[PPT] - Value function and optimal trajectories for a control problem with PowerPoint Presentation

SLIDE 1

Value function and optimal trajectories for a control problem with supremum cost function and state constraints

Hasnaa Zidani

ENSTA ParisTech, University of Paris-Saclay

joint work with: A. Assellaou, & O. Bokanowski, & A. Desilles Workshop ”Numerical methods for Hamilton-Jacobi equations in optimal control and related fields”

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 1 / 29

SLIDE 2

Consider the following state constrained control problem: ϑ(t, y) := inf

max

θ∈[0,t] Φ(yu y(θ))

u ∈ U,

yu

y(s) ∈ K, s ∈ [0, t]

(1)

where yu

y denotes the solution of the controlled differential system:

˙

y(s) := f (y(s), u(s)), a.e s ∈ [0, t], y(0) := y, K is a closed set of Rd and U is a set of admissible control inputs.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 2 / 29

SLIDE 3

Hamilton-Jacobi approach

➤ How can we handle correctly the state constraints ? ➤ What boundary conditions should be considered for the HJB equation? ➤ Trajectory reconstruction and feedback control law. ➤ When Φ ≥ 0, we know that t Φ(yu

y(s))2p ds

1

2p

→ max

s∈[0,t] Φ(yu y(s)),

as p → +∞. So, it is possible to approximate the maximum running cost problem by a Bolza problem. Does this approximation work well in practice?

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 3 / 29

SLIDE 4

Outline

1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 4 / 29

SLIDE 5

Outline

1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 5 / 29

SLIDE 6

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 6 / 29

SLIDE 7

Abort landing problem in presence of windshear

(Miele, Wang and Melvin(1987,1988); Bulirsch, Montrone and Pesch (1991..); Botkin-Turova(2012 ...))

Consider the flight motion of an aircraft in a vertical plane:          ˙ x = V cos γ + wx ˙ h = V sin γ + wh ˙ V = FT

m cos(α + δ) − FD m − g sin γ − ( ˙

wx cos γ + ˙ wh sin γ) ˙ γ = 1

V ( FT m sin(α + δ) + FL m − g cos γ + ( ˙

wx sin γ − ˙ wh cos γ)) where

˙ wx = ∂wx ∂x (V cos γ + wx) + ∂wx ∂h (V sin γ + wh) ˙ wh = ∂wh ∂x (V cos γ + wx) + ∂wh ∂h (V sin γ + wh)

and FT := FT(V ) is the thrust force FD := FD(V , α) and FL := FL(V , α) are the drag and lift forces wx := wx(x) and wh := wh(x, h) are the wind components m, g, and δ are constants.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 7 / 29

SLIDE 8

Controlled system

Consider the state y(.) = (x(.), h(.), V (.), γ(.), α(.)). The control variable u is the angular speed of the angle of attack α. Let T be a fixed time horizon and let U be the set of admissible controls U :=

u : (0, T) → R, measurable, u(t) ∈ U a.e
where U is a compact set.

The controlled dynamics in this case is:                ˙ x = V cos γ + wx, ˙ h = V sin γ + wh, ˙ V = FT

m cos(α + δ) − FD m − g sin γ − ( ˙

wx cos γ + ˙ wh sin γ), ˙ γ = 1

V ( FT m sin(α + δ) + FL m − g cos γ + ( ˙

wx sin γ − ˙ wh cos γ)), ˙ α = u.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 8 / 29

SLIDE 9

Formulation of the optimal control problem

Aim: Maximize the minimal altitude over a time interval: min

θ∈[0,t] h(θ)

while the aircraft stays in a given domain K. Consider the following optimal control problem: (P) : ϑ(t, y) = inf

max

θ∈[0,t] Φ(yu y(θ)),

u ∈ U, and yu

y(s) ∈ K, ∀s ∈ [0, t]

where Φ(yu

y(.)) = Hr − h(.), Hr being a reference altitude, and K is a set of

state constraints.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 9 / 29

SLIDE 10

Formulation of the optimal control problem

Aim: Maximize the minimal altitude over a time interval: min

θ∈[0,t] h(θ)

