Second-order optimality conditions in Pontryagin form for optimal - - PowerPoint PPT Presentation

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Second-order optimality conditions in Pontryagin form for optimal control problems

• J. Fr´

ed´ eric Bonnans∗, Xavier Dupuis∗, and Laurent Pfeiffer∗

∗INRIA Saclay and CMAP, Ecole Polytechnique (France)

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Introduction

Goal: Study of 2nd-order conditions for smooth optimal control problems of ODEs with pure and mixed constraints. Specificity: We consider strong solutions. Classical results are strengthened and expressed with Pontryagin’s multipliers. Our tools: Necessary conditions: use of relaxation Sufficient conditions: use of a decomposition principle Possible applications: shooting methods, discretization methods, characterization of local optimality...

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Setting

General references: [Bonnans, Shapiro ’00], [Kawazaki ’88],... Consider the abstract optimization problem Min

x∈X f (x) subject to g(x) ∈ K,

where f : X → R and g : X → Y are C 2, X and Y are Hilbert spaces, K a convex set of K. Let ¯ x, Robinson qualification condition (RQC) holds iff ∃ε > 0, εB ⊂ g(¯ x) + Dg(¯ x)X − K, where B is the unit ball.

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Setting

For x ∈ X, λ ∈ Y ∗, the Lagrangian is L[λ](x) = f (x) + λ, g(x). The Lagrange multipliers at ¯ x are defined by ΛL = {λ ∈ Y ∗, s.t. λ ∈ NK(g(¯ x)), DxL[λ](¯ x) = 0}. The critical cone C(¯ x) and the quasi-radial critical cone C QR(¯ x) are defined by C(¯ x) =

• h ∈ X, s.t. Df (¯

x)h = 0, Dg(¯ x)h ∈ TK(¯ x)

• C QR(¯

x) =

• h ∈ C(¯

x), s.t. distK(g(¯ x) + θDg(¯ x)h) = o(θ2)

• .

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Necessary optimality conditions

Proposition Assume that ¯ x is a local optimal solution and that RQC holds, then: 1st-order necessary conditions: ΛL(¯ x) is non-empty 2nd-order necessary conditions: for all h in cl(C QR(¯ x)), max

λ∈ΛL

D2

xxL[λ](¯

x)h2 ≥ 0. NB: we will work in the framework of the extended polyhedrecity condition, C(¯ x) = cl(C QR(¯ x)).

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Proof

• Proof. Let h ∈ C QR(¯

x), d ∈ X be such that Dg(¯ x)d + 1 2D2g(¯ x)h2 ∈ TK(g(¯ x)), with a metric regularity result, we show the existence of a mapping x : [0, 1] → X satisfying g(x(θ)) ∈ K and x(θ) = ¯ x + θh + θ2d + o(θ2). Then, f (x(θ)) − f (¯ x) =

• Df (¯

x)d + 1 2D2f (¯ x)h2 θ2 + o(θ2) ≥ 0.

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Proof

It follows that the following problem    Min

d∈X

Df (¯ x)d + 1

2D2f (¯

x)h2 s.t. Dg(¯ x)d + 1

2D2g(¯

x)h2 ∈ TK(g(¯ x)). has a nonnegative value. Its dual has the same value: Max

λ∈ΛL

D2L[λ](¯ x)h2. The result extends to cl(C QR(¯ x)).

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Quadratic growth

The 2nd-order sufficient conditions are: for all h ∈ C(¯ x)\0, Max

λ∈Λ D2 xxL[λ](¯

x)h2 > 0. A quadratic form Q : X → R is said to be a Legendre form if it is weakly lower semi-continuous hn ⇀ 0 and Q(hn) → 0 imply that hn → 0.

