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 |

Dedicated to Prof. Luiz Adauto Medeiros

in his 70th birthday.

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.