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Strong local optimality of singular trajectories Gianna Stefani - - PowerPoint PPT Presentation

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Strong local optimality of singular trajectories Gianna Stefani Dipartimento di Matematica e Informatica, Universit` a di Firenze Nonholonomic mechanics and optimal control Institut Henri Poincar e, November 25-28, 2014 G.Stefani (DiMaI)

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Strong local optimality of singular trajectories

Gianna Stefani

Dipartimento di Matematica e Informatica, Universit` a di Firenze

Nonholonomic mechanics and optimal control

Institut Henri Poincar´ e, November 25-28, 2014

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 1 / 20

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The optimal control problem

Minimize J(ξ, u, T ) subject to      ˙ ξ(t) = (f0 + m

i=1 ui(t)fi) ◦ ξ(t)

ξ(0) ∈ N0, ξ(T ) ∈ Nf u = (u1, . . . , um) ∈ U ⊂ Rm, int U = ∅. state space = M ( n-dimensional), N0, Nf - sub-manifolds C∞ data, L∞ control maps The focus will be on the case when the controlled vector fields f1, . . . , fm generate a non involutive Lie Algebra joint work with F .C. Chittaro for what is written I refer to http://arxiv.org/abs/1404.7336 partial results are in CDC 2013

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 2 / 20

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The optimal control problem

Minimize J(ξ, u, T ) subject to      ˙ ξ(t) = (f0 + m

i=1 ui(t)fi) ◦ ξ(t)

ξ(0) ∈ N0, ξ(T ) ∈ Nf u = (u1, . . . , um) ∈ U ⊂ Rm, int U = ∅. state space = M ( n-dimensional), N0, Nf - sub-manifolds C∞ data, L∞ control maps The focus will be on the case when the controlled vector fields f1, . . . , fm generate a non involutive Lie Algebra joint work with F .C. Chittaro for what is written I refer to http://arxiv.org/abs/1404.7336 partial results are in CDC 2013

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 2 / 20

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SLIDE 4

The optimal control problem

Minimize J(ξ, u, T ) subject to      ˙ ξ(t) = (f0 + m

i=1 ui(t)fi) ◦ ξ(t)

ξ(0) ∈ N0, ξ(T ) ∈ Nf u = (u1, . . . , um) ∈ U ⊂ Rm, int U = ∅. state space = M ( n-dimensional), N0, Nf - sub-manifolds C∞ data, L∞ control maps The focus will be on the case when the controlled vector fields f1, . . . , fm generate a non involutive Lie Algebra joint work with F .C. Chittaro for what is written I refer to http://arxiv.org/abs/1404.7336 partial results are in CDC 2013

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 2 / 20

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The cost Mayer problem on [0, T] fixed

minimize

c0(ξ(0)) + cf(ξ( T )) minimum time problem

minimize the final time

T Reference couple satisfying PMP

  • u: [0,

T] → U reference control,

  • ξ : [0,

T] → M reference trajectory

an hat above denotes reference objects

Reference vector field

  • ft = f0 + m

i=1

ui(t)fi

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 3 / 20

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The cost Mayer problem on [0, T] fixed

minimize

c0(ξ(0)) + cf(ξ( T )) minimum time problem

minimize the final time

T Reference couple satisfying PMP

  • u: [0,

T] → U reference control,

  • ξ : [0,

T] → M reference trajectory

an hat above denotes reference objects

Reference vector field

  • ft = f0 + m

i=1

ui(t)fi

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 3 / 20

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SLIDE 7

The cost Mayer problem on [0, T] fixed

minimize

c0(ξ(0)) + cf(ξ( T )) minimum time problem

minimize the final time

T Reference couple satisfying PMP

  • u: [0,

T] → U reference control,

  • ξ : [0,

T] → M reference trajectory

an hat above denotes reference objects

Reference vector field

  • ft = f0 + m

i=1

ui(t)fi

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 3 / 20

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Aim of the talk

To give necessary conditions and sufficient conditions for the strong local optimality of ( u, ξ) in the case when the couple is totally singular ( u(t) ∈ int U) partially singular ( only some control maps take interior values ) Strong local optimality (SLO) Roughly speaking:

  • ξ is a strong local minimizer if it is a minimizer w.r.t. admissible trajectories ξ

