Implications of change management in public administration

Revista Tinerilor Economişti (The Young Economists Journal)
the optimal trajectories of our system are the geodesics in the framework of sub-
Riemannian geometry [3], [5], [6].
2. Control affine systems
Let us consider the drift less control affine system (called also distributional
n
system) in the space R on the form
with
X
i
,
.
X ( t)
m
i1
i
u ( t)
X
i
( x(
t))
i 1,...,
m vector fields in R n and the controls u u , u ,..., u ) take
(1)
(
1 2 m
n
values in an open subset 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
u F ( x(
t),
u(
t))
dt , (2)
(.) 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
bracket generating (i.e. the vector fields of D and iterated Lie brackets span the entire
n
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 maximization 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
If the equations (4) permit us to find in a unique way u as a smooth function of x and p,
then we can write the Hamiltonian system (5) without any dependence on the control.
This nice situation happens always for distributional systems with quadratic cost
.
v
108