Sensitivity analysis for relaxed optimal control problems with - - PowerPoint PPT Presentation

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Sensitivity analysis for relaxed optimal control problems with final-state constraints

• J. Fr´

ed´ eric Bonnans∗, Laurent Pfeiffer∗, and Oana Serea†

∗INRIA Saclay and CMAP, Ecole Polytechnique (France), †University of Perpignan (France)

May 2012

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Introduction

We consider... a family of optimal control problems (OCP), parametrized by a nonnegative variable θ close to 0, its value functionV (θ), a strong solution (u, y) for the value θ = 0. Our goal: computing a second-order expansion of V (θ) near 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Introduction

The main difficulty: large perturbations in uniform norm for the control variable may occur. Our tools: the abstract methodology of sensitivity analysis, relaxation with Young measures.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

1 Relaxation of optimal control problems

Strong solutions Relaxation

2 Methodology for sensitivity analysis

Upper estimate Lower estimate

3 Application to optimal control

Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

1 Relaxation of optimal control problems

Strong solutions Relaxation

2 Methodology for sensitivity analysis

Upper estimate Lower estimate

3 Application to optimal control

Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Strong solutions

For a control u in L∞([0, T], Rm) and θ ≥ 0, consider the trajectory y[u, θ] solution of the following differential system:

• ˙

yt = f (ut, yt, θ), for a. a. t in [0, T], y0 = y0. Set K = {0nE } × RnI

+. Our family of OCP’s is

Min

u∈L∞ φ(yT[u, θ]),

s.t. Φ(yT[u], θ) ∈ K. Reference problem: θ = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Strong solutions

For a control u in L∞([0, T], Rm) and θ ≥ 0, consider the trajectory y[u, θ] solution of the following differential system:

• ˙

yt = f (ut, yt, θ), for a. a. t in [0, T], y0 = y0. Set K = {0nE } × RnI

+. Our family of OCP’s is

Min

u∈L∞ φ(yT[u, θ]),

s.t. Φ(yT[u], θ) ∈ K. Reference problem: θ = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Strong solutions

A control u is a R−strong solution for the reference problem if there exists η > 0 such that u is solution to the localized problem:    Min

||u||∞≤R φ(yT[u, 0]),

s.t. Φ(yT[u], 0) ∈ K, ||y[u, 0] − y||∞ ≤ η. Now, we fix u, R, and η.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Relaxation

Set: BR, the ball of Rm of radius R PR, the set of probabilities on BR, MR, the set of Young measures on [0, T] × BR, defined by L∞([0, T], PR). For µ ∈ MR, denote by y[µ, θ] the solution to

• ˙

yt =

• BR f (u, yt, θ) dµt(u),

y0 = y0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Relaxation

Set: BR, the ball of Rm of radius R PR, the set of probabilities on BR, MR, the set of Young measures on [0, T] × BR, defined by L∞([0, T], PR). For µ ∈ MR, denote by y[µ, θ] the solution to

• ˙

yt =

• BR f (u, yt, θ) dµt(u),

y0 = y0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Relaxation

Example: for µt = 1

2δu1

t + 1

2δu2

t , the dynamic is:

˙ yt[µ, θ] = 1 2f (u1

t , y[µ, θ], θ) + 1

2f (u2

t , y[µ, θ], θ).

and can be approximated by this control:

T

() with probability 1/2 () with probability 1/2

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Relaxation

Example: for µt = 1

2δu1

t + 1

2δu2

t , the dynamic is:

˙ yt[µ, θ] = 1 2f (u1

t , y[µ, θ], θ) + 1

2f (u2

t , y[µ, θ], θ).

and can be approximated by this control:

T

() with probability 1/2 () with probability 1/2 v()

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Relaxation

We relax the OCP’s as follows    V (θ) = Min

µ∈MRφ(yT[µ, θ]),

s.t. Φ(yT[µ], θ) ∈ K, ||y[µ, θ] − y||∞ ≤ η. Theorem Under some qualification assumptions, the relaxed and the classical OCP’s have the same value. Therefore, µt = δut is a relaxed solution for θ = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

1 Relaxation of optimal control problems

Strong solutions Relaxation

2 Methodology for sensitivity analysis

Upper estimate Lower estimate

3 Application to optimal control

Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

Consider the family of optimization problems: V (θ) = Min

x∈H f (x, θ)

s.t. g(x, θ) ∈ K, where K stands for inequalities and equalities. The Lagrangian is L(x, λ, θ) = f (x, θ) + λ, g(x, θ). Consider x an optimal solution for θ = 0, Λ the set of Lagrange multipliers associated with x.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