”near in graph” to ξ independently of the values of the control functions SLO is a strong property : the optimality has to be independent of the control set, hence the sufficient conditions have to hold true also for unbounded control sets

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 4 / 20

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Aim of the talk

To give necessary conditions and sufficient conditions for the strong local optimality of ( u, ξ) in the case when the couple is totally singular ( u(t) ∈ int U) partially singular ( only some control maps take interior values ) Strong local optimality (SLO) Roughly speaking:

  • ξ is a strong local minimizer if it is a minimizer w.r.t. admissible trajectories ξ

”near in graph” to ξ independently of the values of the control functions SLO is a strong property : the optimality has to be independent of the control set, hence the sufficient conditions have to hold true also for unbounded control sets

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 4 / 20

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Aim of the talk

To give necessary conditions and sufficient conditions for the strong local optimality of ( u, ξ) in the case when the couple is totally singular ( u(t) ∈ int U) partially singular ( only some control maps take interior values ) Strong local optimality (SLO) Roughly speaking:

  • ξ is a strong local minimizer if it is a minimizer w.r.t. admissible trajectories ξ

”near in graph” to ξ independently of the values of the control functions SLO is a strong property : the optimality has to be independent of the control set, hence the sufficient conditions have to hold true also for unbounded control sets

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 4 / 20

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Partially singular trajectories

the control set U is a box given by |ui| ≤ ai (possibly unbounded in some direction) the reference control map the reference control maps are either bang-bang or singular | ui(t)| < ai , i = 1, . . . , m1 and | ui(t)| = ai , i = m1 + 1, . . . , m the subsystem I consider also the subsystem where only the singular controls vary : ˙ ξ(t) =

  • ft + m1

i=1 ui(t)fi

  • ξ(t)

with u1 ∈ U1 ⊂ Rm1, int U1 = ∅.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 5 / 20

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Partially singular trajectories

the control set U is a box given by |ui| ≤ ai (possibly unbounded in some direction) the reference control map the reference control maps are either bang-bang or singular | ui(t)| < ai , i = 1, . . . , m1 and | ui(t)| = ai , i = m1 + 1, . . . , m the subsystem I consider also the subsystem where only the singular controls vary : ˙ ξ(t) =

  • ft + m1

i=1 ui(t)fi

  • ξ(t)

with u1 ∈ U1 ⊂ Rm1, int U1 = ∅.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 5 / 20

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Partially singular trajectories

the control set U is a box given by |ui| ≤ ai (possibly unbounded in some direction) the reference control map the reference control maps are either bang-bang or singular | ui(t)| < ai , i = 1, . . . , m1 and | ui(t)| = ai , i = m1 + 1, . . . , m the subsystem I consider also the subsystem where only the singular controls vary : ˙ ξ(t) =

  • ft + m1

i=1 ui(t)fi

  • ξ(t)

with u1 ∈ U1 ⊂ Rm1, int U1 = ∅.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 5 / 20

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Optimality conditions with unbounded controls

totally singular: U = Rm partially singular: consider the subsystem with U1 = Rm1 From one hand unbounded controls are the natural setting for strong local

  • ptimality and justify the gap between necessary and sufficient conditions when

the control set is bounded. This gap appears only when the Lie algebra generated by the controlled vector fields is not involutive. From the other hand unbounded controls are possibly related to ”optimal trajectories” with jumps. Necessary conditions High Order Maximum Principle will be given Sufficient conditions The sufficient condition are given in terms of regularity conditions and of the coercivity of a suitable quadratic form.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 6 / 20

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Optimality conditions with unbounded controls

totally singular: U = Rm partially singular: consider the subsystem with U1 = Rm1 From one hand unbounded controls are the natural setting for strong local

  • ptimality and justify the gap between necessary and sufficient conditions when

the control set is bounded. This gap appears only when the Lie algebra generated by the controlled vector fields is not involutive. From the other hand unbounded controls are possibly related to ”optimal trajectories” with jumps. Necessary conditions High Order Maximum Principle will be given Sufficient conditions The sufficient condition are given in terms of regularity conditions and of the coercivity of a suitable quadratic form.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 6 / 20

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SLIDE 16

Optimality conditions with unbounded controls

totally singular: U = Rm partially singular: consider the subsystem with U1 = Rm1 From one hand unbounded controls are the natural setting for strong local