while the aircraft stays in a given domain K. Consider the following optimal control problem: (P) : ϑ(t, y) = inf

max

θ∈[0,t] Φ(yu y(θ)),

u ∈ U, and yu

y(s) ∈ K, ∀s ∈ [0, t]

where Φ(yu

y(.)) = Hr − h(.), Hr being a reference altitude, and K is a set of

state constraints.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 9 / 29

SLIDE 11

Outline

1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 10 / 29

SLIDE 12

A general setting

➤ For a given non-empty compact subset U of Rk and a finite time T > 0, define the set of admissible control to be, U :=

u : (0, T) → Rk, measurable, u(t) ∈ U a.e
.

➤ Consider the following control system:

˙

y(s) := f (y(s), u(s)), a.e s ∈ [0, T], y(0) := y, (2) where u ∈ U and the function f is defined and continuous on Rd × U and that it is Lipschitz continuous w.r.t y,

(i) f : Rd × U → Rd is continuous,

(ii) ∃L > 0 s.t. ∀(y1, y2) ∈ Rd × Rd, ∀u ∈ U, |f (y1, u) − f (y2, u)| ≤ L(|y1 − y2|).

➤ The corresponding set of feasible trajectories: S[0,T](y) := {y ∈ W 1,1(0, T; Rd), y satisfies (2) for some u ∈ U},

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 11 / 29

SLIDE 13

A general setting

➤ For a given non-empty compact subset U of Rk and a finite time T > 0, define the set of admissible control to be, U :=

u : (0, T) → Rk, measurable, u(t) ∈ U a.e
.

➤ Consider the following control system:

˙

y(s) := f (y(s), u(s)), a.e s ∈ [0, T], y(0) := y, (2) where u ∈ U and the function f is defined and continuous on Rd × U and that it is Lipschitz continuous w.r.t y,

(i) f : Rd × U → Rd is continuous,

(ii) ∃L > 0 s.t. ∀(y1, y2) ∈ Rd × Rd, ∀u ∈ U, |f (y1, u) − f (y2, u)| ≤ L(|y1 − y2|).

➤ The corresponding set of feasible trajectories: S[0,T](y) := {y ∈ W 1,1(0, T; Rd), y satisfies (2) for some u ∈ U},

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 11 / 29

SLIDE 14

State constrained control problem with maximum cost

➤ Consider the following state constrained control problem: ϑ(t, y) := inf

max

θ∈[0,t] Φ(yu y(θ))

u ∈ U,

yu

y(s) ∈ K, s ∈ [0, t]

(3)

➤ the cost function Φ(·) is assumed to be Lipschitz continuous and K is a closed set of Rd. ➤ For every y ∈ Rd, the set f (y, U) = {f (y, u), u ∈ U} is assumed to be convex. ➤ The function ϑ is lower semicontinuous (lsc) on K, ϑ ≡ +∞ outside K.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 12 / 29

SLIDE 15

State constrained control problem with maximum cost

➤ Consider the following state constrained control problem: ϑ(t, y) := inf

max

θ∈[0,t] Φ(yu y(θ))

u ∈ U,

yu

y(s) ∈ K, s ∈ [0, t]

(3)

➤ the cost function Φ(·) is assumed to be Lipschitz continuous and K is a closed set of Rd. ➤ For every y ∈ Rd, the set f (y, U) = {f (y, u), u ∈ U} is assumed to be convex. ➤ The function ϑ is lower semicontinuous (lsc) on K, ϑ ≡ +∞ outside K.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 12 / 29

SLIDE 16

State constrained control problem with maximum cost

➤ Consider the following state constrained control problem: ϑ(t, y) := inf

max

θ∈[0,t] Φ(yu y(θ))

u ∈ U,

yu

y(s) ∈ K, s ∈ [0, t]

(3)

➤ the cost function Φ(·) is assumed to be Lipschitz continuous and K is a closed set of Rd. ➤ For every y ∈ Rd, the set f (y, U) = {f (y, u), u ∈ U} is assumed to be convex. ➤ The function ϑ is lower semicontinuous (lsc) on K, ϑ ≡ +∞ outside K.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 12 / 29