Consider the family of optimization problems: V (θ) = Min

x∈H f (x, θ)

s.t. g(x, θ) ∈ K, where K stands for inequalities and equalities. The Lagrangian is L(x, λ, θ) = f (x, θ) + λ, g(x, θ). Consider x an optimal solution for θ = 0, Λ the set of Lagrange multipliers associated with x.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

Let d and h be in H, set yθ = x + dθ + hθ2, then, f (yθ, θ) = f (x, 0) + Df (x, 0)(d, 1) θ +

• Dxf (x, 0)h + 1

2D2f (x, 0)(d, 1)2 θ2 + o(θ2), g(yθ, θ) = g(x, 0) + Dg(x, 0)(d, 1) θ +

• Dxg(x, 0)h + 1

2D2g(x, 0)(d, 1)2 θ2 + o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

By a regularity metric theorem, if Dg(x, 0)(d, 1) ∈ TK(g(x, 0)), and Dxg(x, 0)h + 1 2D2g(x, 0)(d, 1) ∈ T 2

K

• g(x, 0), Dg(x, 0)(d, 1)
• ,

then, there exists ˜ xθ = x + dθ + hθ2 + o(θ2) such that g(˜ xθ, θ) = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

By a regularity metric theorem, if Dg(x, 0)(d, 1) ∈ TK(g(x, 0)), and Dxg(x, 0)h + 1 2D2g(x, 0)(d, 1) ∈ T 2

K

• g(x, 0), Dg(x, 0)(d, 1)
• ,

then, there exists ˜ xθ = x + dθ + hθ2 + o(θ2) such that g(˜ xθ, θ) = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

This justifies the two linearized problems: at the first order, Min

d∈H Df (x, 0)(d, 1)

s.t. Dg(x, 0)(d, 1) ∈ TK(...). (LP) Its dual is: Max

λ∈Λ DθL(x, λ, 0).

(LD)

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

at the second order, for d in S(LP),    Min

h∈H

Dxf (x, 0)h + 1

2D2f (x, 0)(d, 1)2

s.t. Dxg(x, 0)h + 1

2D2g(x, 0)(d, 1)2 ∈ T 2 K(...).

(QP(d)) Its dual is: Max

λ∈S(LD)D2L(x, λ, 0)(d, 1)2.

(QD(d)) Finally, V (θ) ≤ V (0) + Val(LP)θ + Min

d∈S(LP)

• Val(QP(d))
• θ2 + o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Upper estimate

at the second order, for d in S(LP),    Min

h∈H

Dxf (x, 0)h + 1

2D2f (x, 0)(d, 1)2

s.t. Dxg(x, 0)h + 1

2D2g(x, 0)(d, 1)2 ∈ T 2 K(...).

(QP(d)) Its dual is: Max

λ∈S(LD)D2L(x, λ, 0)(d, 1)2.

(QD(d)) Finally, V (θ) ≤ V (0) + Val(LP)θ + Min

d∈S(LP)

• Val(QP(d))
• θ2 + o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Lower estimate

Denote by C(x) the critical cone: C(x) =

• d, Dxf (x, 0)d = 0 & Dxg(x, 0)d ∈ TK(g(x, 0))
• and the sufficient second order condition:

   for some α > 0, for all d in C(x), Max

λ∈S(LD)

• D2

x L(x, λ, 0)d2

≥ α|d|2. (SC2) Then, up to a subsequence, with a proof by contradiction, the solutions xθ satisfy: xθ − x θ − →

θ↓0 d ∈ S(LP).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Lower estimate

Denote by C(x) the critical cone: C(x) =

• d, Dxf (x, 0)d = 0 & Dxg(x, 0)d ∈ TK(g(x, 0))
• and the sufficient second order condition:

   for some α > 0, for all d in C(x), Max

λ∈S(LD)

• D2

x L(x, λ, 0)d2

≥ α|d|2. (SC2) Then, up to a subsequence, with a proof by contradiction, the solutions xθ satisfy: xθ − x θ − →

θ↓0 d ∈ S(LP).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Lower estimate

With a Taylor expansion, for all λ in S(LD), V (θ) − V (0) = f (xθ, θ) − f (x, 0) ≥ L(xθ, λ, θ) − L(x, λ, 0) = θDθL(x, λ, 0) + θ2 2