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Quadratic growth

Definition The quadratic growth property holds iff there exists ε > 0 and α > 0 such that for all x ∈ X, if g(x) ∈ K and x − ¯ x ≤ ε, then f (x) − f (¯ x) ≥ αx − ¯ x2. Proposition Assume that the 2nd-order sufficient condition holds and that D2L[λ](¯ x) is a Legendre form for all λ ∈ ΛL. Then, the quadratic growth property holds.

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Proof (by contradiction). Assume that ∃ xn → ¯ x, g(xn) ∈ K s.t. f (xn) − f (¯ x) ≤ o(xn − ¯ x2). Set hn = (xn − ¯ x)/xn − ¯ x and denote by h a weak limit point. We check that h ∈ C(¯ x). Then, for all λ ∈ ΛL(¯ x), f (xn) − f (¯ x) ≥ f (xn) − f (¯ x) + λ, g(xn) − g(¯ x) ≥ L[λ](xn) − L[λ](¯ x) = D2

xxL[λ](¯

x)(xn − ¯ x)2 + o(xn − ¯ x2). Therefore, D2

xxL[λ]h2 ≤ lim inf DxxL[λ]h2 n ≤ 0.

Thus, supλ∈ΛL D2

xxL[λ]h2 = 0 and h = 0. We obtain that hn → 0,

in contradiction with hn = 1.

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1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

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Problem

The optimal control problem is: Min

u∈U, y∈Y φ(yT)

subject to: the dynamic: ˙ yt = f (ut, yt), y0 = y0 final-state constraints: ΦE(yT) = 0, ΦI(yT) ≤ 0 pure constraints: g(yt) ≤ 0 for all t mixed constraints: c(ut, yt) ≤ 0 for a.a. t, where U = L∞(0, T; Rm) and Y = W 1,∞(0, T; Rn). The mappings are C 2 and defined in Rn, RnE , RnI , Rng , Rnc. For u ∈ U, we denote by y[u] the solution to the state equation and set J(u) = φ(yT[u]).

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Local optimal solutions

A control ¯ u is said to be a weak minimum iff ∃ε > 0 such that for all feasible u, u − ¯ u∞ ≤ ε = ⇒ J(u) ≥ J(¯ u) a Pontryagin minimum iff for all R > ¯ u∞, there exists ε > 0 such that for all feasible u, u∞ ≤ R and u − ¯ u1 ≤ ε = ⇒ J(u) ≥ J(¯ u) a bounded strong minimum iff for all R > ¯ u, there exists ε > 0 such that for all feasible u, u∞ ≤ R and y[u] − y[¯ u]∞ ≤ ε = ⇒ J(u) ≥ J(¯ u). Bounded strong min. = ⇒ Pontryagin min. = ⇒ weak min.

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Normal cones to the constraints

We set KΦ = {0}RnΦE × R

nΦI − ,

Kg = C([0, T]; Rng

− ),

Kc = L∞(0, T; Rnc

− ),

and consider their normal cones at (¯ u, ¯ y): NKΦ =

• Ψ = (ΨE, ΨI) ∈ RnE × RnI , ΨIΦI(¯

yT) = 0

• NKg =
• µ ∈ M(0, T; Rng

+ ),

T g(yt) dµt = 0

• NKc =
• ν ∈ L∞(0, T; Rnc),

T c(ut, yt)νt dt = 0

• Let ¯

u be feasible, ¯ y = y[¯ u]. The normal cone of the constraints is N = NKΦ × NKg × NKc.

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Multipliers

Assumption (Inward pointing condition) There exists v ∈ U and ε > 0 such that for a.a. t, c(¯ ut, ¯ yt) + Duc(¯ ut, ¯ yt)vt ≤ −ε. This assumption enables to consider multipliers (for the mixed constraint) in L∞(0, T; Rnc) instead of (L∞(0, T; Rnc))∗. We define for all u ∈ Rm, y ∈ Rn, p ∈ Rn, ν ∈ Rnc the end-point Lagrangian Φ[β, Ψ](yT) = βφ(yT) + ΨΦ(yT), the Hamiltonian H[p](u, y) = pf (u, y), the augmented Hamiltonian Ha[p, ν](u, y) = pf (u, y) + νc(u, y).

