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
Consider the following state constrained control problem:
ϑ(t, y ) := inf
max
θ∈[0,t]
Φ(yuy (θ))
u ∈ U, yyu (s) ∈ K, s ∈ [0, t]
(1)
where yyu denotes the solution of the controlled differential system:
(
ẏ(s) := f (y(s), u(s)),
y(0) := y ,
a.e s ∈ [0, t],
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
Hamilton-Jacobi approach
ä How can we handle correctly the state constraints ?
ä What boundary conditions should be considered for the HJB equation?
ä Trajectory reconstruction and feedback control law.
ä When Φ ≥ 0, we know that
Z
2p1
t
0
Φ(yyu (s))2p
ds
→ max Φ(yyu (s)),
as p → +∞.
s∈[0,t]
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
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
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
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
6 / 29
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:

ẋ = V cos γ + wx



ḣ = V sin γ + w
h
FT

cos(α
+
δ) − FmD − g sin γ − (ẇx cos γ + ẇh sin γ)
V̇
=

m


γ̇ = V1 ( FmT sin(α + δ) + FmL − g cos γ + (ẇx sin γ − ẇh cos γ))
where
ẇx
=
ẇh
=
∂wx
(V cos γ + wx ) +
∂x
∂wh
(V cos γ + wx ) +
∂x
∂wx
(V sin γ + wh )
∂h
∂wh
(V sin γ + wh )
∂h
and
FT
FD
wx
m,
:= FT (V ) is the thrust force
:= FD (V , α) and FL := FL (V , α) are the drag and lift forces
:= wx (x) and wh := wh (x, h) are the wind components
g , and δ are constants.
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
7 / 29
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:


