Revista Tinerilor Economiºti (The Young Economists Journal)

Economic Theories – International Economic Relations
A STUDY ON CONTROL AFFINE SYSTEMS WITH DEGENERATE COST 44
Assoc. Prof. Popescu Liviu Ph.D
University of Craiova
Faculty of Economy and Business Administration,
Craiova, Romania
Abstract: In this paper we study a drift less control affine system with
degenerate cost of Kropina type. We use the Pontryagin Maximum
Principle in order to find the general solution.
JEL classification: C02, C6
Key words: drift less control affine system, degenerate cost, Pontryagin maximum
principle.
1. Introduction
The solution of a drift less control affine system (see [1]) is provided by
Pontryagin's Maximum Principle: that is, the curve c(t)=(x(t),u(t)) is an optimal
trajectory if there exists a lifting of x(t) to the dual space (x(t),p(t)) satisfying
Hamilton’s equations.
In this paper we study a drift less control affine system (see also [5], [6]) with
degenerate cost of Kropina type, such that the rank of distribution is not constant. The
distribution is strong bracket generating (i.e. the vector fields of distribution together
with the first iterated Lie brackets span the entire tangent bundle) and from Chow’s
theorem the system is controllable, that is the system can be brought from any state a to
any other state b.
We find the explicit solution of the problem, using the Hamilton equations on
dual space and a convenient change of variables. In the particular case of quadratic cost,
the optimal trajectories of our system are the geodesics of the so called Grushin plane
(see [2], [4])
2. Control affine systems
Let us consider a drift less control affine system (called also distributional
n
system) in the space R on the form
.
X ( t)
=
m
∑
i=
1
i
u ( t)
X ( x(
t))
with X i , i = 1,...,
m vector fields in n
R and the controls u = ( u1,
u2
,..., um
) take
44 Acknowledgments.This work was supported by the strategic grant
POSDRU/89/1.5/S/61968, Project ID61968 (2009), co-financed by the European Social Fund
within the Sectorial Operational Program Human Resources Development 2007-2013.
117
i
( 1)