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Some references

Maximum cost problems without state constraints (K = Rd): Barron-Ishii (99) Bolza or Mayer problems with state constraints: Soner (86), Rampazzo-Vinter (89), Frankowska-Vinter (00), Motta (95), Cardaliaguet-Quincampoix-Saint-Pierre (97), Altarovici-Bokanowski-HZ (13), Hermosilla-HZ (15) Maximum cost problems with state constraints: Quincampoix-Serea (02), Bokanowski-Picarelli-HZ (13), Assellaou-Bokanowski-Desilles-HZ (CDC’16), Assellaou-Bokanowski-Desilles-HZ (preprint’16)

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 13 / 29

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An auxiliary control problem

➤ Let g be a Lipschitz continuous function such that: ∀y ∈ K, g(y) ≤ 0 ⇔ y ∈ K. (4) ➤ Consider the following auxiliary control problem :

w(t, y, z) := inf

y∈S[0,t](y) max θ∈[0,t]

Φ(y(θ)) − z)
g(y(θ)
,

where a ∨ b = max(a, b).

Theorem

Let (t, y, z) ∈ [0, T] × K × R. The following assertions hold: (i) ϑ(t, y) − z ≤ 0 ⇔ w(t, y, z) ≤ 0, (ii) ϑ(t, y) = min

z ∈ R , w(t, y, z) ≤ 0
.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 14 / 29

SLIDE 19

➤ Define the following Hamiltonian as: H(y, p) := max

u∈U

− f (y, u) · p
∀y, p ∈ Rd.

Proposition

The value function w is the unique Lipschitz continuous viscosity solution of the following Hamilton-Jacobi-Bellman (HJB) equation: min

∂tw(t, y, z) + H(y, ∇yw), w(t, y, z) − Ψ(y, z)
= 0

]0, T]×Rd ×R, w(0, y, z) = Ψ(y, z), Rd × R, where Ψ(y, z) = (Φ(y) − z) g(y).

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 15 / 29

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A particular choice of function g

➤ Let η > 0 and define the following extended set wK: Kη :≡ K + B(0, η). ➤ g(y) := dK(y) the signed distance to K. ➤ Consider the following auxiliary control problem :

w(t, y, z) := inf

y∈S[0,t](y)

max

θ∈[0,t]

Φ(y(θ)) − z)
g(y(θ)

η

,

where a ∧ b = min(a, b).

Theorem

Let (t, y, z) ∈ [0, T] × K × R. The following assertions hold: (i) ϑ(t, y) − z ≤ 0 ⇔ w(t, y, z) ≤ 0, (ii) ϑ(t, y) = min

z ∈ R , w(t, y, z) ≤ 0
.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 16 / 29

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A particular choice of function g

Theorem

The function w is the unique Lipschitz continuous viscosity solution of the following HJB equation: min

∂tw(t, y, z) + H(y, ∇yw), w(t, y, z) − Ψη(y, z)
= 0

]0, T]×Kη×R, w(0, y, z) = Ψη(y, z), Kη × R, w(t, y, z) = η, y ∈ Kη, where Ψη(y, z) =

(Φ(y) − z) g(y)

η.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 17 / 29

SLIDE 22

Link with exit time function

➤ Define the exit time function: T (y, z) := sup

t ∈ [0, T]
ϑ(t, y) ≤ z
=

sup

t ∈ [0, T]
w(t, y, z) ≤ 0
➤ Link with viability theory:

(i) T is the exit time function for Epi(Φ) K × Rd , (ii) T (y, z) = t ⇒ w(t, y, z) = 0, (iii) ϑ(t, y) = inf

z
T (y, z) ≥ t
.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 18 / 29

SLIDE 23

Link with exit time function

➤ Define the exit time function: T (y, z) := sup

t ∈ [0, T]
ϑ(t, y) ≤ z
=

sup

t ∈ [0, T]
w(t, y, z) ≤ 0
➤ Link with viability theory:

(i) T is the exit time function for Epi(Φ) K × Rd , (ii) T (y, z) = t ⇒ w(t, y, z) = 0, (iii) ϑ(t, y) = inf

z
T (y, z) ≥ t
.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 18 / 29

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Outline

1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 19 / 29

SLIDE 25

Optimal trajectories

Proposition

Let y ∈ K such that ϑ(T, y) < ∞. Define z∗ := ϑ(T, y). Let y∗ = (y∗, z∗) be the optimal trajectory for the auxiliary control problem associated with the initial point (y, z∗) ∈ K × R. Then, the trajectory y∗ is

ptimal for the original control problem.