ẋ = V cos γ + wx ,





ḣ = V sin γ + wh ,
V̇ = FmT cos(α + δ) − FmD − g sin γ − (ẇx cos γ + ẇh sin γ),



γ̇ = V1 ( FmT sin(α + δ) + FmL − g cos γ + (ẇx sin γ − ẇh cos γ)),



α̇ = u.
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
8 / 29
Formulation of the optimal control problem
Aim: Maximize the minimal altitude over a time interval:
min h(θ)
θ∈[0,t]
while the aircraft stays in a given domain K.
Consider the following optimal control problem:
u
u
(P) : ϑ(t, y ) = inf max Φ(yy (θ)), u ∈ U, and yy (s) ∈ K, ∀s ∈ [0, t]
θ∈[0,t]
where Φ(yyu (.)) = 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
Formulation of the optimal control problem
Aim: Maximize the minimal altitude over a time interval:
min h(θ)
θ∈[0,t]
while the aircraft stays in a given domain K.
Consider the following optimal control problem:
u
u
(P) : ϑ(t, y ) = inf max Φ(yy (θ)), u ∈ U, and yy (s) ∈ K, ∀s ∈ [0, t]
θ∈[0,t]
where Φ(yyu (.)) = 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
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
A general setting
ä For a given non-empty compact subset U of Rk and a finite time T > 0,
define the set of admissible control to be,
k
U := u : (0, T ) → R , measurable, u(t) ∈ U a.e .
ä Consider the following control system:
(
ẏ(s) := f (y(s), u(s)),
y(0) := y ,
a.e s ∈ [0, T ],
(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 |).
ä 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
A general setting
ä For a given non-empty compact subset U of Rk and a finite time T > 0,
define the set of admissible control to be,
k
U := u : (0, T ) → R , measurable, u(t) ∈ U a.e .
ä Consider the following control system:
(
ẏ(s) := f (y(s), u(s)),
y(0) := y ,
a.e s ∈ [0, T ],
(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 |).
ä 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
State constrained control problem with maximum cost
ä Consider the following state constrained control problem:
ϑ(t, y ) := inf
max Φ(yyu (θ)) u ∈ U, yyu (s) ∈ K, s ∈ [0, t]
(3)
θ∈[0,t]
ä the cost function Φ(·) is assumed to be Lipschitz continuous and K is a
closed set of Rd .
ä For every y ∈ Rd , the set f (y , U) = {f (y , u), u ∈ U} is assumed to be
convex.
ä 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
State constrained control problem with maximum cost
ä Consider the following state constrained control problem:
ϑ(t, y ) := inf
max Φ(yyu (θ)) u ∈ U, yyu (s) ∈ K, s ∈ [0, t]
(3)
θ∈[0,t]
ä the cost function Φ(·) is assumed to be Lipschitz continuous and K is a
closed set of Rd .
ä For every y ∈ Rd , the set f (y , U) = {f (y , u), u ∈ U} is assumed to be
convex.
ä 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
State constrained control problem with maximum cost
ä Consider the following state constrained control problem:
ϑ(t, y ) := inf
max Φ(yyu (θ)) u ∈ U, yyu (s) ∈ K, s ∈ [0, t]
(3)
θ∈[0,t]
ä the cost function Φ(·) is assumed to be Lipschitz continuous and K is a
closed set of Rd .
ä For every y ∈ Rd , the set f (y , U) = {f (y , u), u ∈ U} is assumed to be
convex.
ä 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
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
An auxiliary control problem
ä Let g be a Lipschitz continuous function such that:
∀y ∈ K, g (y ) ≤ 0 ⇔ y ∈ K.
(4)
ä Consider the following auxiliary control problem :
w (t, y , z) :=
inf
_
max Φ(y(θ)) − z) g (y(θ) ,
y∈S[0,t] (y ) θ∈[0,t]
where a ∨ b = max(a, b).
Theorem
Let (t, y , z) ∈ [0, T ] × K × R. The following assertions hold:
(i)
(ii)
Hasnaa Zidani (ENSTA )
ϑ(t, y ) − z ≤ 0 ⇔ w (t, y , z) ≤ 0,
ϑ(t, y ) = min z ∈ R , w (t, y , z) ≤ 0 .
Abort landing Problem
RICAM, 21-25 November, 2016
14 / 29
ä Define the following Hamiltonian as:
H(y , p) := max − f (y , u) · p
u∈U
∀y , p ∈ Rd .
Proposition
The value function w is the unique Lipschitz continuous viscosity solution of the
following Hamilton-Jacobi-Bellman (HJB) equation:
min ∂t w (t, y , z) + H(y , ∇y w ), w (t, y , z) − Ψ(y , z) = 0 ]0, T ]×Rd ×R,
Rd × R,
W
where Ψ(y , z) = (Φ(y ) − z) g (y ).
w (0, y , z) = Ψ(y , z),
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
