Page 26 - IJOCTA-15-3

S.Ezzeroual, B.Sadik / IJOCTA, Vol.15, No.3, pp.396-406 (2025)
P k 2
If we add the condition u ≤ 1, the system A vector v is a symmetry of our system if it
i=1 i
given by equations Equations(2)-(4), becomes: satisfies the following two conditions:
k k L v (∆) ⊂ ∆ and L v (g) = 0. (7)
X X 2
˙ x = u i X i (x), u ≤ 1, where L denotes the Lie derivative. In what
i
i=1 i=1 (5) follows, we outline the necessary steps for com-
x(0) = q 0 , x(t 1 ) = q 1 ,
puting the symmetry algebra of a system de-
t 1 → min . fined by Equations (2)-Eq.(4). We consider
1
By Filipov’s theorem, the existence of optimal ∆ = {∆ q ⊂ T q M | q ∈ M}, where
∆ q = span{X 1 (q), . . . , X k (q)}. Additionally,
solutions for the optimal control problem Equa-
P n
g = a
tion (5), is guaranteed. i=1 i dq i ⊗ dq i , where a i are some con-
4
The Pontryagin Maximum Principle provides stants, represents a Riemannian metric such
that g(X i , X j ) = δ ij . Furtheremore, we suppose
necessary conditions for the optimality of trajec-
that ∆ is a contact distribution, meaning that
tory solutions of sub-Riemannian control prob-
1
ker(ω) = ∆, where ω ∈ Λ (M) is a 1-form in the
lems. Following this principle, we first compute
trajectories, called extremals, of a dynamic sys- manifold M. In terms of the local coordinates
tem in the cotangent bundle of the variety M. q 1 , . . . , q n , the contact form can be written as
n
P
e
Then the projections of the extremals on the state ω = i=1 i dq i , where e i are scalar functions of
the coordinates q 1 , . . . , q n . An arbitrary vector
space M constitute the optimal solutions and are
field v can be expressed as
called geodesics. A point γ(t 1 ) is called a Maxwell
v = h 1 (q 1 , . . . , q n )X 1 +. . .+h n (q 1 , . . . , q n )X n and
point along a geodesic γ if there exists another ge-
it is a symmetry vector if it satisfies the two pre-
odesic eγ ̸≡ γ such that γ(t 1 ) = eγ(t 1 ). This means
ceding conditions Equation (7). We recall that in
that there exists a geodesic bγ coming to the point
local coordinates, each X j is of the form
q 1 = γ (t 1 ) earlier than γ.
P n j j
X j = g ∂q i , where g are some scalar func-
i=1 i i
tions defined on M. Our approach begins by
3. Computing infinitesimal symmetries
calculating L v (X 1 ), . . . , L v (X k ):
of a sub-Riemannian problem
L v (X j ) = [v, X j ]
Let (M, ∆, g) be a sub-Riemannian manifold,
= [h 1 X 1 + . . . + h n X n , X j ]
where M is endowed with a Lie group structure
of dimension n that is left-invariant. This means = −[X j , h 1 X 1 + . . . + h n X n ]
that = −h 1 [X j , X 1 ] − (X j h 1 )X 1 − . . .
∆ ab = L a∗ ∆ b ,
− h j [X j , X j ]
g b (v, w) = g ab (L a∗ v, L a∗ w), ∀a, b ∈ M.
− (X j h j )X j − . . . − h n [X j , X n ]
where L a∗ denotes the differential of the left trans-
− (X j h n )X n
lation by a. We consider the sub-Riemannian sys-
n n ! n
tem Equations(2)-(4) on the Lie group (M, ∆, g) X ∂ X j ∂h 1 X 1 ∂
= −h 1 s i − g g i
and we assume it is controllable. Then Lie q ∆ = ∂q i i ∂q i ∂q i
i=1 i=1 i=1
T q M for every q ∈ M. Therefore,

n ! n
X j ∂h j X j ∂
Lie q (∆) = span{X 1 (q), . . . , X k (q), − . . . − g i g i
(6) ∂q i ∂q i
X k+1 (q), . . . , X n (q)} i=1 i=1
n n !
for all q ∈ M, where ∆ is given by the vector fields X ∂ X j ∂h n
X 1 , . . . , X k and X k+1 , . . . , X n ∈ Lie(∆). Further- − . . . − h n t i ∂q i − g i ∂q i
i=1 i=1
more, suppose the problem admits optimal solu-
n
tions. X n ∂
g i
Definition 1. (Agrachev and Barilari. 16) Let i=1 ∂q i
M be a 2m + 1 dimensional manifold. A sub- where s i and t i are some scalar functions defined
Riemannian structure on M is said to be contact on M. Then, applying the condition
if ∆ is a contact distribution, i.e. ∆ = ker ω, ω(L v (X j )) = 0 yields an equation of the form:
1
where ω ∈ Λ M satisfies ( V m dω) ∧ ω ̸= 0. n n n
X X ∂h 1 X ∂h n
b i h i + c i + . . . + d i = 0, (8)
By reference to Olver, 17 which provides us ∂q i ∂q i
i̸=j i=1 i=1
with an important result regarding the condi-
where b i , c i and d i are some scalar functions.
tions that a symmetry vector must satisfy, we
Extending the procedure to the all vector fields
have that:
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