  • ptimality and justify the gap between necessary and sufficient conditions when

the control set is bounded. This gap appears only when the Lie algebra generated by the controlled vector fields is not involutive. From the other hand unbounded controls are possibly related to ”optimal trajectories” with jumps. Necessary conditions High Order Maximum Principle will be given Sufficient conditions The sufficient condition are given in terms of regularity conditions and of the coercivity of a suitable quadratic form.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 6 / 20

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SLIDE 17

Optimality conditions with unbounded controls

totally singular: U = Rm partially singular: consider the subsystem with U1 = Rm1 From one hand unbounded controls are the natural setting for strong local

  • ptimality and justify the gap between necessary and sufficient conditions when

the control set is bounded. This gap appears only when the Lie algebra generated by the controlled vector fields is not involutive. From the other hand unbounded controls are possibly related to ”optimal trajectories” with jumps. Necessary conditions High Order Maximum Principle will be given Sufficient conditions The sufficient condition are given in terms of regularity conditions and of the coercivity of a suitable quadratic form.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 6 / 20

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SLIDE 18

Optimality conditions with unbounded controls

totally singular: U = Rm partially singular: consider the subsystem with U1 = Rm1 From one hand unbounded controls are the natural setting for strong local

  • ptimality and justify the gap between necessary and sufficient conditions when

the control set is bounded. This gap appears only when the Lie algebra generated by the controlled vector fields is not involutive. From the other hand unbounded controls are possibly related to ”optimal trajectories” with jumps. Necessary conditions High Order Maximum Principle will be given Sufficient conditions The sufficient condition are given in terms of regularity conditions and of the coercivity of a suitable quadratic form.

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 6 / 20

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Notation

The symplectic structure on π: T ∗M → M allows to lift vector fields on M to functions and vector fields on T ∗M Hamiltonians associated to fi , i = 0, 1, . . ., m Fi : ℓ ∈ T ∗M → ℓ , fi(πℓ) ∈ R Hamiltonian

  • Fi : T ∗M → T T ∗M

Hamiltonian vector field

Reference Hamiltonian ( possibly time-dependent )

  • Ft = F0 + m

i=1

ui(t)Fi and

  • F t =

F0 + m

i=1

ui(t) Fi Maximized Hamiltonian F max : ℓ → supu∈U (F0 + m

i=1 uiFi) (ℓ)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 7 / 20

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Notation

The symplectic structure on π: T ∗M → M allows to lift vector fields on M to functions and vector fields on T ∗M Hamiltonians associated to fi , i = 0, 1, . . ., m Fi : ℓ ∈ T ∗M → ℓ , fi(πℓ) ∈ R Hamiltonian

  • Fi : T ∗M → T T ∗M

Hamiltonian vector field

Reference Hamiltonian ( possibly time-dependent )

  • Ft = F0 + m

i=1

ui(t)Fi and

  • F t =

F0 + m

i=1

ui(t) Fi Maximized Hamiltonian F max : ℓ → supu∈U (F0 + m

i=1 uiFi) (ℓ)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 7 / 20

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Notation

The symplectic structure on π: T ∗M → M allows to lift vector fields on M to functions and vector fields on T ∗M Hamiltonians associated to fi , i = 0, 1, . . ., m Fi : ℓ ∈ T ∗M → ℓ , fi(πℓ) ∈ R Hamiltonian

  • Fi : T ∗M → T T ∗M

Hamiltonian vector field

Reference Hamiltonian ( possibly time-dependent )

  • Ft = F0 + m

i=1

ui(t)Fi and

  • F t =

F0 + m

i=1

ui(t) Fi Maximized Hamiltonian F max : ℓ → supu∈U (F0 + m

i=1 uiFi) (ℓ)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 7 / 20

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Pontryagin Maximum Principle

if ξ is a minimizer then there are p0 ∈ {0, 1} and an absolutely continuous solution λ: [0, T] → T ∗M of ˙ λ(t) =