15 / 29
A particular choice of function g
ä Let η > 0 and define the following extended set w K:
Kη :≡ K + B(0, η).
ä g (y ) := dK (y ) the signed distance to K.
ä Consider the following auxiliary control problem :
w (t, y , z) :=
inf
y∈S[0,t] (y )
h
^ i
_
max Φ(y(θ)) − z) g (y(θ)
η ,
θ∈[0,t]
where a ∧ b = min(a, b).
Theorem
Let (t, y , z) ∈ [0, T ] × K × R. The following assertions hold:
(i)
(ii)
Hasnaa Zidani (ENSTA )
ϑ(t, y ) − z ≤ 0 ⇔ w (t, y , z) ≤ 0,
ϑ(t, y ) = min z ∈ R , w (t, y , z) ≤ 0 .
Abort landing Problem
RICAM, 21-25 November, 2016
16 / 29
A particular choice of function g
Theorem
The function w is the unique Lipschitz continuous viscosity solution of the
following HJB equation:
min ∂t w (t, y , z) + H(y , ∇y w ), 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 6∈ Kη ,
W
V
where Ψη (y , z) = (Φ(y ) − z) g (y )
η.
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
17 / 29
Link with exit time function
ä Define the exit time function:
T (y , z)
:=
=
sup t ∈ [0, T ]
sup t ∈ [0, T ]
ϑ(t, y ) ≤ z
w (t, y , z) ≤ 0
ä Link with viability theory:
(i)
(ii)
(iii)
Hasnaa Zidani (ENSTA )
T is the exit time function for Epi(Φ)
\
K × Rd ,
T (y , z) = t ⇒ w (t, y , z) = 0,
ϑ(t, y ) = inf z T (y , z) ≥ t .
Abort landing Problem
RICAM, 21-25 November, 2016
18 / 29
Link with exit time function
ä Define the exit time function:
T (y , z)
:=
=
sup t ∈ [0, T ]
sup t ∈ [0, T ]
ϑ(t, y ) ≤ z
w (t, y , z) ≤ 0
ä Link with viability theory:
(i)
(ii)
(iii)
Hasnaa Zidani (ENSTA )
T is the exit time function for Epi(Φ)
\
K × Rd ,
T (y , z) = t ⇒ w (t, y , z) = 0,
ϑ(t, y ) = inf z T (y , z) ≥ t .
Abort landing Problem
RICAM, 21-25 November, 2016
18 / 29
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
Optimal trajectories
Proposition
Let y ∈ K such that ϑ(T , y ) < ∞. Define z ∗ := ϑ(T , y ).
Let b
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
optimal for the original control problem.
Let b
y∗ = (y∗ , z∗ ) be an optimal trajectory for the exit time problem
associated with the initial point (y , z) ∈ K × R. Then, b
y∗ is also optimal for
the auxiliary control problem.
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
20 / 29
Reconstruction of optimal trajectories - Algorithm A.
ä For n ≥ 1, consider (t0 = 0, t1 , ..., tn−1 , tn = T ) a uniform partition of [0, T ]
with ∆t = Tn .
ä Let {yn (·), zn (·)} be a trajectory defined recursively on the intervals (ti−1 , ti ],
with zn (·) := z = ϑ(0, y ) and yn (0) = y .
ä [Step 1] Knowing ykn = yn (tk ), choose the optimal control at tk s.t.:
n
n
n
uk ∈ arg min w (tk , yk + ∆tf∆t yk , u , z) + λn C (u, ∆t) .
u∈U
ä [Step 2] Define un (t) := ukn , ∀t ∈ (tk , tk+1 ] and yn (t) on (tk , tk+1 ] as the
solution of
ẏ(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
Reconstruction of optimal trajectories - Algorithm A.
ä For n ≥ 1, consider (t0 = 0, t1 , ..., tn−1 , tn = T ) a uniform partition of [0, T ]
with ∆t = Tn .
ä Let {yn (·), zn (·)} be a trajectory defined recursively on the intervals (ti−1 , ti ],
with zn (·) := z = ϑ(0, y ) and yn (0) = y .
ä [Step 1] Knowing ykn = yn (tk ), choose the optimal control at tk s.t.:
n
n
n
uk ∈ arg min w (tk , yk + ∆tf∆t yk , u , z) + λn C (u, ∆t) .
u∈U
ä [Step 2] Define un (t) := ukn , ∀t ∈ (tk , tk+1 ] and yn (t) on (tk , tk+1 ] as the
solution of
ẏ(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
Reconstruction of optimal trajectories - Algorithm A.
ä For n ≥ 1, consider (t0 = 0, t1 , ..., tn−1 , tn = T ) a uniform partition of [0, T ]
with ∆t = Tn .
ä Let {yn (·), zn (·)} be a trajectory defined recursively on the intervals (ti−1 , ti ],
with zn (·) := z = ϑ(0, y ) and yn (0) = y .
ä [Step 1] Knowing ykn = yn (tk ), choose the optimal control at tk s.t.:
n
n
n
uk ∈ arg min w (tk , yk + ∆tf∆t yk , u , z) + λn C (u, ∆t) .
u∈U
ä [Step 2] Define un (t) := ukn , ∀t ∈ (tk , tk+1 ] and yn (t) on (tk , tk+1 ] as the
solution of
ẏ(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
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 ȳ(·) there exists a control law
ū(·) such that (ȳ(·), z̄(·), ū(·)) is optimal for the auxiliary control problem.