Let y∗ = (y∗, z∗) be an optimal trajectory for the exit time problem associated with the initial point (y, z) ∈ K × R. Then, y∗ is also optimal for the auxiliary control problem.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 20 / 29

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Reconstruction of optimal trajectories - Algorithm A.

➤ For n ≥ 1, consider (t0 = 0, t1, ..., tn−1, tn = T) a uniform partition of [0, T] with ∆t = T

n .

➤ Let {yn(·), zn(·)} be a trajectory defined recursively on the intervals (ti−1, ti], with zn(·) := z = ϑ(0, y) and yn(0) = y. ➤ [Step 1] Knowing y n

k = yn(tk), choose the optimal control at tk s.t.:

un

k ∈ arg min u∈U

w(tk, y n

k + ∆tf∆t

y n

k , u

, z) + λnC(u, ∆t)
.

➤ [Step 2] Define un(t) := un

k, ∀t ∈ (tk, tk+1] and yn(t) on (tk, tk+1] as the

solution of ˙ y(t) := f (y(t), un(t)) a.e t ∈ (tk, tk+1], with initial condition yn(tk) at tk and zn(·) := z.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 21 / 29

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Reconstruction of optimal trajectories - Algorithm A.

➤ For n ≥ 1, consider (t0 = 0, t1, ..., tn−1, tn = T) a uniform partition of [0, T] with ∆t = T

n .

➤ Let {yn(·), zn(·)} be a trajectory defined recursively on the intervals (ti−1, ti], with zn(·) := z = ϑ(0, y) and yn(0) = y. ➤ [Step 1] Knowing y n

k = yn(tk), choose the optimal control at tk s.t.:

un

k ∈ arg min u∈U

w(tk, y n

k + ∆tf∆t

y n

k , u

, z) + λnC(u, ∆t)
.

➤ [Step 2] Define un(t) := un

k, ∀t ∈ (tk, tk+1] and yn(t) on (tk, tk+1] as the

solution of ˙ y(t) := f (y(t), un(t)) a.e t ∈ (tk, tk+1], with initial condition yn(tk) at tk and zn(·) := z.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 21 / 29

SLIDE 28

Reconstruction of optimal trajectories - Algorithm A.

➤ For n ≥ 1, consider (t0 = 0, t1, ..., tn−1, tn = T) a uniform partition of [0, T] with ∆t = T

n .

➤ Let {yn(·), zn(·)} be a trajectory defined recursively on the intervals (ti−1, ti], with zn(·) := z = ϑ(0, y) and yn(0) = y. ➤ [Step 1] Knowing y n

k = yn(tk), choose the optimal control at tk s.t.:

un

k ∈ arg min u∈U

w(tk, y n

k + ∆tf∆t

y n

k , u

, z) + λnC(u, ∆t)
.

➤ [Step 2] Define un(t) := un

k, ∀t ∈ (tk, tk+1] and yn(t) on (tk, tk+1] as the

solution of ˙ y(t) := f (y(t), un(t)) a.e t ∈ (tk, tk+1], with initial condition yn(tk) at tk and zn(·) := z.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 21 / 29

SLIDE 29

Theorem

Let {yn(·), zn(·), un(·)} be a sequence generated by algorithm A for n ≥ 1. Then, the sequence of trajectories {yn(·)}n has cluster points with respect to the uniform convergence topology. For any cluster point ¯ y(·) there exists a control law ¯ u(·) such that (¯ y(·), ¯ z(·), ¯ u(·)) is optimal for the auxiliary control problem. ➤ Let w ∆ be a numerical approximate solution such that, |w ∆(t, y, z) − w(t, y, z)| ≤ E1(∆t, ∆y), where E1(∆t, ∆y) → 0 as ∆t, ∆y → 0. ➤ Let {Yn(.), un(.)} be the sequence generated by the algorithm A with w ∆. ➤ Then, (Y n)n converges to an optimal trajectory for the auxiliary control problem.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 22 / 29