• D(x,θ)2L(x, λ, 0)

xθ − x θ , 1 2 + o(θ2) ≥ θDθL(x, λ, 0) + θ2 2

• Min

d∈S(LP)D(x,θ)2L(x, λ, 0)(d, 1)2

+ o(θ2). Finally, V (θ) ≥ V (0) + Val(LP)θ + Min

d∈S(LP)

• Val(QP(d))
• θ2 + o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Lower estimate

With a Taylor expansion, for all λ in S(LD), V (θ) − V (0) = f (xθ, θ) − f (x, 0) ≥ L(xθ, λ, θ) − L(x, λ, 0) = θDθL(x, λ, 0) + θ2 2

• D(x,θ)2L(x, λ, 0)

xθ − x θ , 1 2 + o(θ2) ≥ θDθL(x, λ, 0) + θ2 2

• Min

d∈S(LP)D(x,θ)2L(x, λ, 0)(d, 1)2

+ o(θ2). Finally, V (θ) ≥ V (0) + Val(LP)θ + Min

d∈S(LP)

• Val(QP(d))
• θ2 + o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

1 Relaxation of optimal control problems

Strong solutions Relaxation

2 Methodology for sensitivity analysis

Upper estimate Lower estimate

3 Application to optimal control

Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Multipliers

Let us define: the end-point Lagrangian Φ[λ], Φ[λ](y, θ) = φ(y, θ) + λ, Φ(y, θ), the Hamiltonian H, H[p](u, y, θ) = p, f (u, y, θ), the costate pλ associated with λ in NK(Φ(yT, 0)), the solution to

• ˙

pt = −Hy[pt](ut, yt, 0) pT = ΦyT [λ](yT, 0).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Let λ ∈ NK(Φ(yT, 0)), λ is said to be: a Lagrange multiplier if for a.a. t, DuH[pλ

t ](ut, yt, 0) = 0,

a Pontryagin multiplier if for a.a. t, for all u in BR, H[pλ

t ](ut, yt, 0) ≤ H[pλ t ](u, yt, 0).

The associated sets ΛL and ΛP satisfy ΛP ⊂ ΛL. Theorem (Pontryagin’s principle) Under a qualification condition, if u is an R-strong solution, then ΛP = ∅. In the sequel: “f [t] = f (ut, yt, 0)”.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

Define ξθ, the solution to ˙ ξθ

t =

Dyf [t]ξθ

t + Dθf [t],

ξθ

0 =

0, z[v], the standard linearization

• ˙

zt[v] = Dyf [t]zt[v] + Duf [t]vt, z0[v] = 0. ξ[µ], the Pontryagin linearization ˙ ξt[µ] = Dyf [t]ξt[µ] +

• UR f (u, yt, 0) − f [t] dµt(u),

ξ0[µ] = 0,

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

We consider two kinds of paths, the “standard” ones: uθ = u + vθ, for which φ(yT[uθ, θ], θ) − φ(yT, 0) = Dφ(yT, 0)(zT[v] + ξθ

T, 1)θ + o(θ),

the “Pontryagin” ones: µθ = (1 − θ)µ + θµ, for which φ(yT[µθ, θ], θ) − φ(yT, 0) = Dφ(yT, 0)(ξT[µ] + ξθ

T, 1)θ + o(θ),

Denote by:

• CS =

cone

• z[v], v ∈ L∞(0, T; Rm)
• ,

CP = cone

• ξ[µ], µ ∈ MR
• .

Note that: CS ⊂ CP.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

We consider two kinds of paths, the “standard” ones: uθ = u + vθ, for which φ(yT[uθ, θ], θ) − φ(yT, 0) = Dφ(yT, 0)(zT[v] + ξθ

T, 1)θ + o(θ),

the “Pontryagin” ones: µθ = (1 − θ)µ + θµ, for which φ(yT[µθ, θ], θ) − φ(yT, 0) = Dφ(yT, 0)(ξT[µ] + ξθ

T, 1)θ + o(θ),

Denote by:

• CS =

cone

• z[v], v ∈ L∞(0, T; Rm)
• ,

CP = cone

• ξ[µ], µ ∈ MR
• .

Note that: CS ⊂ CP.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

At the first-order, we can consider: the standard linearized problem and its dual    Min

ξ∈CS

Dφ(yT, 0)(ξ + ξθ

T, 1)

s.t. DΦ(yT, 0)(ξ + ξθ

T, 1) ∈ TK(Φ(yT, 0)).

