)]]dτ3dτ2dτ1
+(∫
Σ2
[gτ2
v(τ2), gτ1
v(τ1)]dτ2dτ1
)◦(∫ 1
0
g tv(t)dt
)+(∫ 1
0
g tv(t)dt
)◦(∫
Σ2
[gτ2
v(τ2), gτ1
v(τ1)]dτ2dτ1
)+(∫ 1
0
g tv(t)dt
)◦(∫ 1
0
g tv(t)dt
)◦(∫ 1
0
g tv(t)dt
)
Computation of D3uF
New notation G (v) = F (u + v).
The end-point mapping is defined via a flow that can be expressed as aright (or left) chronological exponential and can be interpreted as anoperator acting on C∞(M).
In this sense we have the expansion
G (v) = Id + d0G (v) +1
2d2
0G (v) +1
6d3
0G (v) + error
The third order term is (we do not explain the notation)
d30G (v) =
∫Σ3
[gτ3
v(τ3), [gτ2
v(τ2), gτ1
v(τ1)]]dτ3dτ2dτ1
+(∫
Σ2
[gτ2
v(τ2), gτ1
v(τ1)]dτ2dτ1
)◦(∫ 1
0
g tv(t)dt
)+(∫ 1
0
g tv(t)dt
)◦(∫
Σ2
[gτ2
v(τ2), gτ1
v(τ1)]dτ2dτ1
)+(∫ 1
0
g tv(t)dt
)◦(∫ 1
0
g tv(t)dt
)◦(∫ 1
0
g tv(t)dt
)
Computation of D3uF
New notation G (v) = F (u + v).
The end-point mapping is defined via a flow that can be expressed as aright (or left) chronological exponential and can be interpreted as anoperator acting on C∞(M).
In this sense we have the expansion
G (v) = Id + d0G (v) +1
2d2
0G (v) +1
6d3
0G (v) + error
The third order term is (we do not explain the notation)
d30G (v) =
∫Σ3
[gτ3
v(τ3), [gτ2
v(τ2), gτ1
v(τ1)]]dτ3dτ2dτ1
+(∫
Σ2
[gτ2
v(τ2), gτ1
v(τ1)]dτ2dτ1
)◦(∫ 1
0
g tv(t)dt
)+(∫ 1
0
g tv(t)dt
)◦(∫
Σ2
[gτ2
v(τ2), gτ1
v(τ1)]dτ2dτ1
)+(∫ 1
0
g tv(t)dt
)◦(∫ 1
0
g tv(t)dt
)◦(∫ 1
0
g tv(t)dt
)
Comments
1. The above representation for the third differential is not unique.
2. When v ∈ dom(D30G ) the only surviving term is the geometric one:
d30G (v) =
∫Σ3
[gτ3
v(τ3), [gτ2
v(τ2), gτ1
v(τ1)]]dτ3dτ2dτ1
3. After scalarization with λ orthogonal to Im(d0G ) the representationbecomes unique.
Comments
1. The above representation for the third differential is not unique.
2. When v ∈ dom(D30G ) the only surviving term is the geometric one:
d30G (v) =
∫Σ3
[gτ3
v(τ3), [gτ2
v(τ2), gτ1
v(τ1)]]dτ3dτ2dτ1
3. After scalarization with λ orthogonal to Im(d0G ) the representationbecomes unique.
Comments
1. The above representation for the third differential is not unique.
2. When v ∈ dom(D30G ) the only surviving term is the geometric one:
d30G (v) =
∫Σ3
[gτ3
v(τ3), [gτ2
v(τ2), gτ1
v(τ1)]]dτ3dτ2dτ1
3. After scalarization with λ orthogonal to Im(d0G ) the representationbecomes unique.
Third order necessary conditions
Theorem. Let (γ, u) be a strictly singular minimizing extremal.
Assume that:
a) we have corank(u) = 1;
b) dom(D3uF ) is of finite codimension in ker(duF ).
Then any adjoint curve λ : [0, 1]→ T ∗M satisfies for i , j , ` = 1, . . . , k
〈λ(t), [fi , [fj , f`]](γ(t)) + [f`, [fj , fi ]](γ(t))〉 = 0 t ∈ [0, 1]. (∗)
Third order necessary conditions
Theorem. Let (γ, u) be a strictly singular minimizing extremal.
Assume that:
a) we have corank(u) = 1;
b) dom(D3uF ) is of finite codimension in ker(duF ).
Then any adjoint curve λ : [0, 1]→ T ∗M satisfies for i , j , ` = 1, . . . , k
〈λ(t), [fi , [fj , f`]](γ(t)) + [f`, [fj , fi ]](γ(t))〉 = 0 t ∈ [0, 1]. (∗)
Third order necessary conditions
Theorem. Let (γ, u) be a strictly singular minimizing extremal.
Assume that:
a) we have corank(u) = 1;
b) dom(D3uF ) is of finite codimension in ker(duF ).
Then any adjoint curve λ : [0, 1]→ T ∗M satisfies for i , j , ` = 1, . . . , k
〈λ(t), [fi , [fj , f`]](γ(t)) + [f`, [fj , fi ]](γ(t))〉 = 0 t ∈ [0, 1]. (∗)
Comments
1. In the second order case, using the notion of index it is possible tocover the case of general corank.
2. In the second order case, assumption b) reads
dom(D2uF ) = ker(duF ) is of finite codimension in X
and it is trivially satisfied.
3. Assumption b) does the following job. Contradicting (∗), there isenough room to construct a function v ∈ L2([0, 1];Rk) and in factv ∈ dom(D3
0G ) such that
D30G (v) 6= 0.
This contradicts the third order open mapping theorem.
Comments
1. In the second order case, using the notion of index it is possible tocover the case of general corank.
2. In the second order case, assumption b) reads
dom(D2uF ) = ker(duF ) is of finite codimension in X
and it is trivially satisfied.
3. Assumption b) does the following job. Contradicting (∗), there isenough room to construct a function v ∈ L2([0, 1];Rk) and in factv ∈ dom(D3
0G ) such that
D30G (v) 6= 0.
This contradicts the third order open mapping theorem.
Comments
1. In the second order case, using the notion of index it is possible tocover the case of general corank.
2. In the second order case, assumption b) reads
dom(D2uF ) = ker(duF ) is of finite codimension in X
and it is trivially satisfied.
3. Assumption b) does the following job. Contradicting (∗), there isenough room to construct a function v ∈ L2([0, 1];Rk) and in factv ∈ dom(D3
0G ) such that
D30G (v) 6= 0.
This contradicts the third order open mapping theorem.
That’s all.
Thank you for your attention.
Hoping in better times.