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Lagrange multipliers

Let λ = (β, Ψ, µ, ν), with β ∈ R+ and (Ψ, µ, ν) ∈ N(¯ u), we call associated costate the solution pλ ∈ BV (0, T; Rn) to

• dpt =

DyHa[pt, νt] dt + Dyg(yt) dµt, pT+ = DyT Φ[β, Ψ](yT). We define the set ΛL of generalized Lagrange multipliers as follows: ΛL =

• (β, Ψ, µ, ν), β ≥ 0, (Ψ, µ, ν) ∈ N(¯

u), DuHa[pλ

t , νt](¯

ut, ¯ yt) = 0, for a.a. t

• \ {(0, 0, 0, 0)}.

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Pontryagin multipliers

We define the multimapping U(t) = cl

• u ∈ Rm, c(u, ¯

yt) < 0

• .

This is the set of feasible controls, for the mixed constraint. We define the set ΛP of Pontryagin multipliers as follows: ΛP =

• λ = (β, Ψ, µ, ν) ∈ ΛL,

H[pλ

t ](u, ¯

yt) ≥ H[pλ

t ](¯

ut, ¯ yt), for a.a. t, for all u ∈ U(t)

• NB: It is not possible to define U(t) by

U(t) =

• u ∈ Rm, c(u, ¯

yt) ≤ 0

• .

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1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

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Necessary conditions

Technical difficulties in stating 2nd-order necessary conditions: Two spaces arise naturally,

L∞ for the Taylor expansions L2 for the definition of the Hessian of the Lagrangian

Finding quasi-radial critical directions to ensure the extended polyhedricity condition.

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Touch points

We say that τ ∈ [0, T] is a touch point for the constraint gi iff it is isolated among {t, gi(¯ yt) = 0}. It is reducible iff τ ∈ (0, T),

d2 dt2 gi(¯

yt) is defined for t close to τ, continuous at τ and d2 dt2 gi(t, ¯ yt)|t=τ < 0. We define: Tg,i :=

• ∅

if gi is of order 1, {touch points for gi} if gi is of order at least 2. The pure constraint gi is of order qi if the associated state variable is originally controled by its qi-th derivative.

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Contact sets and linearization

Contact sets of the pure constraints ∆0

g,i :={t ∈ [0, T] : gi(t, ¯

yt) = 0} \ Tg,i, ∆ε

g,i :={t ∈ [0, T] : dist(t, ∆0 g,i) ≤ ε}

Contact sets of the mixed constraints ∆δ

c,i := {t ∈ [0, T] : ci(¯

ut, ¯ yt) ≥ −δ} For a given v ∈ L2(0, T; Rm) := U2, we define the linearized state equation z[v] ∈ W 1,2(0, T; Rn) := Y2, solution to ˙ zt[v]t = Df (¯ ut, ¯ yt)(vt, zt), z0[v] = 0.

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Supplementary assumptions

Assumption (Structure of contact sets) For 1 ≤ i ≤ ng, the set Tg,i is finite and contains only reducible touch points, ∆0

g,i is the reunion of finitely many intervals and gi is of finite

• rder qi.

Assumption (Surjectivity) There exists δ, ε > 0 such that the linear mapping from U2 to nc

i=1 L2(∆δ c,i) × ng i=1 W qi,2(∆ε g,i) defined by

v →   

• Dci(¯

ut, ¯ y)(vt, zt[v])|∆δ

c,i

• 1≤i≤nc
• Dgi(¯

yt)zt[v]|∆ε

g,i

• 1≤i≤ng

   is onto.

