Page 28 - IJOCTA-15-3

S.Ezzeroual, B.Sadik / IJOCTA, Vol.15, No.3, pp.396-406 (2025)
2
F = {u 1 X 1 +u 2 X 2 |u = (u 1 , u 2 ) ∈ R } is symmet-
˙
ric. Computing the vector field X 3 = [X 1 , X 2 ], we h 1 = {h u (λ), h 1 }
get X 3 = [X 1 , X 2 ] = − sinh z∂ x − cosh z∂ y . Since = u 1 {h 1 , h 1 } + u 2 {h 2 , h 1 }
X 1 , X 2 , and X 3 are linearly independent, we have
= u 2 h 3
Lie q (F) = span(X 1 (q), X 2 (q), X 3 (q)) = T q (SH(2)), ˙
h 2 = −u 1 h 3
for every point q. This implies that system Equa-
Then, Abnormal extremals satisfy the Hamilton-
tion (11), is controllable. After transforming this
ian system:
system into an equivalent time-optimal problem
with controls ˙
h 1 = u 2 h 3
2
2
u + u ⩽ 1, one can observe that
1 2 ˙
h 2 = −u 1 h 3
|u 1 X 1 + u 2 X 2 | ⩽ c|q|, q ∈ M.
˙ q = u 1 X 1 + u 2 X 2
This gives the inequality |q(t)| ⩽ q 0 exp(tc).
Therefore, the attainable sets satisfy the following and the hypothesis h u (λ) −→ max implies
a priori bound:
h 1 (λ t ) = h 2 (λ t ) = 0
(≤ t 1 ) ⊂ {q ∈ M | |q| ⩽ q 0 exp(t 1 c)} .
A q 0
1
Thus, according to Filipov’s theorem ( ), optimal Therefore, condition Equation (12) gives h 3 (λ t ) ̸=
controls exist. 0 and the first two equations of the Hamiltonian
system yield u 1 (t) = u 2 (t) = 0. So, abnormal
trajectories are constant.
4.2. Computing extremal trajectories - Normal case
The maximality condition
We begin by applying the Pontryagin Maximum
Principle, 4 which allows to determine the ex-
2
2
1
tremal trajectories of system Equation(11). For u 1 h 1 + u 2 h 2 − (u + u ) −→ max
2 1 2
this purpose, we define the functions h i (λ) = yields u 1 = h 1 and u 2 = h 2 , and then the Hamil-
∗
⟨λ, X i ⟩ i = 1, 2, 3, where λ ∈ T M. These func- 1 2 2
tonnianis H = (h + h ). Thus, we get the sys-
tions (h 1 , h 2 , h 3 ), form a coordinate system on the tem: 2 1 2
∗
fibers of T M. Consequently, we adopt the global
coordinates (q, h 1 , h 2 , h 3 ) to describe the struc- h 1 = h 2 h 3
˙
∗
ture of T M. The Hamiltonian from the Pon-
˙
tryagin Maximum Principle (PMP) is given by: h 2 = −h 1 h 3
ν 2 2 ˙
v
h (λ) = u + u 2 + u 1 h 1 (λ) + u 2 h 2 (λ), h 3 = h 1 h 2
u
1
2
˙ q = h 1 X 1 + h 2 X 2
2
where, u = (u 1 , u 2 ) ∈ R , and v ∈ {−1, 0}. Then
4
the Pontryagin maximum principle for the prob- and Hamiltonian system in the normal case is
given by the equations:
lem under consideration reads as follows.
4
˙
˙
˙
Theorem 1. ( ) Let u(t) and q(t), t ∈ [0, t 1 ], be h 1 = h 2 h 3 , h 2 = −h 1 h 3 , h 3 = h 1 h 2 (13)
an optimal control and the corresponding optimal
˙ x = h 1 cosh z, ˙ y = h 1 sinh z, ˙ z = h 2 . (14)
trajectory in problem Equation (11). Then there
∗
exist a Lipschitzian curve λ t ∈ T M, π (λ t ) =
q(t), t ∈ [0, t 1 ], and a number v ∈ {−1, 0} for In this case, the initial covector λ lies on the initial
which the following conditions hold for almost all cylinder defined by
t ∈ [0, t 1 ]: ∗ 1
C = T M ∩ {H(λ) = }.
q 0
˙
⃗ v
λ t = h (λ t ) , 2
u(t) This set can be explicitly described as
v
h ν (λ t ) = max h (λ t ) ,
u(t) 2 u 3 2 2
u∈R C = {(h 1 , h 2 , h 3 ) ∈ R | h + h = 1},
2
1
(v, λ t ) ̸= 0 (12) representing a cylinder in the cotangent space
where the Hamiltonian takes the constant value
Our analysis considers two distinct cases: 1 . We introduce the following change of variables:
2
- Abnormal case
h 1 = cos(α), h 2 = sin(α).
In this case v = 0. Thus, the Hamiltonian of
PMP for the system takes the form: With these coordinates, the vertical system Equa-
h u (λ) = u 1 h 1 (λ) + u 2 h 2 (λ) tion (13) satisfies the equations:
400