SLIDE 30

Theorem

Let {yn(·), zn(·), un(·)} be a sequence generated by algorithm A for n ≥ 1. Then, the sequence of trajectories {yn(·)}n has cluster points with respect to the uniform convergence topology. For any cluster point ¯ y(·) there exists a control law ¯ u(·) such that (¯ y(·), ¯ z(·), ¯ u(·)) is optimal for the auxiliary control problem. ➤ Let w ∆ be a numerical approximate solution such that, |w ∆(t, y, z) − w(t, y, z)| ≤ E1(∆t, ∆y), where E1(∆t, ∆y) → 0 as ∆t, ∆y → 0. ➤ Let {Yn(.), un(.)} be the sequence generated by the algorithm A with w ∆. ➤ Then, (Y n)n converges to an optimal trajectory for the auxiliary control problem.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 22 / 29

SLIDE 31

Theorem

Let {yn(·), zn(·), un(·)} be a sequence generated by algorithm A for n ≥ 1. Then, the sequence of trajectories {yn(·)}n has cluster points with respect to the uniform convergence topology. For any cluster point ¯ y(·) there exists a control law ¯ u(·) such that (¯ y(·), ¯ z(·), ¯ u(·)) is optimal for the auxiliary control problem. ➤ Let w ∆ be a numerical approximate solution such that, |w ∆(t, y, z) − w(t, y, z)| ≤ E1(∆t, ∆y), where E1(∆t, ∆y) → 0 as ∆t, ∆y → 0. ➤ Let {Yn(.), un(.)} be the sequence generated by the algorithm A with w ∆. ➤ Then, (Y n)n converges to an optimal trajectory for the auxiliary control problem.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 22 / 29

SLIDE 32

Theorem

Let {yn(·), zn(·), un(·)} be a sequence generated by algorithm A for n ≥ 1. Then, the sequence of trajectories {yn(·)}n has cluster points with respect to the uniform convergence topology. For any cluster point ¯ y(·) there exists a control law ¯ u(·) such that (¯ y(·), ¯ z(·), ¯ u(·)) is optimal for the auxiliary control problem. ➤ Let w ∆ be a numerical approximate solution such that, |w ∆(t, y, z) − w(t, y, z)| ≤ E1(∆t, ∆y), where E1(∆t, ∆y) → 0 as ∆t, ∆y → 0. ➤ Let {Yn(.), un(.)} be the sequence generated by the algorithm A with w ∆. ➤ Then, (Y n)n converges to an optimal trajectory for the auxiliary control problem.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 22 / 29

SLIDE 33

Reconstruction of optimal trajectories using the exit time

Algorithm B.

➤ For n ≥ 1, consider (t0 = 0, t1, ..., tn−1, tn = T) a uniform partition of [0, T] with ∆t = T

n .

➤ Let {yn(·), zn(·)} be a trajectory defined recursively on the intervals (ti−1, ti], with zn(·) := z = ϑ(T, y) and yn(t0) = y. ➤ [Step 1] Knowing y n

k = yn(tk), choose the optimal control at tk s.t.:

un

k ∈ arg max u∈U

T
yn(tk) + ∆tf∆t
yn(tk), u
, z
+ ∆t

T

,

➤ [Step 2] Define un(t) := un

k, ∀t ∈ (tk, tk+1] and yn(t) on (tk, tk+1] as the

solution of ˙ y(t) := f (y(t), un(t)) a.e t ∈ (tk, tk+1], with initial condition yn(tk) at tk and zn(·) := z. with initial condition yn(tk) at tk and zn(·) := z.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 23 / 29

SLIDE 34

Reconstruction of optimal trajectories using the exit time

Algorithm B.

➤ For n ≥ 1, consider (t0 = 0, t1, ..., tn−1, tn = T) a uniform partition of [0, T] with ∆t = T

n .