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Critical cones and quadratic form

We consider the critical cone C2 in U2 C2 :=                (v, z) ∈ U2 × Y2 : z = z[v] Dφ(¯ yT)(zT) ≤ 0 DΦ(¯ yT)(z0, zT) ∈ TKΦ(Φ(¯ y0, ¯ yT)) Dc(¯ ut, ¯ yt)(vt, zt) ∈ TKc(c(¯ u, ¯ y)) Dg(¯ yt)zt ∈ TKg (g(¯ y))                and the strict critical cone C S

2 :=

     (v, z) ∈ C2(¯ u, ¯ y) : Dci(¯ ut, ¯ yt)(vt, zt) = 0 t ∈ ∆0

c,i

1 ≤ i ≤ nc Dgi(¯ yt)zt = 0 t ∈ ∆0

g,i

1 ≤ i ≤ ng      . NB: If there exists λ ∈ ΛL such that νi

t > 0 for a.a. t in ∆0 c,i and

such that supp(µi) ∩ ∆0

g,i = ∆0 g,i, then C S 2 = C2.

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Quadratic form

For λ = (β, Ψ, ν, µ) ∈ ΛL, we define the Hessian of the the Lagrangian Ω[λ]: U2 × Y2 → R by Ω[λ](v, z) := T D2Ha[pλ

t , νt](¯

ut, ¯ yt)(vt, zt)2 dt + D2Φ[β, Ψ](¯ yT)(zT)2 +

• [0,T]

D2g(¯ yt)(zt)2 dµt −

• τ∈Tg,i

1≤i≤ng

µi(τ)(d/dt g(¯ yτ)zτ)2 d2/dt2 g(¯ yτ) .

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Weak conditions 2nd-order necessary conditions

Theorem ([Stefani, Zezza ’96], [Bonnans, Hermant ’09],...) Assume that ¯ u is weak minimum, and that the assumptions on the structure of contact sets and surjectivity hold. Then, for all (v, z) in C s

2(¯

u, ¯ y), there exists λ ∈ ΛL such that Ω[λ](v, z) ≥ 0.

• Proof. A reduction approach is used. The extended polyhedricity

condition is satisified thanks to the surjectivity condition.

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Reduction

For all τ ∈ Ti,g, consider the mapping Θε

i,τ(y) =

sup

t∈[τ−ε,τ+ε]

gi(yt). The pure constraint gi(yt) ≤ 0 is reformulated as follows, for ε > 0 small enough: for all τ ∈ Ti,g, Θε

i,τ(y) ≤ 0

for all t ∈ [0, T]\

• ∪τ∈Tg,i [τ − ε, τ + ε]
• , gi(yt) ≤ 0.

The mappings Θε

i,τ are twice Fr´

echet-differentiable and allow to

• btain the terms
• Dg(1)

i

(¯ yτ)zτ 2 g(2)

i

(¯ uτ, ¯ yτ) in the Hessian.

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1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

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Castaing representation

Goal: 2nd-order necessary conditions for Pontryagin minimum Method: we apply weak conditions to auxiliary problems. Remember that U(t) was defined as U(t) = cl

• u ∈ Rm, c(u, ¯

yt) < 0

• .

Proposition There exists a sequence (called Castaing representation of U) (uk)k in U such that For all k, ∃ε > 0, such that for a.a. t, c(uk

t , ¯

yt) < −ε For a.a. t, {uk

t }k is dense in U(t).

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Relaxation of the dynamic

Let N > 0, let α ∈ L∞(0, T; RN

+) be such that N i=1 αi t ≤ 1.

Denote by y[u, α] the solution y to the relaxed state equation: ˙ yt =

• 1 −

N

• i=1

αi

t

• f (u, yt) +

N

• i=1

αi

tf (ui, yt).

Proposition ([Dmitruk ’07], [Gamkrelidze ’78],...) For all ε > 0, for all α ≥ 0 such that N

i=1 αi t ≤ 1, for all u, there

exists ˜ u such that: y[u, α] − y[˜ u]∞ ≤ ε ˜ u − u1 = O(N

i=1 αi∞).