(SLP) Max

λ∈ΛL

T DθH[pλ

t ][t] dt + DθΦ[λ](yT, 0)

• .

(SLD) the Pontyragin linearized problem and its dual    Min

ξ∈CP

Dφ(yT, 0)(ξ + ξθ

T, 1)

s.t. DΦ(yT, 0)(ξ + ξθ

T, 1) ∈ TK(Φ(yT, 0)).

(PLP) Max

λ∈ΛP

T DθH[pλ

t ][t] dt + DθΦ[λ](yT, 0)

• .

(PLD)

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

At the first-order, we can consider: the standard linearized problem and its dual    Min

ξ∈CS

Dφ(yT, 0)(ξ + ξθ

T, 1)

s.t. DΦ(yT, 0)(ξ + ξθ

T, 1) ∈ TK(Φ(yT, 0)).

(SLP) Max

λ∈ΛL

T DθH[pλ

t ][t] dt + DθΦ[λ](yT, 0)

• .

(SLD) the Pontyragin linearized problem and its dual    Min

ξ∈CP

Dφ(yT, 0)(ξ + ξθ

T, 1)

s.t. DΦ(yT, 0)(ξ + ξθ

T, 1) ∈ TK(Φ(yT, 0)).

(PLP) Max

λ∈ΛP

T DθH[pλ

t ][t] dt + DθΦ[λ](yT, 0)

• .

(PLD)

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

Note that: Val(PLP) ≤ Val(SLP). Theorem Under a qualification condition, the following estimate holds: V (θ) ≤ V (0) + Val(PLP)θ + o(θ). Two difficulties arise now: problem PLP does not have necessarily solutions impossible to compute second-order expansions with Pontryagin paths. Therefore, we suppose that Val(PLP) = Val(SLP).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

Note that: Val(PLP) ≤ Val(SLP). Theorem Under a qualification condition, the following estimate holds: V (θ) ≤ V (0) + Val(PLP)θ + o(θ). Two difficulties arise now: problem PLP does not have necessarily solutions impossible to compute second-order expansions with Pontryagin paths. Therefore, we suppose that Val(PLP) = Val(SLP).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

At the second order, we “mix” linearizations. For v ∈ S(SLP), set µθ = (1 − θ2) (u + vθ)

• 1st-order term

+ θ2µ

• 2nd-order term

, with µ ∈ MR to be optimized. The dual of the second-order linearized problem is: Max

λ∈S(LD)Ωθ[λ](v),

(QD(v)) where Ωθ is the “Hessian of the Lagrangian”: Ωθ[λ](v) = D2Φ[λ](yT, 0)(zT[v] + ξθ

t , 1)

+ T D2H[pλ

t ][t](vt, zt[v] + ξθ t , 1)2 dt.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Linearizations of the problem

At the second order, we “mix” linearizations. For v ∈ S(SLP), set µθ = (1 − θ2) (u + vθ)

• 1st-order term

+ θ2µ

• 2nd-order term

, with µ ∈ MR to be optimized. The dual of the second-order linearized problem is: Max

λ∈S(LD)Ωθ[λ](v),

(QD(v)) where Ωθ is the “Hessian of the Lagrangian”: Ωθ[λ](v) = D2Φ[λ](yT, 0)(zT[v] + ξθ

t , 1)

+ T D2H[pλ

t ][t](vt, zt[v] + ξθ t , 1)2 dt.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Theorem The following expansion is an upper estimate of V (θ): V (0) + θVal(SLP) + θ2 2

• Min

v∈S(SLP)

Max

λ∈S(LD)Ωθ[λ](v)

• + o(θ2).

We can extend: the first-order linearized problem, the Hessian Ωθ, the theorem, by replacing v by a Young measure ν on [0, T] × Rm with a generalized L2 − norm (in a space denoted by M2), V (θ) ≤ V (0)+θVal(SLP)+θ2 2

• Min

ν∈S(SLP′)

Max

λ∈S(LD)Ωθ[λ](ν)

• +o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Theorem The following expansion is an upper estimate of V (θ): V (0) + θVal(SLP) + θ2 2

• Min

v∈S(SLP)

Max

λ∈S(LD)Ωθ[λ](v)

• + o(θ2).