➤ Let {yn(·), zn(·)} be a trajectory defined recursively on the intervals (ti−1, ti], with zn(·) := z = ϑ(T, y) and yn(t0) = y. ➤ [Step 1] Knowing y n

k = yn(tk), choose the optimal control at tk s.t.:

un

k ∈ arg max u∈U

T
yn(tk) + ∆tf∆t
yn(tk), u
, z
+ ∆t

T

,

➤ [Step 2] Define un(t) := un

k, ∀t ∈ (tk, tk+1] and yn(t) on (tk, tk+1] as the

solution of ˙ y(t) := f (y(t), un(t)) a.e t ∈ (tk, tk+1], with initial condition yn(tk) at tk and zn(·) := z. with initial condition yn(tk) at tk and zn(·) := z.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 23 / 29

SLIDE 35

Reconstruction of optimal trajectories using the exit time

Algorithm B.

➤ For n ≥ 1, consider (t0 = 0, t1, ..., tn−1, tn = T) a uniform partition of [0, T] with ∆t = T

n .

➤ Let {yn(·), zn(·)} be a trajectory defined recursively on the intervals (ti−1, ti], with zn(·) := z = ϑ(T, y) and yn(t0) = y. ➤ [Step 1] Knowing y n

k = yn(tk), choose the optimal control at tk s.t.:

un

k ∈ arg max u∈U

T
yn(tk) + ∆tf∆t
yn(tk), u
, z
+ ∆t

T

,

➤ [Step 2] Define un(t) := un

k, ∀t ∈ (tk, tk+1] and yn(t) on (tk, tk+1] as the

solution of ˙ y(t) := f (y(t), un(t)) a.e t ∈ (tk, tk+1], with initial condition yn(tk) at tk and zn(·) := z. with initial condition yn(tk) at tk and zn(·) := z.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 23 / 29

SLIDE 36

Outline

1 Motivation: Abort landing problem in presence of ”Wind Shear” 2 Optimal control problem with maximum-cost and state constraints 3 Optimal trajectories 4 Numerical simulations

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 24 / 29

SLIDE 37

Numerical schemes

Finite Difference scheme    W n+1

I,j

= max

W n

I,j + ∆tH(yI, D+W n(yI, zj), D−W n(yI, zj)), ΨI,j

W N

I,j = ΨI,j,

Semi Lagrangian scheme

W n+1

I,j

= mina∈U

W n

yI + f (yI, a)∆t, zj ΨI,j W N

I,j = ΨI,j

Same error estimates for both schemes under adequate CFL conditions. Application on the Wind Shear problem (Boeing 727 aircraft model data) Simulations on a grid with NG = 403 × 202 × 10 nodes (where 30 is the number of points per axis for the first three components, namely, x, h and v, 20 is the number of the points for the angles γ and α an 10 is the number of points for the additional variable z)

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 25 / 29

SLIDE 38

Approximation by Bolza problems

t Φ(yu

y(s))2p ds

1

2p

→ max

s∈[0,t] Φ(yu y(s)),

as p → +∞. Criterion Optimal cost value Bolza, 2p = 6 907.59 Bolza, 2p = 8 807.64 Bolza, 2p = 10 728.08 Bolza, 2p = 12 686.29 Maximum running cost 611.72

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 26 / 29

SLIDE 39

Numerical simulations

✤ Quite similar optimal trajectories by using either algorithms A and B. However, using the exit time function is much more cheaper (in term of data storage). ✤ The control variable enters linearly: f (y, u) = uf0(y) + f1(y). The Hamiltonian has a simple form (assume u ∈ [−1, 1]): H(x, p) := −f1(y) · p + |f0(y) · p|. However, the optimal control strategy may be of Bang-bang type or may include ”singular arcs” when the gradient of the value function is close to 0.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 27 / 29

SLIDE 40

Figure: Reconstruction of the state variables by adding a penalization term λC(u, un) = |u − un|, with λ = 0.0, 1.0 and 2.0.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 28 / 29

SLIDE 41

... thank you for your attention.

Hasnaa Zidani (ENSTA ) Abort landing Problem RICAM, 21-25 November, 2016 29 / 29