  • F t(λ(t))

such that π λ(t) = ξ(t) , p0 and λ are not both zero, and transversality

  • λ(0)|T

ξ(0)N0 = p0dc0(

ξ(0)),

  • λ(

T)|T

ξ( T )Nf = −p0dcf(

ξ( T)) Mayer

  • λ(0)|T

ξ(0)N0 = 0

  • λ(

T)|T

ξ( T )Nf = 0

minimum time maximality

  • Ft ◦

λ(t) = F max ◦ λ(t) Mayer

  • Ft ◦

λ(t) = F max ◦ λ(t) ≡ p0 minimum time

  • λ is a Pontryagin extremal and it is also called the adjont convector

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 8 / 20

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Pontryagin Maximum Principle

if ξ is a minimizer then there are p0 ∈ {0, 1} and an absolutely continuous solution λ: [0, T] → T ∗M of ˙ λ(t) =

  • F t(λ(t))

such that π λ(t) = ξ(t) , p0 and λ are not both zero, and transversality

  • λ(0)|T

ξ(0)N0 = p0dc0(

ξ(0)),

  • λ(

T)|T

ξ( T )Nf = −p0dcf(

ξ( T)) Mayer

  • λ(0)|T

ξ(0)N0 = 0

  • λ(

T)|T

ξ( T )Nf = 0

minimum time maximality

  • Ft ◦

λ(t) = F max ◦ λ(t) Mayer

  • Ft ◦

λ(t) = F max ◦ λ(t) ≡ p0 minimum time

  • λ is a Pontryagin extremal and it is also called the adjont convector

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 8 / 20

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Pontryagin Maximum Principle

if ξ is a minimizer then there are p0 ∈ {0, 1} and an absolutely continuous solution λ: [0, T] → T ∗M of ˙ λ(t) =

  • F t(λ(t))

such that π λ(t) = ξ(t) , p0 and λ are not both zero, and transversality

  • λ(0)|T

ξ(0)N0 = p0dc0(

ξ(0)),

  • λ(

T)|T

ξ( T )Nf = −p0dcf(

ξ( T)) Mayer

  • λ(0)|T

ξ(0)N0 = 0

  • λ(

T)|T

ξ( T )Nf = 0

minimum time maximality

  • Ft ◦

λ(t) = F max ◦ λ(t) Mayer

  • Ft ◦

λ(t) = F max ◦ λ(t) ≡ p0 minimum time

  • λ is a Pontryagin extremal and it is also called the adjont convector

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 8 / 20

slide-25
SLIDE 25

Pontryagin Maximum Principle

if ξ is a minimizer then there are p0 ∈ {0, 1} and an absolutely continuous solution λ: [0, T] → T ∗M of ˙ λ(t) =

  • F t(λ(t))

such that π λ(t) = ξ(t) , p0 and λ are not both zero, and transversality

  • λ(0)|T

ξ(0)N0 = p0dc0(

ξ(0)),

  • λ(

T)|T

ξ( T )Nf = −p0dcf(

ξ( T)) Mayer

  • λ(0)|T

ξ(0)N0 = 0

  • λ(

T)|T

ξ( T )Nf = 0

minimum time maximality

  • Ft ◦

λ(t) = F max ◦ λ(t) Mayer

  • Ft ◦

λ(t) = F max ◦ λ(t) ≡ p0 minimum time

  • λ is a Pontryagin extremal and it is also called the adjont convector

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 8 / 20

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Consequences for singular trajectories

totally singular trajectories Fi ◦ λ(t) ≡ 0 , i = 1, . . . , m F0i ◦ λ(t) := {F0, Fi} ◦ λ(t) ≡ 0 , i = 1, . . . , m partially singular trajectories Fi ◦ λ(t) ≡ 0 , i = 1, . . . , m1 F0i ◦ λ(t) := {F0, Fi} ◦ λ(t) ≡ 0 , i = 1, . . . , m1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 9 / 20

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Consequences for singular trajectories

totally singular trajectories Fi ◦ λ(t) ≡ 0 , i = 1, . . . , m F0i ◦ λ(t) := {F0, Fi} ◦ λ(t) ≡ 0 , i = 1, . . . , m partially singular trajectories Fi ◦ λ(t) ≡ 0 , i = 1, . . . , m1 F0i ◦ λ(t) := {F0, Fi} ◦ λ(t) ≡ 0 , i = 1, . . . , m1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 9 / 20

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Further necessary conditions