We can extend: the first-order linearized problem, the Hessian Ωθ, the theorem, by replacing v by a Young measure ν on [0, T] × Rm with a generalized L2 − norm (in a space denoted by M2), V (θ) ≤ V (0)+θVal(SLP)+θ2 2

• Min

ν∈S(SLP′)

Max

λ∈S(LD)Ωθ[λ](ν)

• +o(θ2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Decomposition principle

The critical cone C∞ is the set of v in L∞(0, T; Rm) such that DyT φ(yT, 0)zT[v] ≤ 0, DyT Φ(yT, 0)zT[v] ∈ TK(Φ(yT, 0)). We also define the quadratic form Ω[λ](v) by Ω[λ](v) = D(yT )2Φ[λ](yT, 0)(zT[ν]) + T D(u,y)2H[pλ

t ](ut, yt, 0)(vt, zt[ν])2 dµt(u) dt.

Like previously, we can extend C∞ to a cone C2 ⊂ M2 and Ω to elements of M2.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Decomposition principle

Let θk ↓ 0 and µk be feasible points of the perturbed problems with θ = θk, such that d2(µk, µ) → 0. We decompose µk as follows: µA,k, where the perturbations are small in L∞−norm, µB,k, where the perturbations are large on a subset Bk. Theorem Assume that: meas(Bk) → 0 and d∞(u, µA,k) → 0, then, a lower estimate of V (θk) − V (0) is

• Bk

H[pλ

t ](u, yt) − H[pλ t ][t] dµB,k(u) dt

+ Ω[λ](µA,k − u) + o(d2(µ, µk)2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Decomposition principle

Let θk ↓ 0 and µk be feasible points of the perturbed problems with θ = θk, such that d2(µk, µ) → 0. We decompose µk as follows: µA,k, where the perturbations are small in L∞−norm, µB,k, where the perturbations are large on a subset Bk. Theorem Assume that: meas(Bk) → 0 and d∞(u, µA,k) → 0, then, a lower estimate of V (θk) − V (0) is

• Bk

H[pλ

t ](u, yt) − H[pλ t ][t] dµB,k(u) dt

+ Ω[λ](µA,k − u) + o(d2(µ, µk)2).

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Decomposition principle

The strong second-order sufficient condition (SC2) is: there exists α > 0 such that

1 for almost all t, for all u ∈ BR,

sup

λ∈S(LDθ)

• H[pλ

t ](u, yt, 0) − H[pλ t ](ut, yt, 0)

• ≥ α|u − ut|2,

2 for all ν in C2,

sup

λ∈S(LDθ)

Ω[λ](ν) ≥ α||ν||2

2.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

We consider solutions µθ to the perturbed problem. Theorem IF (SC2) holds, for all sequence θk ↓ 0, the sequence µθk −u

θk

has a limit point for the narrow topology in S(SLP′). Moreover, ≥ V (0) + θVal(LPθ) + θ2 2

• Min

ν∈S(SLP)

Max

λ∈S(LDθ)Ωθ[λ](ν)

• + o(θ2).

is a lower estimate of V (θ). Sketch of the proof: we neglect the large perturbations by contradiction, d2(µ, µθ

k) = O(θk)

conclusion with decomposition principle.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

We consider solutions µθ to the perturbed problem. Theorem IF (SC2) holds, for all sequence θk ↓ 0, the sequence µθk −u

θk

has a limit point for the narrow topology in S(SLP′). Moreover, ≥ V (0) + θVal(LPθ) + θ2 2

• Min

ν∈S(SLP)

Max

λ∈S(LDθ)Ωθ[λ](ν)

• + o(θ2).

is a lower estimate of V (θ). Sketch of the proof: we neglect the large perturbations by contradiction, d2(µ, µθ

k) = O(θk)

conclusion with decomposition principle.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Bibliography I

J.F. Bonnans, A. Shapiro, Perturbation analysis of optimization problems, Springer-Verlag, New York, 2000.

• M. Valadier,

Young measures. In Methods of nonconvex analysis, Lecture Notes in Math., 1446:152-188, 1990. L.C. Young, Lectures on the calculus of variations and optimal control theory. W.B. Saunders Co., 1969.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control

Bibliography II

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

• J. Convex Anal., 17(3-4):885-913, 2010.

J.F. Bonnans, L.P., O.S. Serea, Sensitivity analysis for relaxed OCP’s with final-state constraints, Inria Research Report 7977, submitted.

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