• Proof. Build ˜

u by oscillating between u, u1,...,uN at “frequencies” (1 −

i αi t), α1 t ,...,αN t .

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Relaxation of the problem

We can consider the relaxed problem: Min

u∈U, α∈L∞(0,T;RN), y∈Y φ(yT)

subject to: the relaxed dynamic, the initial constraints, α ≥ 0. Actually, we need a small “margin” on the constraints. The relaxed problem considered is: Min

u∈U, α∈L∞(0,T;RN), y∈Y, θ∈R θ

subject to:      φ(yT) − φ(¯ yT) ≤ θ g(yt) ≤ θ for all t y = y[u, α] c(ut, yt) ≤ θ for a.a. t ΦE(yT) = 0, ΦI(yT) ≤ θ αi

t ≥ −θ for a.a. t.

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Relaxation of the problem

Note that: The reduced constraints (with Θi,τ) should have been taken into account. A discussion on the qualification of the equality constraint ΦE has to be done. Proposition If (¯ u, ¯ y) is a Pontryagin minimum to the original problem, then (¯ u, ¯ α = 0, ¯ y, ¯ θ = 0) is a weak minimum to the relaxed problem. Proof by contradiction. Clearly (¯ u, ¯ α = 0, ¯ y, ¯ θ = 0) is feasible. If it not a weak minimum, we build feasible relaxed controls (uk, yk, αk, θk) converging uniformly to (¯ u, ¯ y, ¯ α, 0) with θk < 0. We get the contradiction with the associated ˜ uk.

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Multipliers

The normal cone NR to the constraints of the relaxed problem is

• (β, Ψ, µ, ν, γ) : β ∈ R+, (Ψ, µ, ν) ∈ N(¯

u), γ ∈ L(0, T; RN

+)

• .

Moreover, the new augmented Hamiltonian is Ha

R[p, ν, γ](u, α, y)

= (1 −

• i

αi)pf (u, y) +

• i

αipf (ui, y) + νc(u, y) −

• i

γiαi. It follows that DyHa

R[p, ν, γ](¯

ut, ¯ αt, ¯ yt) = DyHa[p, ν](¯ ut, ¯ yt), DuHa

R[p, ν, γ](¯

ut, ¯ αt, ¯ yt) = DuHa[p, ν](¯ ut, ¯ yt), DαiHa

R[p, ν, γ](¯

ut, ¯ αt, ¯ yt) = H[p](ui

t, ¯

yt) − H[p](¯ ut, ¯ yt) − γi

t.

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Multipliers

Proposition The set of Lagrange multipliers ΛN

L to the relaxed problem is the

set of pairs (λ, γ) satisfying λ ∈ ΛL for a.a. t, for all i = 1, ..., N, H[pλ

t ](ui t, ¯

yt) − H[pλ

t ](¯

ut, ¯ yt) = γi

t ≥ 0

β +

i Ψi I + i

T

0 νi t dt + i

T dµi

t + i

T

0 γi t = 1.

To the limit when N → ∞, we will obtain Pontryagin multipliers (of the original problem).

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2nd-order necessary conditions in Pontryagin form

Theorem (Bonnans, Dupuis, Pfeiffer ’13) Let (¯ u, ¯ y) be a Pontryagin minimum. Then, under the same assumptions as for the weak necessary conditions, for all (v, z) ∈ C S

2 , there exists λ ∈ ΛP such that

Ω[λ](v, z) ≥ 0.

• Proof. For all N, (¯

u, ¯ α = 0, ¯ y, 0) is a weak minimum, and there exists λN ∈ ΛN

L such that Ω[λN](v, z) ≥ 0. Any weak limit point of

(λN)N satisfies the theorem and belongs to ΛP. NB: extension of [Maurer, Osmolovskii ’12].

