Full paper in PDF:
$%J.-L. Lions and E. Zuazúa, The cost of controlling unstable systems: time irreversible systems,
Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 481–523.%$
The Cost of Controlling Unstable Systems:
Time Irreversible Systems
Jacque-Louis LIONS and Enrique ZUAZÚA
Collège de France
3 rue d’Ulm
75231 Paris Cedex 05 -- France | Departamento de Matemática Aplicada
Universidad Complutense
28040 Madrid -- Spain |
| |
Received: December 9, 1996
Dedicated to Prof. Luiz Adauto Medeiros
in his 70th birthday.
ABSTRACT
We discuss the cost of controlling parabolic equations of the form
in a bounded smooth domain
of
with
homogeneous Dirichlet boundary conditions where
is fixed. The
control
acts on the system through the open and non-empty subset
of
. As
the system becomes more and more instable. We analyze the
dependence of the control
on the parameter
in the context of various
different control problems. We show that the norm of the control diverges when
the trajectory is driven from an initial state
to the null state at
any time
. However, we prove that the control converges to zero when
the null initial state
is driven into a ball of an arbitrary radius
around a given terminal state
or when the solution fulfills exactly
a finite number of constraints at time
.
2000 Mathematics Subject Classification: 93B05, 35K05, 35B35.