Revista Tinerilor Economiºti (The Young Economists Journal)

Revista Tinerilor Economişti (The Young Economists Journal)
n
values in an open subset Ω ⊂ R . The vector fields X i generate a nonholonomic
n
(nonintegrable) distribution D ⊂ R such that the rank of D is not necessarily constant.
n
Let x0 and x 1 be two points of R . An optimal control problem consists of
finding those trajectories of the distributional system which connect x0 and x 1,
while
minimizing the cost
min F ( x(
t),
u(
t))
dt , (2)
u(.)
∫ I
where F is a positive homogeneous cost on D .
The controlled paths are obtained by integrating the system (1). If D is assumed
to be strong bracket generating (i.e. the vector fields of D and first iterated Lie brackets
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span the entire R ), by a well-known theorem of Chow the system (1) is controllable,
that is for any two points x 0 and x 1 there exists an optimal curve which connects these
points.
1 2
We consider the Lagrangian function of the form L = F and it results that is
2
2-homogeneous positive function. Necessary conditions for a trajectory to be an
extreme are given by Pontryagin Maximum Principle. The Hamiltonian reads as
H ( x,
p,
u)
=< p,
X > −L(
x,
u)
, (3)
where p is the momentum variable on the dual space. The maximixation conditions
with respect to the control variables u, namely
H ( x(
t),
p(
t),
u(
t))
= max H ( x(
t),
p(
t),
v)
leads to the equations
∂H
( x,
p,
u)
= 0 , (4)
∂u
and the extreme trajectories satisfy the Hamilton’s equations
. ∂H
. ∂H
x = , p = − .
(5)
∂p
∂x
3. Application
Let us consider in the two dimensional space
(Grushin plane)
with
and minimizing the cost
where
.
1 2
( t)
u X 1 u X 2
.
118
v
2
R the drift less control affine system
X = +
(6)
min
u(.)
⎛1
⎞ ⎛0
⎞
X 1 = ⎜ ⎟ , X = ⎜ ⎟
⎝0
⎠ ⎝ x⎠
∫ I
2 ,
F ( u(
t))
dt , (7)
1 2 2 2 1
= ( u ) + ( u ) u , is the positive homogeneous cost (Kropina metric).
F +