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1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

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Conditions

Definition We say that the second-order sufficient conditions in Pontryagin form hold iff ∃ α > 0 such that for some λ∗ = (β∗, Ψ∗, ν∗, µ∗) ∈ ΛP, for a.a. t, for all u ∈ U(t), H[pλ∗

t ](u, ¯

yt) − H[pλ∗

t ](¯

ut, ¯ yt) ≥ α|u − ¯ ut|2

2,

for all (v, z) ∈ C2\{0}, sup

λ∈ΛP

Ω[λ](v, z) > 0.

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Supplementary assumptions

Assumption (Metric regularity) There exists ε > 0 such that for all (u, y) in U × Y with y − ¯ y ≤ ε, if (u, y) satisfies the mixed constraint, then there exists ˆ u such that c(ˆ ut, ¯ yt) ≤ 0, for a.a. t and u − ˆ u∞ = O(y − ¯ y∞). Assumption (Legendre form) For all λ ∈ ΛP, Ω[λ] is a Legendre form. This is satisfied if for all λ ∈ ΛP, ∃ γ > 0 such that for a.a. t, γ ≤ D2

uuHa[pλ t , νt](¯

ut, ¯ yt).

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Quadratic growth

Definition We say that the quadratic growth for bounded strong solutions holds at (¯ u, ¯ y) iff ∀R > 0, ∃ε > 0 such that for all feasible (u, y), u − ¯ u∞ ≤ R and y − ¯ y∞ ≤ ε = ⇒ J(u) − J(¯ u) ≥ u − ¯ u2

2.

Theorem (Bonnans, Dupuis, Pfeiffer ’13) Assume that the following assumptions hold: the reducibility of touch points of Tg,i the sufficient 2nd-order conditions the last two extra conditions. Then, the quadratic growth for bounded strong solutions holds.

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Convergence of controls

Proof (by contradiction). Let R > 0. Assume that there exists a sequence of feasible trajectories (uk, yk) such that: uk − ¯ u∞ ≤ R and yk − ¯ y∞ → 0 J(uk) − J(¯ u) ≤ o(uk − ¯ u2

2).

With an integration by parts, we show that J(uk) − J(¯ u) ≥ βφ(yk

T) − βφ(¯

yT) + ΨΦ(yk

T) − ΨΦ(¯

yT) ≥ DyT Φ[β∗, Ψ∗](¯ yT)(yk

T − ¯

yT) + o(1) ≥ T H[pλ∗

t ](uk t , ¯

yt) − H[pλ∗

t ](¯

ut, ¯ yt) dt + o(1) ≥ αuk − ¯ u2

2 + o(1).

Therefore uk − ¯ u2 → 0.

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Decomposition principle

We set R1,k = uk − ¯ u1 and R2,k = uk − ¯

• u2. Note that

R2,k → 0 and thus R1,k → 0. The idea now is to decompose the perturbations uk − ¯ u into “large” and “small” perturbations. We set: Ak =

• t ∈ [0, T], |uk

t − ¯

ut| ≤ R1/4

1,k

• and

Bk = Ac

k.

Note that: meas(Bk) → 0. We define uA,k

t

=

• uk

t

if t ∈ Ak , ¯ ut if t ∈ Bk. Note that: uA,k − ¯ u∞ → 0.

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Decomposition principle

We set R1,A,k = uA,k − ¯ u1 R2,A,k = uA,k − ¯ u2 R1,B,k = uk − uA,k1 R2,B,k = uk − uA,k2. and set zA,k = z[uA,k − ¯ u]. We obtain that R1,B,k = o(R2,B,k) and yk − (¯ y + zA,k)∞ = o(R2,k).