Goh Condition {Fi, Fj} ◦ λ(t) = λ(t), [fi, fj]( ξ(t)) = 0, i, j = 1, . . . , m set m = m1 for the partially singular case Generalized Legiandre Condition the quadratic form on Rm v →

m

  • i,j=1

vivj{Fi, {Fj, Ft}} ◦ λ(t) ≤ 0 a.e. t ∈ [0, T] set m = m1 for the partially singular case

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 10 / 20

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Further necessary conditions

Goh Condition {Fi, Fj} ◦ λ(t) = λ(t), [fi, fj]( ξ(t)) = 0, i, j = 1, . . . , m set m = m1 for the partially singular case Generalized Legiandre Condition the quadratic form on Rm v →

m

  • i,j=1

vivj{Fi, {Fj, Ft}} ◦ λ(t) ≤ 0 a.e. t ∈ [0, T] set m = m1 for the partially singular case

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 10 / 20

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High Order Maximum Principle

Suppose that ξ is a totally (partially) singular minimizer for the problem with U = Rm (U1 = Rm1 ), then there is an adjoint convector λ which satisfies the previous properties and HOGC: F ◦ λ(t) ≡ 0 ∀ f ∈ Lie{f1, . . . , fm} (Lie{f1, . . . , fm1}) The result was known for u ∈ C∞ and we proved for u ∈ L∞ in the minimum time problem. The proof is based on high order cones of needle-like variations with ”good properties” and we used a result by R.M.Bianchini which apply both to Mayer and minimum-time problems. Open problem Prove other high order conditions, possibly for u ∈ L1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 11 / 20

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SLIDE 31

High Order Maximum Principle

Suppose that ξ is a totally (partially) singular minimizer for the problem with U = Rm (U1 = Rm1 ), then there is an adjoint convector λ which satisfies the previous properties and HOGC: F ◦ λ(t) ≡ 0 ∀ f ∈ Lie{f1, . . . , fm} (Lie{f1, . . . , fm1}) The result was known for u ∈ C∞ and we proved for u ∈ L∞ in the minimum time problem. The proof is based on high order cones of needle-like variations with ”good properties” and we used a result by R.M.Bianchini which apply both to Mayer and minimum-time problems. Open problem Prove other high order conditions, possibly for u ∈ L1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 11 / 20

slide-32
SLIDE 32

High Order Maximum Principle

Suppose that ξ is a totally (partially) singular minimizer for the problem with U = Rm (U1 = Rm1 ), then there is an adjoint convector λ which satisfies the previous properties and HOGC: F ◦ λ(t) ≡ 0 ∀ f ∈ Lie{f1, . . . , fm} (Lie{f1, . . . , fm1}) The result was known for u ∈ C∞ and we proved for u ∈ L∞ in the minimum time problem. The proof is based on high order cones of needle-like variations with ”good properties” and we used a result by R.M.Bianchini which apply both to Mayer and minimum-time problems. Open problem Prove other high order conditions, possibly for u ∈ L1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 11 / 20

slide-33
SLIDE 33

High Order Maximum Principle

Suppose that ξ is a totally (partially) singular minimizer for the problem with U = Rm (U1 = Rm1 ), then there is an adjoint convector λ which satisfies the previous properties and HOGC: F ◦ λ(t) ≡ 0 ∀ f ∈ Lie{f1, . . . , fm} (Lie{f1, . . . , fm1}) The result was known for u ∈ C∞ and we proved for u ∈ L∞ in the minimum time problem. The proof is based on high order cones of needle-like variations with ”good properties” and we used a result by R.M.Bianchini which apply both to Mayer and minimum-time problems. Open problem Prove other high order conditions, possibly for u ∈ L1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 11 / 20

slide-34
SLIDE 34

From High Order Maximum Principle

by deriving HOGC along the adjoint covector {F0, F} ◦ λ(t) ≡ 0, ∀ f ∈ Lie{f1, . . . , fm}:= L Generalized Legiandre Condition for ℓ ∈ T ∗M define the Generalized Legiandre quadratic form on Rm L(ℓ) : v →

m

  • i,j=1

vivj{Fi, {Fj, F0}}(ℓ) the GLC becomes L ◦ λ(t) ≤ 0 t ∈ [0, T] partially singular case set m = m1 in the above equations