ä 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.
ä Let {Yn (.), un (.)} be the sequence generated by the algorithm A with w ∆ .
ä 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
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 ȳ(·) there exists a control law
ū(·) such that (ȳ(·), z̄(·), ū(·)) is optimal for the auxiliary control problem.
ä 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.
ä Let {Yn (.), un (.)} be the sequence generated by the algorithm A with w ∆ .
ä 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
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 ȳ(·) there exists a control law
ū(·) such that (ȳ(·), z̄(·), ū(·)) is optimal for the auxiliary control problem.
ä 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.
ä Let {Yn (.), un (.)} be the sequence generated by the algorithm A with w ∆ .
ä 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
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 ȳ(·) there exists a control law
ū(·) such that (ȳ(·), z̄(·), ū(·)) is optimal for the auxiliary control problem.
ä 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.
ä Let {Yn (.), un (.)} be the sequence generated by the algorithm A with w ∆ .
ä 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
Reconstruction of optimal trajectories using the exit time
- Algorithm B.
ä For n ≥ 1, consider (t0 = 0, t1 , ..., tn−1 , tn = T ) a uniform partition of [0, T ]
with ∆t = Tn .
ä Let {yn (·), zn (·)} be a trajectory defined recursively on the intervals (ti−1 , ti ],
with zn (·) := z = ϑ(T , y ) and yn (t0 ) = y .
ä [Step 1] Knowing ykn = yn (tk ), choose the optimal control at tk s.t.:
^ ukn ∈ arg max
T yn (tk ) + ∆tf∆t yn (tk ), u , z + ∆t
T ,
u∈U
ä [Step 2] Define un (t) := ukn , ∀t ∈ (tk , tk+1 ] and yn (t) on (tk , tk+1 ] as the
solution of
ẏ(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
Reconstruction of optimal trajectories using the exit time
- Algorithm B.
ä For n ≥ 1, consider (t0 = 0, t1 , ..., tn−1 , tn = T ) a uniform partition of [0, T ]
with ∆t = Tn .
ä Let {yn (·), zn (·)} be a trajectory defined recursively on the intervals (ti−1 , ti ],
with zn (·) := z = ϑ(T , y ) and yn (t0 ) = y .
ä [Step 1] Knowing ykn = yn (tk ), choose the optimal control at tk s.t.:
^ ukn ∈ arg max
T yn (tk ) + ∆tf∆t yn (tk ), u , z + ∆t
T ,
u∈U
ä [Step 2] Define un (t) := ukn , ∀t ∈ (tk , tk+1 ] and yn (t) on (tk , tk+1 ] as the
solution of
ẏ(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
Reconstruction of optimal trajectories using the exit time
- Algorithm B.
ä For n ≥ 1, consider (t0 = 0, t1 , ..., tn−1 , tn = T ) a uniform partition of [0, T ]
with ∆t = Tn .
ä Let {yn (·), zn (·)} be a trajectory defined recursively on the intervals (ti−1 , ti ],
with zn (·) := z = ϑ(T , y ) and yn (t0 ) = y .
ä [Step 1] Knowing ykn = yn (tk ), choose the optimal control at tk s.t.:
^ ukn ∈ arg max
T yn (tk ) + ∆tf∆t yn (tk ), u , z + ∆t
T ,
u∈U
ä [Step 2] Define un (t) := ukn , ∀t ∈ (tk , tk+1 ] and yn (t) on (tk , tk+1 ] as the
solution of
ẏ(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
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
Numerical schemes
Finite Difference scheme

W n+1 = max W n + ∆tH(y , D + W n (y , z ), D − W n (y , z )), Ψ
I
I j
I j
I ,j
I ,j
I ,j
 N
WI ,j = ΨI ,j ,
Semi Lagrangian scheme
(
W
WIn+1
= mina∈U W n yI + f (yI , a)∆t, zj
ΨI ,j
,j
WIN,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
Approximation by Bolza problems
Z
2p1
t
Φ(yyu (s))2p
0
ds
→ max Φ(yyu (s)),
Criterion
Bolza, 2p = 6
Bolza, 2p = 8
Bolza, 2p = 10
Bolza, 2p = 12
Maximum running cost
Hasnaa Zidani (ENSTA )
as p → +∞.
s∈[0,t]
Optimal cost value
907.59
807.64
728.08
686.29
611.72
Abort landing Problem
RICAM, 21-25 November, 2016
26 / 29
Numerical simulations
D 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).
D 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
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
... thank you for your attention.
Hasnaa Zidani (ENSTA )
Abort landing Problem
RICAM, 21-25 November, 2016
29 / 29