Implications of change management in public administration
Implications of change management in public administration
Implications of change management in public administration
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Revista T<strong>in</strong>erilor Economişti (The Young Economists Journal)<br />
the optimal trajectories <strong>of</strong> our system are the geodesics <strong>in</strong> the framework <strong>of</strong> sub-<br />
Riemannian geometry [3], [5], [6].<br />
2. Control aff<strong>in</strong>e systems<br />
Let us consider the drift less control aff<strong>in</strong>e system (called also distributional<br />
n<br />
system) <strong>in</strong> the space R on the form<br />
with<br />
X<br />
i<br />
,<br />
.<br />
X ( t)<br />
<br />
m<br />
<br />
i1<br />
i<br />
u ( t)<br />
X<br />
i<br />
( x(<br />
t))<br />
i 1,...,<br />
m vector fields <strong>in</strong> R n and the controls u u , u ,..., u ) take<br />
(1)<br />
(<br />
1 2 m<br />
n<br />
values <strong>in</strong> an open subset R . The vector fields X<br />
i<br />
generate a nonholonomic<br />
n<br />
(non<strong>in</strong>tegrable) distribution D R such that the rank <strong>of</strong> D is not necessarily constant.<br />
n<br />
Let x0<br />
and x<br />
1<br />
be two po<strong>in</strong>ts <strong>of</strong> R . An optimal control problem consists <strong>of</strong><br />
f<strong>in</strong>d<strong>in</strong>g those trajectories <strong>of</strong> the distributional system which connect x0<br />
and x<br />
1<br />
, while<br />
m<strong>in</strong>imiz<strong>in</strong>g the cost<br />
m<strong>in</strong><br />
u F ( x(<br />
t),<br />
u(<br />
t))<br />
dt , (2)<br />
(.) I<br />
where F is a positive homogeneous cost on D .<br />
The controlled paths are obta<strong>in</strong>ed by <strong>in</strong>tegrat<strong>in</strong>g the system (1). If D is assumed to be<br />
bracket generat<strong>in</strong>g (i.e. the vector fields <strong>of</strong> D and iterated Lie brackets span the entire<br />
n<br />
R ), by a well-known theorem <strong>of</strong> Chow the system (1) is controllable, that is for any<br />
two po<strong>in</strong>ts x<br />
0<br />
and x<br />
1 there exists an optimal curve which connects these po<strong>in</strong>ts.<br />
1 2<br />
We consider the Lagrangian function <strong>of</strong> the form L F and it results that is<br />
2<br />
2-homogeneous positive function. Necessary conditions for a trajectory to be an<br />
extreme are given by Pontryag<strong>in</strong> Maximum Pr<strong>in</strong>ciple. The Hamiltonian reads as<br />
H(<br />
x,<br />
p,<br />
u)<br />
p,<br />
X L(<br />
x,<br />
u)<br />
, (3)<br />
where p is the momentum variable on the dual space. The maximization conditions with<br />
respect to the control variables u, namely<br />
H( x(<br />
t),<br />
p(<br />
t),<br />
u(<br />
t))<br />
max H(<br />
x(<br />
t),<br />
p(<br />
t),<br />
v)<br />
leads to the equations<br />
H<br />
( x,<br />
p,<br />
u)<br />
0 , (4)<br />
u<br />
and the extreme trajectories satisfy the Hamilton‟s equations<br />
.<br />
H<br />
.<br />
H<br />
x , p .<br />
(5)<br />
p<br />
x<br />
If the equations (4) permit us to f<strong>in</strong>d <strong>in</strong> a unique way u as a smooth function <strong>of</strong> x and p,<br />
then we can write the Hamiltonian system (5) without any dependence on the control.<br />
This nice situation happens always for distributional systems with quadratic cost<br />
.<br />
v<br />
108