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 12 / 20

slide-35
SLIDE 35

From High Order Maximum Principle

by deriving HOGC along the adjoint covector {F0, F} ◦ λ(t) ≡ 0, ∀ f ∈ Lie{f1, . . . , fm}:= L Generalized Legiandre Condition for ℓ ∈ T ∗M define the Generalized Legiandre quadratic form on Rm L(ℓ) : v →

m

  • i,j=1

vivj{Fi, {Fj, F0}}(ℓ) the GLC becomes L ◦ λ(t) ≤ 0 t ∈ [0, T] partially singular case set m = m1 in the above equations

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 12 / 20

slide-36
SLIDE 36

From High Order Maximum Principle

by deriving HOGC along the adjoint covector {F0, F} ◦ λ(t) ≡ 0, ∀ f ∈ Lie{f1, . . . , fm}:= L Generalized Legiandre Condition for ℓ ∈ T ∗M define the Generalized Legiandre quadratic form on Rm L(ℓ) : v →

m

  • i,j=1

vivj{Fi, {Fj, F0}}(ℓ) the GLC becomes L ◦ λ(t) ≤ 0 t ∈ [0, T] partially singular case set m = m1 in the above equations

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 12 / 20

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SLIDE 37

Sufficient conditions

The written results concern the minimum-time problem and a totally singular arc. The Mayer problem is now in progress. Partially singular trajectories The conditions apply when the bang reference controls do not switch:

  • ft = f0 + m

i=m1+1(±ai)fi + m1 i=1

ui(t)fi := ˜ f0 + m1

i=1

ui(t)fi.

The general case is under study

The sufficient conditions for ξ to be a strong local minimizer of the subsystem ˙ ξ(t) =

  • ˜

f0 + m1

i=1 ui(t)fi

  • ξ(t)

give the strong-local optimality of ξ w.r.t. the admissible trajectories of the original system. The result can be obtained thanks to the Hamiltonian approach

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 13 / 20

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SLIDE 38

Sufficient conditions

The written results concern the minimum-time problem and a totally singular arc. The Mayer problem is now in progress. Partially singular trajectories The conditions apply when the bang reference controls do not switch:

  • ft = f0 + m

i=m1+1(±ai)fi + m1 i=1

ui(t)fi := ˜ f0 + m1

i=1

ui(t)fi.

The general case is under study

The sufficient conditions for ξ to be a strong local minimizer of the subsystem ˙ ξ(t) =

  • ˜

f0 + m1

i=1 ui(t)fi

  • ξ(t)

give the strong-local optimality of ξ w.r.t. the admissible trajectories of the original system. The result can be obtained thanks to the Hamiltonian approach

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 13 / 20

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SLIDE 39

Sufficient conditions

The written results concern the minimum-time problem and a totally singular arc. The Mayer problem is now in progress. Partially singular trajectories The conditions apply when the bang reference controls do not switch:

  • ft = f0 + m

i=m1+1(±ai)fi + m1 i=1

ui(t)fi := ˜ f0 + m1

i=1

ui(t)fi.

The general case is under study

The sufficient conditions for ξ to be a strong local minimizer of the subsystem ˙ ξ(t) =

  • ˜

f0 + m1

i=1 ui(t)fi

  • ξ(t)

give the strong-local optimality of ξ w.r.t. the admissible trajectories of the original system. The result can be obtained thanks to the Hamiltonian approach

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 13 / 20

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SLIDE 40

Sufficient conditions

The written results concern the minimum-time problem and a totally singular arc. The Mayer problem is now in progress. Partially singular trajectories The conditions apply when the bang reference controls do not switch:

  • ft = f0 + m

i=m1+1(±ai)fi + m1 i=1

ui(t)fi := ˜ f0 + m1

i=1

ui(t)fi.