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Decomposition principle

Finally, define a control ˜ uk which is such that for a.a. t, c(˜ uk, yk

t ) ≤ 0

and ˜ uk − ¯ u∞ = O(R1,k). Theorem (Bonnans, Dupuis, Pfeiffer ’13) For all λ ∈ ΛP, the following expansion holds: J(uk) − J(¯ u) ≥

• Bk

H[pλ

t ](uk t , ¯

yt) − H[pλ

t ](˜

ut, ¯ yt) dt + Ω[λ](uA,k − ¯ u, zA,k) + o(R2

2,k).

NB: Extension of [Bonnans, Osmolovskii ’10].

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Proof of the decomposition principle

We use this dualization of the constraints: J(uk) − J(¯ u) ≥ βφ(yk

T) − βφ(¯

yT) + ΨΦ(yk

t ) − ΨΦ(¯

yT) + T (g(yt) − g(¯ yt)) dµt +

• Ak

νt(c(uA,k

t

, yk

t ) − c(¯

ut, ¯ yt)) dt +

• Bk

νt(c(˜ uk, yk

t ) − c(¯

ut, ¯ yt)) dt. The result follows with the previous estimates.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Comments

Difficulties arise from the mixed constraints: The mixed constraints are partially dualized when large perturbations occur. The quadratic growth of the Hamiltonian may not be satisfied for uk, but only for ˆ uk. Two additional terms must be removed:

• Bk

H[pλ

t ](ˆ

uk

t , ¯

yt) − H[pλ

t ](uk t , ¯

yt) dt and

• Bk

H[pλ

t ](¯

ut, ¯ yt) − H[pλ

t ](˜

uk

t , ¯

yt) dt.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Quadratic growth: end of the proof

Using assumption of metric regularity and the quadratic growth of the Hamiltonian, we find that

• Bk

H[pλ

t ](uk t , ¯

yt) − H[pλ

t ](˜

ut, ¯ yt) dt ≥ αR2

2,B,k + o(R2 2,k).

With the decomposition principle, we obtain that R2,B,k = O(R2,A,k). Finally, the difference of Hamiltonians being nonnegative, we have: Ω[λ](uA,k − ¯ u, zA,k) = o(R2

2,A,k).

The end of the proof is classical.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Theorem (Bonnans, Dupuis, Pfeiffer 13’) If the technical assumptions of the necessary and the sufficient conditions (in particular, Ω[λ] is a Legendre form) hold, then the quadratic growth for bounded strong solutions ⇐ ⇒ the 2nd-order sufficient conditions in Pontryagin form.

• Proof. =

⇒ Apply the 2nd-order necessary conditions to the

• ptimal control problem:

Min

u,y

φ(yT) + α T ut − ¯ ut2

2 dt,

with the same constraints as before.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Bibliography I

J.F. Bonnans, X. Dupuis, L. Pfeiffer, Second-order sufficient conditions for bounded strong solutions to optimal control problems, In preparation. J.F. Bonnans, X. Dupuis, L. Pfeiffer, Second-order necessary conditions in Pontryagin form for

• ptimal control problems,

In preparation.

• H. Kawazaki,

An envelope-like effect of infinitely many inequality constraints

• n second-order necessary conditions for minimization

problems,

• Math. Programming, 1988.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Bibliography II

J.F. Bonnans, A. Shapiro, Perturbation analysis of optimization problems, Springer-Verlag, New York, 2000. N.P. Osmolovski˘ ı, H. Maurer, Applications to regular and bang-bang control, Advances in Design and Control, SIAM, 2012.

• G. Stefani, P. Zezza,

Optimality conditions for a constrained optimal control problem. SIAM J. of Control Optim., 1996. J.F. Bonnans, A. Hermant, Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints, Annales de l’I.H.P., 2009.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions

Bibliography III

A.V. Dmitruk, An approximation theorem for a nonlinear control system with sliding modes,

• Math. Programming, 1988.

J.F. Bonnans, N. Osmolovski˘ ı, Second-order analysis of optimal control problems with control and initial-final state constraints.

• J. Convex Anal., 2010.

Thank you for your attention!