The general case is under study

The sufficient conditions for ξ to be a strong local minimizer of the subsystem ˙ ξ(t) =

  • ˜

f0 + m1

i=1 ui(t)fi

  • ξ(t)

give the strong-local optimality of ξ w.r.t. the admissible trajectories of the original system. The result can be obtained thanks to the Hamiltonian approach

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 13 / 20

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SLIDE 41

From now on consider the system ˙ ξ(t) = (f0 + m

i=1 ui(t)fi) ◦ ξ(t)

ξ(0) ∈ N0, ξ(T ) ∈ Nf, u ∈ Rm with the following regularity assumption RA1 L has constant dimension r ≥ m Ix := integral manifold of L through x

For the minimum time problem require also

N0 ⊂ I

ξ(0)

NT ⊂ I

ξ( T )

For the partially singular case set f0 = ˜ f0 and m = m1

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 14 / 20

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SLIDE 42

Regularity assumptions I

The adjoint convector λ : [0, T] → T ∗M satisfies RA2 The High Order Maximum Principle in the normal form. RA3 The Strong Generalized Legendre Condition L ◦ λ(t) : v →

m

  • i,j=1

vivj{Fi, {Fj, F0}} ◦ λ(t) < 0, t ∈ [0, T] ⇓

  • u ∈ C∞([0,

T], Rm)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 15 / 20

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SLIDE 43

Regularity assumptions I

The adjoint convector λ : [0, T] → T ∗M satisfies RA2 The High Order Maximum Principle in the normal form. RA3 The Strong Generalized Legendre Condition L ◦ λ(t) : v →

m

  • i,j=1

vivj{Fi, {Fj, F0}} ◦ λ(t) < 0, t ∈ [0, T] ⇓

  • u ∈ C∞([0,

T], Rm)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 15 / 20

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SLIDE 44

Regularity assumptions I

The adjoint convector λ : [0, T] → T ∗M satisfies RA2 The High Order Maximum Principle in the normal form. RA3 The Strong Generalized Legendre Condition L ◦ λ(t) : v →

m

  • i,j=1

vivj{Fi, {Fj, F0}} ◦ λ(t) < 0, t ∈ [0, T] ⇓

  • u ∈ C∞([0,

T], Rm)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 15 / 20

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SLIDE 45

Regularity assumptions II

Fundamental sub-manifolds of T ∗M Σ = {ℓ ∈ T ∗M : F(ℓ) = 0 , ∀ f ∈ L} S = {ℓ ∈ Σ: {F0, F}(ℓ) = 0 , ∀ f ∈ L}

  • λ(t) is in S

( ⇐ = HOMP ) properties coming from RA1–RA3 the codimension of Σ is r

  • F0 is tangent to Σ only on S

Lie{ F1, . . . , Fm} is tangent to Σ

  • F1, . . . ,

Fm are transversal to S thus codim S in Σ is ≥ m RA4 The codimension of S in Σ is = m

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 16 / 20

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SLIDE 46

Regularity assumptions II

Fundamental sub-manifolds of T ∗M Σ = {ℓ ∈ T ∗M : F(ℓ) = 0 , ∀ f ∈ L} S = {ℓ ∈ Σ: {F0, F}(ℓ) = 0 , ∀ f ∈ L}

  • λ(t) is in S

( ⇐ = HOMP ) properties coming from RA1–RA3 the codimension of Σ is r

  • F0 is tangent to Σ only on S

Lie{ F1, . . . , Fm} is tangent to Σ

  • F1, . . . ,

Fm are transversal to S thus codim S in Σ is ≥ m RA4 The codimension of S in Σ is = m

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 16 / 20

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SLIDE 47

Regularity assumptions II

Fundamental sub-manifolds of T ∗M Σ = {ℓ ∈ T ∗M : F(ℓ) = 0 , ∀ f ∈ L} S = {ℓ ∈ Σ: {F0, F}(ℓ) = 0 , ∀ f ∈ L}

  • λ(t) is in S

( ⇐ = HOMP ) properties coming from RA1–RA3 the codimension of Σ is r

  • F0 is tangent to Σ only on S

Lie{ F1, . . . , Fm} is tangent to Σ

  • F1, . . . ,

Fm are transversal to S thus codim S in Σ is ≥ m RA4 The codimension of S in Σ is = m

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 16 / 20

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SLIDE 48

Regularity assumptions II

Fundamental sub-manifolds of T ∗M Σ = {ℓ ∈ T ∗M : F(ℓ) = 0 , ∀ f ∈ L} S = {ℓ ∈ Σ: {F0, F}(ℓ) = 0 , ∀ f ∈ L}

  • λ(t) is in S

( ⇐ = HOMP ) properties coming from RA1–RA3 the codimension of Σ is r

  • F0 is tangent to Σ only on S

Lie{ F1, . . . , Fm} is tangent to Σ

  • F1, . . . ,

Fm are transversal to S thus codim S in Σ is ≥ m RA4 The codimension of S in Σ is = m

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 16 / 20

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SLIDE 49

Hamiltonian approach

The hamiltonian approach consists in lifting trajectories to the cotangent bundle and compare the costs there a super-Hamiltonian (possibly time–dependent) Ht s.t. Ht ≥ F max, Ht ◦ ˆ λ(t) = F max ◦ ˆ λ(t) a suitable horizontal Lagrangian sub–manifold Λ s.t. Λ = {ℓ = dα(x), x ∈ M} πHt : Λ → M loc. invertible [0, T] × Λ [0, T] × T ∗M [0, T] × M id ×H id ×π (id ×πH)−1 ω = (id × H)∗(pdq − Htdt) is exact on [0, T] × Λ ⇒

  • ω = 0

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 17 / 20

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SLIDE 50

Leading ideas Use the regularity conditions to define Ht. Require the coercivity of a suitable second order approximation. This allows to define Λ and to prove that πHt|Λ is locally invertible. The idea is to choose Λ on Σ and to define an Ht whose flow is tangent to Σ RA1–RA4 imply that we can define H0 s.t. H0(ℓ)

  • ≥ F max
  • n Σ

= F max

  • n S

and

  • H0 is tangent to Σ

The super-Hamiltonian Ht is obtained by modifying the reference Hamiltonian Ht = H0 +

m

  • i=1
  • ui(t)Fi

(m = m1 in the partially singular case)

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 18 / 20

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SLIDE 51

2nd Order Condition - Minimum Time Problem

pullback vector fields define: gi

t =

S−1

t∗ f i 1 ◦

St, = ⇒ ˙ gi

t =

S−1

t∗ [f0, fi] ◦

St, i = 1, . . . , m the Hilbert space Let W be the subspace of the Hilbert space T

ξ(0))M × L2([0,

T], Rm) defined by ˙ ζ(t) =

m

  • i=1

wi(t)˙ gi

t(

ξ(0)) , ζ(0) = δx ∈ L( ξ(0)) = T

ξ(0)I ξ(0) ,

ζ( T) = 0 A5 - second order condition the quadratic form J′′ : (δx, w) → 1 2

m

  • i=1

 

  • T

2wi(t)Lζ(t)L ˙

gi

tβ(

ξ(0)) +

m

  • j=1

wi(t)wj(t)Rij(t) dt   is coercive on W

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 19 / 20

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SLIDE 52

The result

If all the regularity assumptions RA1,RA2,RA3, RA4 hold true and the second

  • rder approximation is coercive, i.e. A5 is satisfied

then

  • ξ is a strong-local optimizer for the minimum-time problem between I

ξ(0) and

I

ξ( T )

  • ξ is strictly optimal between

ξ(0) and I

ξ( T ) and between I ξ(0) and

ξ( T )

for the partially singular case I

ξ(0) and I ξ( T ) are integral manifolds of the Lie - algebra

defined by the subsystem

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 20 / 20

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SLIDE 53

The result

If all the regularity assumptions RA1,RA2,RA3, RA4 hold true and the second

  • rder approximation is coercive, i.e. A5 is satisfied

then

  • ξ is a strong-local optimizer for the minimum-time problem between I

ξ(0) and

I

ξ( T )

  • ξ is strictly optimal between

ξ(0) and I

ξ( T ) and between I ξ(0) and

ξ( T )

for the partially singular case I

ξ(0) and I ξ( T ) are integral manifolds of the Lie - algebra

defined by the subsystem

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 20 / 20

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SLIDE 54

The result

If all the regularity assumptions RA1,RA2,RA3, RA4 hold true and the second

  • rder approximation is coercive, i.e. A5 is satisfied

then

  • ξ is a strong-local optimizer for the minimum-time problem between I

ξ(0) and

I

ξ( T )

  • ξ is strictly optimal between

ξ(0) and I

ξ( T ) and between I ξ(0) and

ξ( T )

for the partially singular case I

ξ(0) and I ξ( T ) are integral manifolds of the Lie - algebra

defined by the subsystem

G.Stefani (DiMaI) Strong local optimality of singular trajectories hhp - Paris - November 2014 20 / 20